Effect of a damper on the wind-induced oscillations of a tall mast

EFFECT OF A DAMPER ON THE WIND-INDUCED OSCILLATIONS

OF A TALL MAST

BY

TECHNISCHE HOGESCHOOL DElFT lUCHTVAART- EN RUIMTEVAARTTECtlK~._

BIBLIOTHEEK

Kluyverweg 1 - OELFT

B. ETKIN and J. S. HANSEN

•

. I

EFFECT OF A DAMPER ON THE WIND-INDUCED OSCILLATIONS

OF A TALL MAST

BY

B. ETKIN and J. S. HANS EN

Submitted November,

1981

J

.

..

Acknowledgements

The research reported herein was supported by NSERC Grant No. A0339 and by agrant from CKVL Radio in Hontreal. The extensi ve programming and computing was carried out by the following research

assistants: G. Heppler; N. McNeilj A. von Flotow; G. Sincarsin

and D. F. Golla. A number of substantive contributions to the work, beyond simple programming, was made by these research assistants.

Special thanks are due to our colleague Dr. p. C. Hughes, who

found a significant errOr in our theory in time for us to correct it before publication.

. ,;

ABSTRACT

A study of the wind-induced oscillations in the downwind

direction has been performed for a taU slender mast that incorporates

a unique form of damper. The damper consists of a hinged extension

to the main mast, rotation of which is opposed by springs that

provide stiffness and by viscous dampers. In that portion of the

domain. of hinge parameters (stiffness and damping) where conventional

(approximate) analysis shows substantial beneficial e.ffects of the

damper, a more exact analysis shows that the benefit is not in fact realized. The "exact" analysis is so termed because it treats the natural modes of the linear vibration problem exactly, in contra-distinction to conventional vibration analysis, which approximates

complex modes by real modes. The cross vibrations associated with

Contents Acknow1edgements ii Abstract iii Symbo1s _v 1. I NTROD UC TI ON 1 ·

'"

Ilo THEORY 3 2 0 1 Exact Theory 3

202 Finite Element Model and Solution for the Modes 9

203 The Generalized Forces and Response Spectra 10

204 Approximate Theory 15

IIl. RESULTS OF CALCULATIONS ₁₇

IVo VORTEX-INDUCED VIBRATIONS ₁₈

Vo CONCLUSIONS

19

REFERENCES ₂₀

FIGURES

B

...

C c d

f.

(t) 1 F(x)

...

F(x) f(x) G (5) g g(X) K .. 1J KT L Q,(x,t) ï(x) i ' (x, t) M· . 1J

...

M m m. J m(x) n Symbo1s

bending stiffness (EI) damping matrix (20 2,1) drag coefficient

damping coefficient diameter of mast

genera1ized force (201,19)

complex mode shape see (2.3,13)

mode shape function see (2.3,10)

acce1eration of gravity mode shape function stiffness matrix (2.1,9) norma1ized hinge stiffness stiffness matrix (2.2,1) rotationa1 stiffness of hinge 1ength of nast

integra1 sca1e of turbu1ence externa1 force per unit 1ength

loca1 time-average force per unit 1ength

f1uctuating component of externa1 force per unit 1ength mass matrix (2.1,11)

mass matrix (2.2,1) mass per unit 1ength

bending moment in j-th mode mass of mast above station x damping constant

q. (t) 1 R(T) t u

v

v W(x, t) W(x) w(x,t) x y r .. 1J P 2 cr 4>(w) w w n ( ) ( ) ( ) I genera1ized coordinates corre1ation function (2.3,6) time eigenvector see (2.3,9) vector

Î..

wind speed

average wind at station x wind perturbation

vertica1 distance a10ng mast from fixed end horizontal def1ection of mast

N vector of joint displacements and rotations norma1ized hinge damping

damping matrix (2.1,10) eigenva1ue dummy variables densi ty of air turbu1ence intensity time de1ay spectra1 density (2.3,6) phase ang1e wave number radian frequency undamped frequency

a/at

y

1ap1ace transform of variab1e

1. INTRODUCTION

The wind-induced oscillations of tall slender structures have been the subject of many investigations (see for example Refs. 1-4). The damping present in the structure (in the material, in the joints, and in the foundation) is always an important factor in determining the ampli tude of the response to wind. This is the case for both the turbulence-induced oscillations in the plane of the mean wind (in which case there is usually significant aerodynamic damping as weIl) and for the vortex induced oscillations normal to the wind (when the aerodynamic damping is nonlinear, and negative

[5]).

Artificial dampers of various kinds have been used to limit the motions and stresses that might otherwise be excessive.

The method almost universally employed for analyzing the vibrations is to represent them as a sum of damped normal modes, each independently excited by the wind. Each mode is characterized by its shape, its

frequency, and its damping (i.e. rate of decay). The equation for the mode describes a motion in which all elements of the mast move in unison, either in phase or in antiphase - i.e. with suitable time origin,

y(x, t)

=

f(x) e -nt coswt (1.1)

where f(x) is the mode shape, and n, ware the positive damping and frequency constants, respectively. A tower cantilevered at its base ~as (N-l) nodes or stationary points in its N-th mode (not counting

the fixed root as a node). The nodes are of course the zeros of f(x). The analytical problem for the engineer consists essentially of

determining the mode shapes.and frequencies, making an acceptable engineering estimate of the damping, computing the generalized forces that drive the modes, and then calculating responses. Now the aerodynamic pressures that produce the forces are random functions of both time and position along the tower, hen ce the responses are random processes as weIl. Thus statistical methods must be used for the analysis. The mode shapes themselves strongly influence the generalized forces and hence the responses, and thus are an important ingredient of the analysis. The methods now available for generating the mathematical model of the driven system and for calculating mode shapes and frequencies are weIl developed and powerful. The information about the turbulent wind and the local forces it produces is good, albeit not perfect, and the methods available for applying this illformation are, like those for mode shapes, weIl developed and powerful.

The essential point of this paper is that the conventional approach to modes, is even within the context of classtcal linear theor , only an approximation. It is vali wenever the damping is weak or when it is appropriately distributed along the tower, conditions usually satisfied weIl enough in practice. However, they may be exceptions. When there are powerful concentrated dampers appliéd to few points of the tower, the modes

can theoretically display a character very different from that described above. The classical mode, wi th its "motion in unison" feature, no longer occurs, and the simplest possible physical motion is one in which there are two shape functions for each mode, f(x) and g(x), that occur 90° out of phase with one another in the form

-nt

y(x,t)

=

2e [f(x)coswt - g(x)sinwt] (1. 2)

where y(x,t) is the deflection of the mast.

Thus the mode still has a unique frequency wand damping n, but the points of the mast do not move in phase, and there are in general no fixed nodes.

Equation (1.2) can of course alternatively be written as -nt y(x,t) = 2e h(x)cos[wt + ~(x)] where 2 2 2 h (x) = f (x) + g (x) (1. 3)

is the amplitude of the oscillation at height x and Hx) is i ts phase angle, variabIe along the mast.

We have been involved with a practical case of a tall slender mast (about 60 m high, 1 m dia) which has an artificial damper. The damper takes the form of an appendage to the main mast coupled to it by a hinge that incorporates both stiffness and viscous damping (Fig. 1,). Being very slender (LiD ~ 60), without guy wires, and of all-welded construction,

the inherent structural damping is low: It was for this reason (particularly

wi th vortex-induced cross-wind oscillations in mind) that the damper was

added. The topmost portion of smaller diameter (about 6 m long) is hinged

to the main mast by a universal joint that has 3 sets of springs and dash-pots disposed at 120°. This subsystem then satisfies the basic criterion for a damper, i.e. a secondary inertial system coupled to the main system via a dissipative mechanism.

In the following, we have solved the eigenvalue problem 'exactly' in that the hinge damping has been included in the determination of the mode shapes. This leads to modes described by (1.2) instead of the usual

(1.1). We have found that there are significant differences between the

modes obtained and their conventional-approximation counterparts, and that final results for btnding-moments and stresses are likewise significantly different.

As an incidental, of both theoretical and practical interest, we have formulated the modal forced-response problem as a first-order complex differential equation instead of the usual second-order real equation.

,~

This enab1es the random forcing to be treated more convenient1y when the exact modes are being used. If one tries to use second-order rea1 equations with exact mode shapes, it becomes necessary to include spectra of time-derivatives of the turbu1ence in the .

genera1ized forces. This is an embarrassment since such spectra diverge at high frequency for the usua1 engineering mode1s of atmospheric turbu1ence.

11. THEORY

2.1 Exact Theory

The differentia1 equation for a vertica1 mast subjected to wind load is the c1assica1 beam equation modified (when required) to

a110w for the additiona1 bending moments produced by gravity when the mast is deflectedo With the customary assumptions of small deflection

and 1inear viscous damping, this equation (with gravity inc1uded) is

"

(By") + g (lily') (2.1,la)

The domain of x is seen to exc1ude the singu1arity at the hinge. In this equation ( )'~d/dX, (·)~d/dt; B is the bending stiffness EI; m(x) and c(x) are mass and damping force per unit 1ength respective1y; m(x) is the tota1 mass above x; g is gravitationa1 acce1eration; and ~'(x) is the loca1 wind force per unit length. The presence of 1umped masses and 1umped dampers wou1d be accomodated by delta functions in m(x) and c(x). The damping might come from the structure, the foundation, aerodynamic forces, or added damping devices. The gravity term

m

gy' is recognized as the shearing force at x contributed by the weight of the mast above x.

It is assumed that the hinge at x

H is vanishing1y weight1ess, so that the fo11owing conditions app1y (H

the upper and lower sides of the hinge). +

small and

and H represent

(2.1,lb)

(2.1,lc)

(2.1,ld)

Condition (201,lb) is obvious; (2.1,lc) states that the bending moment is continuous across the hinge and is governed by the rotatiqna1 stiff-ness kH and rotationa1 damping cH; (2.1,ld) states th at the gravity

shear is discontinuous across the hinge because of the change of slope of the beamo The solution of the autonomous equation, i.e. when ~'=O, yie1ds the eigenva1ues and eigenfunctions, or natura1 modes, of the

structure. (In general,these modes are not orthogonal). These special solutions are of the form

y(x,t) = F(x)eÀt _(2.1,2)

in which F(x) is a complex eigenfunction (mode shape) and À is a complex eigenvalue. They satisfy the characteristic equation

"

(BF") + g(riiF')

'"

'

+ (cÀ + mÀ 2 )F = 0

and additional conditions corresponding to (2.1, lb, c, d)

(2.l,3b)

(BF") = (BF") = k (F' - F' ) + c À(F' - F' )

H+ H- H H+ H- H H+ H- (2.l,3c)

' " , ' " ,

(BF") _H+ + (gmF') _H+

=

(SF") _H- + (gmF' )H- _(2.l,3d)

In our case, all the modes are damped oscillations, so all the F's and À's occur in conjugate complex pairs. Only this case is considered herein.

The Natural Modes

Let F. and F~ be the conjugate pair of eigenfunctions associated with the efgenvalÓes, À. and À~ of the j-th mode, and let

J J

À. = n. + iw.

J J J (201,4)

F. = f. + ig.

J J J (2.1,5)

The real mode is then (dropping the subscript j for convenience)

y(x, t)

=

F(x)e Àt + F*(x)e À*t

=

2ent[f(x)coswt - g(x)sinwt]

.:.

This equation describes the simplest possible physical motion of the mast, a harmonie motion in which all partieles oscillate with the same frequency and damping, but not the same phase t (unless g happens to be proportional to f). The points of zero displacement

(nodes) oscillate.between the zeros of f(x) and the zeros of g(x). The initial conditions required to start the pure mode are evidently:

displacement: y(x,O)

=

2f(x)

velócity: y(x,O)

=

2nf(x) - 2wg(x)

Finding the modes is a major component of the engineering analysis, and as we shall see in the following, it is preferable to find them not by solving (2.1,3), but rather from an equivalent finite matrix

equation derived by the finite-element methode

~rthogonality

Conditionstt

The well-known orthogonality conditions on normal modes of undamped systems are of the form

J

Lm(X)F. (x)F. (x)dx =

°

1 J

o

i ~ j

We need the corresponding conditions for the present case in order to derive the uncoupled equations of forced motion. To obtain them, we write (2.1,3a) for the j-th mode, multiply by Fk, and integrate over the length of the mast (integrating separately from

°

to xH_ and xH+ to L).

L L L

J

Fk (BFj')

"

dx +

f

CFkFjdx +

À~

f

o _o 0

The first two integrals are evaluated by parts, using the hinge conditions (2.1,3b, c, d), the boundary conditions

F

₌

F'

₌

°

at x

₌

,

F"

=

(BF")

₌

°

at x

=

and noting that m

₌

°

at x

₌

t The phase angle locally is

0 L L (fixed end) (free end) ~(x) = tan- l g(x) f(x) (2.1,7)

tt The orthogonality conditions derived herein are essentially the same as those obtained by Foss [7].

The resu1t is where 2 K· l + À.r·k + À.M.k = 0 J ' J J J J

r

_{J '} .k L L =

f

BF~ F~

dx - gJ

~ F~ F~

dx +

kHAF~ AF~

o L o =

f

cF.Fkdx + _cH

AF~ AF~

o J L

=

f

mF/kdx o

We may inter-change subscripts to get

and, since ~, ~, ~ are all symmetric matrices, then

2

Kjk + _{Àkr jk} + ÀkMjk

=

0

The sum and difference of (2.1,8) and (2.1,12b) yie1d

and (a) (b) Using (2.1,13) we e1iminate r jk from (2.1,14) to yie1d (2.1,8) (2.1,9) (2.1,10) (2.1,11) (2.1,12) (2.1,13) (2.1,14) (2.1,15)

---

---,

•

Equations (2.1,13) and (2.1,15) are the required generalization of the usual ortho~ona1ity re1ations, and reduce to them when c = r = 0, i.e., to Kjk

= Mjk

=

°

for ~ :f k. _{\IIhen either r jk} cc Mjk or rjk ex: Kjk' we get

Kjk = _{r jk} = _Mjk

=

0, J :f k.

Uncoup1ed Eguations of Forced Motion

We assume that the genera1 forced motion is a superposition of natura1 modes, i.e.,

[ y(X,t)]

~

y(x,t) =j:l [ F. (x) ] ÀJp.CX) J J q. (t) J (2.1,16)

Here the q's are time on1y. Wh en

the genera1ized coordinates, and are functions of (2.1,16) is substituted into (2.1,1) the resu1t is

I

{(BF'.')" q.

j=1 J J

- ,

.

.

}

+ g(mF!) q. + cF.q. + mÀ.F.q. = ~'(x,t)

J J JJ J J J (2.1,17)

Mu1tip1ying through by Fk and integrating fr om x =

°

to L yie1ds

I

{K·kq· + (rJ·k + ÀjHjk)qj} =

~(t)

(2.1,18) j=1 J J where L =

J

Fk(x)~'(x,t)dx

o

Equation (2.1,18) can be rearranged as

(2.1,19)

I'

{

Kjkqj + (r jk + ÀjMjk)qj} + \kqk + (rkk + ÀkMkk)qk =

~(t)

(2.1,20) j=l

where

L

'

denotes summation over all j except j k. From (2.1,13) r·_l + _{À.M· k}

=

_{-ÀkM· k} J~ JJ ' J' 50 (2.1,20) becomes

I'{

K·kq· - ÀkM·k

ci .}

+ Kkkqk + j=1 J J J J (2.1,21)

From (2.1,16) we note that 00 00

Y

=

L

F.q. =

L

À.F.q. j=l J J j=l J J J

•

Mu1tip1y through by mÀkF k to get 00 00 50 that 00

L

j=l or 00 00 Th us

Now by using (2.1,22) we rewrite (2.1,21) as

r'{K·kq· - ÀJ.\MJ·kqJ·}+ Kkkqk + (rkk +

Àk~k)qk

j=l J J

(2.1,22)

(2.1,23)

But the summation in (2.1,23) vanishes because of the orthogona1ity re1ation (2.1,15), 50 (2.1,23) becomes an uncoup1ed equation of motion

for the k-th mode. It is further simp1ified by using (2.1,8) to e1iminate rkk with the resu1t (with subscript k now omitted in the interest of

simplicity).

I

q -

Àq = (2.1,24)

Equation (2.1,24) is one main resu1t ofthis paper. It describes an infinite set of complex uncoup1ed ordinary differentia1 equations -first order and 1inear - that rep1ace the origina1 partia1 differentia1 equation.

.

'

If, as is usually the case, we approximate the true solution with a truncated set of modes, N in number, then there are N equations like (2.1,24). TIle problem of solving for the forced response then reduces to finding the modes (eigenvalues and eigenfunctions) from , ... hich

r,

K

and 11 eau readily be computed, calculating the driving forces or'(t) , and solving Eq. (2.1,24) for the q's.

2.2 Finite Element Model and Solution for the Modes

The solution of the eigenvalue problem posed by (2.1,3) is not routine, whereas the eigenvalue problem for finite matrices is. We have therefore chosen to formulate the "fini te-element" equations of the system (Appendix A). These combineto form the matrix system equation:

&ï.

+

f~

+

K

t.= 0 (2.2,1)

where y is the N vector of joint displacements and rotations, M, C and

K

are all symmetrie NxN matrices of mass, damping, and stiffness

respectively. This equation is written in canonical first-order form

by defining ~= L~'.,

rJ

T (2.2,2) v = _1: so that

.

A z + B z = 0 (2.2,3) where

o

M -M

o

A

=

B

=

M C

o

K

are 2Nx2N symmetrie matrices. The eigensolutions of (2.2,3) are of the form

Àt

z = u e (2.2,4)

where u is an eigenvector and À is an eigenvalue. u and À are found

as the-non-trivial solutions of

(~ + !D~

=

Q

(2.2,5)

by methods routinely available on modern computers. In practical

vibration problems there are usually 2N distinct u. and À., either

-J J

The Àj found from the fini te matrix system are approximations to the exact val ues for the mast referred to in Section 2. l, and the ~ are a vector (column matrix) representation of F(x) and F'(x).

Cornrnent

Insofar as the the main point of this paper is concerned, i.e., \vhether the conventional approximation to modes in any gi ven case is valid or not, a preliminary assessment can be made as soon as the ~. have been found. For any one mode ~ may be visualized as a set ot rotating complex nurnbers (with moduli ajk exp(i<pjk) as depicted in Fig. 2. Each represents one element of ~, i. e. one joint dis-placement or rotation, and the whole set rotates counterclockwise at angular rate w while shrinking according to e-nt • (There is as wel! an image set corresponding to )J.~, rotating clockwise, such that the surn ~j + ~j is rea!.) If the moae is "classical" all the vectors shO\m are colhnear. The extent to which they are not is a measure

of the effect being examined herein. The two functions f(x) and g(x) of Section 2.1 correspond to the proj ections of the vectors in Fig. 2 along and perpendicular to a reference line selected to normalize the eigenvector • (The normalization procedure should be such as to yield g (x) = 0 when there is no damping. )

2.3 The Generalized Forces and Response Spectra

The force produced by the wind is distributed along the mast, and is continuously variabIe in both space and time. Two very distinct cases are of interest:

(1) forces parallel to the mean wind, (2) forces perpendicular to the mean wind.

The aerodynamics of these two forces is very different •. The first is essentially a drag force locally proportional to the square of the relative velocity, and fits weIl within the theoretical structure discussed above. The second is primarily associated with vortex shedding from the mast, and in practical cases is usually amplitude dependent, leading to limit-cycle oscillations. This force, which includes a negative darnping, is not readily described by linear functions, and is at best only imperfectly known. We have not there-fore considered it worthwhile to apply the present refinement related to the treatment of darnping to that case, and deal only with the first.

Vibration Parallel to the Wind

The local force per unit length of mast is expressed in terms of the local wind, a drag coefficient, and the diametert - i.e., aerodynarnic "strip theory" is used. This is known to be an approxi-mation to the reality, especially near the tip of the mast, but is widely accepted and used as a reasortable one. Thus

l.

1 2

2

=

Cd

'2

pW d

\fuen both 2 and Ware expressed in terms of average values and perturbations, weget where Q; (x) 2

=

Q. + 2' W

=

W + w 1 -2

=

Cd '2 pW d

and to first order in perturbation quantities

2'(x,t) = CdPWdw(x,t)

(2.3,1)

(2.3,2) Any of Cd' W, d can be functions of x. In the case studied herein,

W and Cd are taken to be constants, and d takes on only two different va lues - one for the main mast, one for the damper.

Response Spectra

The contribution of the j-th oscillatory mode to the deflection at x is given by

y.(x,t) = F.(x)q.(t) + F~ (x)q~(t)

J J J J J (2.3,3)

and is of course realo The mean-square-value of y. (x) (ensemble or time)

is therefore J 2 <y. (x) > J 2 2 2

=

F. (x) <q. > + F.*(x) <q.* > + 2F. (x)F.*(x) <q. q.*> J J J J J J J J

=

2Re{F.2(X) <q.2>}+ 2/F.(x)/2 <q. q.*> J J . . J J J (2.3,4)

The total

~y2(x»

with all modes participating is the sum of

e~res:ion (2.3~4) over all j, plus cross terms such as <qjqk> in

WhlCh J f k. Slnce the modes are uncoupled and have different

frequencies (except in degenerate cases) these cross terms are bound to be small, and are normally neglected in analyses of this kind. We neglect them here as weIl.

2 2 In order to evaluate <Yj (t) > by means of (2.3,4) we need <qj >

and <qjqj>. When qj (t) is a complex number, as it is here, the

appropriate theorem that relates mean products to spectral densities

is

00

I <uv*>

=

1

11>u/w)dw

(2.3,5)

Here u and vare any two complex random functions of time and

<P~v

is the usual cross-spectral density, defined as the Fourier Integral of the cross-correlation: "" 1 f -iwT <p (w) = - R (-r)e (h uv 27T uv -00 . R (T)

=

f""

<P (w) eiwT dw uv uv -()()

The correlation for complex variables is in turn defined by

R (T) = <u(t)v*(t + T»

uv

(2.3,6)

If (u,v) are responses to inputs (U,V) related by the transfer functions

(Hl (s), H

2(s)) such that

yes)

=

H

2 (s)

yes)

then the theorem that relates input and output cross-spectra is

(2.3,7)

From Eq. (203,5), by identifying u with qj and v with qj or q / we get

(the subscript is now dropped fOT convenience)

C>O <q2>

~""J

cP qq* (w) dw 00 <qq*>::

J

4> (w) dw ",\00 qq (2.3,8)

The Lap1ace Transform of the basic moda1 differentia1 equation (2.1,24) is (s - À)q(s) = V(À)1?(s) where À V(À)

=

-~-MI. 2_K

which we can write convenient1y as

where q(s) = G(s)V(À)~(s) 1 G(s)

= - - -

_s -À (2.3,9) (2.3,10)

By app1ying (2.3,7) we find the re1ations between the cross-spectra of interest to be:

(2.3,11)

~

_qq (w) = IG(iw) 12IV(À)

12~~~(w)

r

To computethe q spectra we need those of

t:'

that occur on the RHS of (2.3,11). These are derived from (2.1,19), (2.3,2) and (2.3,7), i.e.,

"F(t) = JLF(X)R-I(X,t)dX (2.1,19) o L =

J

F(x)w(x,t)dx (2.3,12) o where F(x) = CdPWdF(x) (2.3,13)

The Lap1ace transforms of ~ (t) and w(x,t) are therefore re1ated by

L

~(s)

=

f

F(x)w(x,s)dx (2.3,14)

o

and their spectra by L L

~t;w)

=

J

J

F(~)F*(n)~ww(~x,w)d~dn

o • 0 (2.3,15) L L

~(w)=

f

J

F(~)F(n)~ww(~x,w)d~dn

o 0 (2.3,16)

An expression for ~ww(~x,w) is given by Surry [6] for the von-Karman model of homogeneous ,turbulence, i.e.,

where ~ (lIx,w) ww -;---,-~- = ~ (w) ww a = 1.339 2

II(

sJ

5/6

r

(5/6)

I

_

l

'2

KS/ 6

C

s

)

-[ t}

1/2

Here r denotes the gamma function, and K the modified Bessel function of the second kind. The basic (two-sided) power spectral density of

.. ..

the wind (von Karman model)is

~ (w) =

ww

1

(2.3,18) 7TW

in which L = _{integral scale of the turbulence,}

Q = w/W = wave number,

00 2

turbulence intensity

_{_ooI}

cp (w)dw

a

=

₌

ww

Comment

Once the eigenvalue problem has been solved, yielding {F(x), À,

M, K} for each mode, and the wind and turbulence have been specified, i.e. the set {W(x), L, a} then the results for each mode are calculated from the sequence illustrated bel ow:

(2.3, 17) (2.3, 18) ((2.3, 4)

I'"

'2

"

..

(2.3, 15) _-"

..,..~,,

7'~

* ~

-(2.3, 16) [(2.3, 8) J ...

~

_{qq qq} ,

~ *~

'

., (2. 3, 11) ] ~ (~x,W)----~~1 ww <qq>, <qq*> ...

_---1

<y. > J

The mean-square value of the desired response variabIe contributed by the j-th mode is thus obtained, and the addition of the contributions from all the modes of interest gives the final result:

2 N 2

<y (x»

=

I

<y. (x» j=l J

Quantities of interest other than e1ements of the state vector can be ca1cu1ated in a simi1ar way. For examp1e the bending moment at station x contributed by mode j is Hence 2 <m. (x» J m.(x,t) = B(x)y~(x,t) J J B(x)[F~(x) q.(t) + F~*(x) q~(t)] J J J J (2.3, 20)

In the practical computation of the above equations, it is in fact preferabIe to rearrange the order of the integrations in (2.3, 15

&

16) and (2.3,8)0 Thus

00 00

IG(iw)

12~ (~x,w)d

_ww

w

}d~d~

(2.3,21) In (2.3,21) we have used the fact that, with G (iw) from (2.3,10),

the inner integrand is an even function of w and so integrate on1y from 0 to 00. Simi1ar1y

00

<q2> = 2V2(À)

JLJ

F(~)F(~)

{

J

Re

G2(iw)~ww(~x,w)dw }d~d~

o 0

(2.3,22)

Here again, the integra1 in w is from 0 to 00 because the rea1 part of

the integrand is even in wand the imaginary part is odd. 2.4 Appro~imate Theory

In the usua1 engineering approximation, undamped modes are used as the basis for expanding the sol~tion. I Thus c (x) is set equa1 to

zero in (201,1), or equiva1ent1y, C

=

0 in (2 02,1). The resu1t of

solving the eigenva1ue prob1em for-the simp1ified equation is a set of rea1 undamped normal modes fu(x) in terms of which the motion is expressed as

00

where fUk and qk are rea1. On substitution of (2.4,1) into (2.1,1)

multiplication by fu'(x), and integration over x we get

J

L

{Kjkqk + r jkqk + MjkClk

(2.4,1)

in which K,

r,

ti are defined as in (2.1,9) to (2.1,11) with fu(x) instead of F(x) in the integra1s, and

but of course

.

fL

j: .

(t) = fu' (x) ,Il,' (x, t) dx

J 0 J

(2.4,3)

By virtue of the orthogona1ity properties of the modes fu. (x) both M' k

1 J

and Kjk are zero for j

t

k. Not so, however, for r._k• Neverthe1ess, the usual engineering approximation is to ignore

th~

intermodal coupling, and to assume r_jk

=

0 for j ~ k. Then (2.4,2) separates into an infinite set of uncoupled second order equations:

M .. q. +

r ..

q.

+ K •. q. =

t..

(t)

JJ J JJ J JJ J J j = 1,00 (2.4,4)

Here q. is the generalized coordinate giving the "amplitude" of the j-th mode artd (2.4,4) is commonly written in the convenient form (subscripts omitted) q + 2çw

<{

+ n 2 (:V / w q = J (t) M n (a) where w 2 = -K (b) n M and ç = 1

r

(c)

2"

lKM

A major issue in the approximate solution arises at this juncture. How to estimate areasonabIe value for

r

(or ç)? The damping is usually the least wel1-defined aspect of the system, coming as it of ten does primarily from internal hysteresis in the structural material, slip in the joints, working of non-structura1 elements

(2.4,5)

and founqations. Faced with the attendant uncertainties, it is . eminently reasonably to treat ç in a somewhat arbitrary but conservative fashiono Thus very of ten ç is simply assumed on the basis of empirical

evidence on similar structures. In a case such as the present one, however, when a.damper is added precisely to produce a known controlled value of ç, one would expect a more rational approach to be used in estimating it.

To get the approximate ç, we use the value of r .. obtained from (2.1,10) where undamped normal modes replace the comPlex modes, Le.,

f

L 2

r ..

=

c(x) f (x)dx

JJ U·

o J

and substitute it into (2.4,Sc).

(This is equivalent, in the matrix formulation of Section 2.2, to keeping only the diagonal terms in U-I

ê

U where U is the eigenmatrix that diagonalizes

M

and

K

simultaneously) ~

~ t

I I

I'

III. RESULTS OF CALCULATIONS

The calculations were carried out for the mast illustrated in

Fig. lat a mean wind speed of 115 fps, uniform along the length of the mast. The design of the hinge elements is such th at it can be

regarded as consisting of a linear torsion spring and a linear viscous

damper. The equations were formulated as described in Section II,t

and solved on a digital computer.

The resul ts for some typical moue shapes are shown in Figs. 3 (a) and (b). The shapes for zero hinge damping (Fig. 3a) are j ust as one

\Voulu expect. Even with aerodynamic damping present, the imaginary

parts practically vanish and the moues are virtually real. The lowest-frequency mode (Mode 1) is seen to consist primarily of oscillation of the damper, with little motion of the main mast, whereas the remaining modes, of higher frequency, all entail significant motion of both the mast anu the damper. All modes shO\I large rotation of the damper, alld hence one might expect significant increases in the damping coefficients

of all moues as the hinge damping is increased. This is in fact realized.

For example, the values of ç for the first five modes at KT = 0.8 are as follows:

Table 1. Damping Coefficient ç, KT = 0.8

Mode 1 2 3 4 5 a=O .051~ .037 . • 0077 .0029 .0019 a=l.O .88 .067 .04 .0197 .0114

In interpreting these results, it should be noted that the hinge stiffness KT and damping a are normalized in such a way that KT = 1 corresponds tothe stiffness at which the frequency of the one-degree of freedom oscillation of the damper as an inverted pendulum equals the

fundamental frequency of the mast with the hinge locked, and a=l

corresponds to critical damping for the same one-degree of freedom

vibration. The ç values at a=O are entirely a result of aerodynamic

damping, and large increases are seen to be provided by the hinged

damper. The arrows in the table signify that the frequency of the

lowest beam mode, which has the lowest frequency at a=O, crosses over

that of the damper mode, which has the lowest frequency at a=1.09

(This cross over can entrap an unwary analyst. If it is not recognized

it might be supposed that the damping of the lowest beam mode increases from 0.051 to 0.88 as the hinge damping goes from 0 to 1, whereas in fact it only increases from 0.051 to 0.067).

tIn writing the computer code, the complex equations of Section 2.3 were in fact replaced by their real and imaginary parts.

Figure 3(b) shows that for typical values of KT and a for which the Jamper is strongly excited, the imaginary part of the mode shape

F (x) is qui te significant. I t will be recalled (see Fig. 2) that

the di vision of F(x) into its real and imaginary parts is arbitrary, depending in fact on the choice of a reference direct ion for the real

axis in the Argand diagram. \ve have chosen this direction so as to

minimize fg2 (x) dx .;. f I F (x) 12Jx. In this way the imaginary part

vanishes when there is zero damping. The principal significance of

a non-negligible g(x) is th at it results in a significant addition

to the general ized force dri ving the mode as compared \á th g = 0,

and hence in a larger response of the beam.

Since the principal concern with a mast of this type is struc-tural integrity, we have calculated the effect of the damper on the base bending moment when the mast is exposed to a turbulent wind of

35.05 mis average value over its whole length, with a turbulence

scale of L = 45.7m and intensity 02 = 14.9 (m/s)2. Some typical

results are plotted in Fig. 4 in the form 0 vs a for KT = 0.6. The

ordinate D is the dynamic component of the response, defined as

o

=

40 m

-m

where Om = rms value of fluctuating part of the bending moment, m =

steady base bending moment at 35.05 mis. The factor 4 is

representa-tive of the value that would be used in design.

This figure displays the most important conclusion of this paper, that is, that the approximate solution is appreciably unconservati ve. Whereas one might expect a significant alleviation of the dynamic response to wind on the basis of the approximate analysis, an expecta-tion in conformi ty wi th intuiexpecta-tion, in fact the exact a11alysis shows that there is li ttle to be gained by using the damper.

IV. VORTEX- mOUCED VIBRATIOHS

The vibrations induced by the phenomenon of vort ex shedding (Ref. 5) are primarily cross-wind, and fundamentally different in character

from those discussed above. They consist of a resonant response of

the mast to vortex shedding at one of its natural frequencies, when the wind speed is in a cri tical range, and invol ve strong nonlinear

coupling between aerodynamic forces and mast motion. A useful model

of the phenomenon is to regard i t as the steady-state forced oscilla-tion of a linear second-order system with negative aerodynamic damping. This negative damping falls to zero at some limiting amplitude, but in the presence of additional damping provided by the structure or an artificial damper, the steady limit-cycle amplitude is smaller than

that which would exist in the absence of such damping. In this

situa-tion , one would expect the artificial damper to have a stronger effect

on the response than we found above for downwind oscillations. However,

v.

CONCLUSIONS

1. The addition of a concentrated damper to a slender vertical structure

driven into vibrat;ion by a turbulent wind can impinge significantly on the validity and usefulness of the usual engineering approximation to the resulting motion and stresses. The "usu al engineering approximation" is characterized by the neglect of the off-diagonal terms in the damping matrix of the system that is generated by using undamped orthogonal modes as generalized coordinates.

2. A method of analys;.s has been presented that does not use this

approximation. It leads to first-order complex equations for the modes

instead of second-order rea 1 equations. The modes themselves are complex, i.e. each is composed of two sub-modes 90° out of phase.

3. The computed results for arealistic structure show that the

reduction in stress anticipated on intuitive grounds by adding the damper

REFERENCES

1. Davenport, A. G., "Response of Slender, Line-1ike Structures to Gusty Winds", Jour. Inst. Civ o Eng. v o 23, Nov. 1962.

2. Etkin, B. "Theory of the Response of a Slender Vertica1 Structure to a Turbulent Wind with Shear" Proc. NASA Conf. on Wind Load Prob1ems of Launch Vehic1es, Lang1ey Field Va. June 1966.

3. Korenev, B. G., "Vibrations of Tower Structures and Certain Methods of Damping Their Vibrations", Proc. Conf. Wind Effects on Buildings and Structures, Ottawa, Can. Sept. 1967, University of Toronto Press.

4. Proceedings of Conf. on Wind Engineering, Colorado, 1979 ed. J. E. Cermak, Pergamon Press. Vo1.2, Tech. session VI: Dynamic Response -Ta11 Bui1dings and Towers (12 papers)

5. Vickery, B. J., - "A Model for the Prediction of the Response of Chimneys to Vortex Shedding", Proc. 3rd IntI. Symp. on Design of Industria1 Chimneys, pp. 157-162, Munich 1978.

60 Surry, D., "Some Effects of Intense Turbu1ence on the Aerodynamics

of a Circu1ar Cylinder at Subcritica1 Reyno1ds Number"o Jour. F1uid Mech. Vol. 52 Part 3 1972.

7. Foss, K. A., "Coordinates Which Uncoup1e the Equations of Motion of Damped Linear Dynamic Systems", Jour. App1. Mech. Sept. 1959, pp. 361-364.

-.

6.10m 58.0m 182.9m above ground ~ .~---. Damper (.254m dia.) Hinge

Main mast (.914m dia.)

Roof top

Tm

ajk ei cl>jk

(wt

+

cl>jk)

---~~~----~---

Re

-1.0

-.5

240

200

160

120

80

40

o

MODE I

f

=

.279

hz

.5

1.0

-1.0

-.5

240

120

\

80

40

o

MODE 2

f

=

.453

hz

.5

1.0

-1.0

-

.

5

240

160

o

MODE 3

f

=

1.59

hz

.5

1.0

-1.0

.... 240

_....

Im--.!

....

-.5

200

160 120

80

40

o

MODE I

f

=

.417

hz

.5

1.0

-1.0

-

.

5

240,r

80

40

o

MODE 2

f

= .462

hz

.5

1.0

-1.0

-

.

5

240

40J

o

MODE 3

f

=

1.59 hz

.5

1.0

1.2

t

\

\

1.0 \

.8

o

.6

.4

.2

o

\

\

\ \

"

"

"

...

1.2

Exact

...

fAPprox .

....

-

---

---.4

.6

.8

HINGE DAMPING,

a

----1.0

APPENDIX

DERIVATION OF THE DISCRETIZED EQUATIONS OF MOTION

The equations of motion for the mast-antenna system have been developed using a modified farm of Lagrange's equation:

~

_[

~

_{1 -}

l!:

+

~

=

H. Ct) ;

dt ". da d. J

oaj aj

j = 1, N

where the system has been discretized using the finite element

technique~ In the above, L is the Lagrangian, R is the Rayleigh

dissipation function, H.(t) is the generalized force, aj are the generalized coordinatesJand N is the number of degrees of freedom af ter discretization. The Lagrangian is defined by L

=

T-V where T, V are the kinetic and potential energies respectively. For this problem the kinetic energy T is

L

T =

~

I

-

m(x) [y(x,t)]2 dx o

The potential energy is composed of the strain energy due to bending over the length of the beam UL, the strain energy at the hinge Uh and the gravitational potential energy V of the mass of the mast as it

deflects. Thus m 1

IL

2 U L =

'2

B(X)(Y") dx 0 Uh =

~

kH(YH_ _YH+ ' ) 2 V =

-I

L { m(x)g JX [y'(X,)]2 dX '} dx w o 0

It is to be noted that in the finite element representation the above farm for V was not particularly convenient sa an alternate and

equi valentW farm

L L Vw = -

I

{[y'(X)]2

I

m(X')gdX'} dx o x = _

IL~(X)

_{[y,cx)]2 dx} o \'las used.

The Rayleigh dissipation function R contains contributions that are distributed over the length, RL, and one contributed by the hinge, RH. That is

where . L . 1

f

0 2 RL =

2"

c(x) (y) dx 0 1 • _{• , ) 2}

111

=

_2"

_{cH(Y'H_ Y H+}

In the above expressions for UH and ~, k_H and c

H are the torsional spring and damping terms while YH- , Yl'l+ are the slopes below and above the hinge respectively. It is to be noted that the inclusion of a hinge inherently implies a discontinuity in y' and thus y" is not defined at this point. Therefore, the integrals involving y" must be treated with care. It is to be further noted that point masses on the system can be incorporated in the Lagrangian merely by including

appropriate delta functions in the definition of m(x).

The displacement y(x) is discretized by subdividing the domain [a,L] into subintervals of length ~~i. This is shown in Fig. Al. The trial functions are chosen to be cubics in the local coordinate ~, where Thus x-~. 1 J.-~ = L-L 1 J. J.-= x-L 1 J.-M. J. a~~<l

Further, the displacements and slopes y., y! at each node 'i' are taken as the generalized coordinates. Thus tfie

r~lationships

between the a's and Yi' yi for the i-th element is

a 1 a a a _Yi-l 0 al a ~~. a a _y! 1 J. J.-= a 2 -3 -2LU. J. 3 -/:'L J. y. J. a 3 2 /:,L J. -2 ~~ y! J. or

a=,!l.

Thus the element contributions to T, V, R, H for the i-th element become 1

f~i

.

2 Ti

=

2"

m(x) [y(x,t)] dx ~i-l 2 B(x) [y"(x,t)] - fLm(X,)gdX,[y'(X,t)]2} dx x

1 _rVi • 2 R.

₌

_{c(x) [y(x, t)] dx} 1 2 R.. I _l -H

₌

IR.i _{R.(x,t)y(x,t)dx} R.. I

1-These expressions may be sirnplified considerably by noting that m(x), B(x), c(x) are constant (w.r.t. x) over the domain of a given element

(and if the element is small enough R.(x,t) can be assumed constant as weIl)o TIle main use of the finite-element equations derived in this appendix is to provide the eigenvalues and eigenvectors of the system. For this purpose we do not need H(t), 50 the remainder of this formula-tion is for the free-vibraformula-tion case, in which H=O. The constant values of m(x) , etc., are designated as m., B., c. respectively. Making use

1 1 1

of this simplification and then transforming into local coordinates yields T. 1

v.

1 1

I

l . 2

=

2

mi ~R.i [y(~,t)] d~ 0 1 B. 1

I

l

[y"(~,t)]2d~

=

(~R..)3 2 1 0 1 { n

2~L

L

1 m.g~R.. J J + } mig

Il~[y,(~,t)]2d~

o

The development of the element mass, stiffness and damping matrices follows by substituting the assumed form for y(~,t) into the above integrals. This yields

In addition to the ahove, the contrihutions to elemental T

k, Vk, Rk resulting from point masses' as weIl as the hinge parameters must also he included. For point masses applied at node 'k' we have the elemental kinetic energy.

I • 2 Tk =

2"

~ _(Yk) 0 0 0 0 I ·T 0 0 0 0

=

-~Y ₂ -0 0 1 0 Y 0 0 0 0

A1so the contributions from the hinge are

and

y' ] [

1

H+ -1

With the above expressions for T. , V. and R. we can form the quantities

l. l. l.

n n n

T =

I

T. ; V =

_L

V. ; R =

_I

R.

i=l l. i=l l. i=l l,

and then deve10p the equations of motion as

,. ,. ,.

where ~, ~, ~ are the mass, damping and stiffness matrices, respective1y, of the system and where it is noted that the appropriate boundary

" • •

CD

I.

~ I

-,

6/

₂

_®

T

1

2

6/.

CD

_I

_I

~

'I. ~

UTli\S Report No. 257

Institute for Aerospaee Studies, University of Toronto (UTli\S)

4925 Dufferin Street, Downsview, Ontario, Canada, M3H 5T6

EFFECT OF A DAMPER ON THE WIND-INDUCED OSCILLATIONS OF A TALL MAST

Etkin, 8., Hansen, J. S.

1. Wind engineering 2. Wind loads 3. Vibration 4. Dampers. vibration

I. Etkin. 8., Hansen. J. S. I I. UTIAS Report No. 257

~

A study of the wind-indueed oscillations in the downwind direction has been performed for a tal!

slender mast that ineorporates a unique form of damper. The damper consists of a hinged extension

to thc main Blast, rotation of whieh is opposed by springs that provide stiffness and by viseous

dampers. In that portion of the domain of hinge parameters (stiffness and damping) whcre

eonven-tional (approximate) analysis shows substantial beneficial effects of the damper. a more exact

analysis shows that thc benefit is not in fact realized. Thc "exact" analysis is 50 termed because it treats the natural modes of the linear vibration problem exactly. in contradistinction

to conventional vibration analysis, which approximates complex modes by real modes. The cross

vibrations associated with vortex-shedding are briefly discussed.

.,

UTIAS Report No. 2S 7

Institute for Aerospaee Studies, University of Toronto (UTlAS) 4925 Dufferin Street, Downsview, Ontario, Canada, M3H 5T6

EFFECT OF A DAMPER ON '!'HE WIND-INDUCED OSCILLATlONS OF A TALL MAST

Etkin, 8., Hansen, J. S.

1. Wind engineering 2. Wind loads 3. Vibration 4. Dampers, vibration

I. Etkin, 8., Hansen, J. S. II' UTIAS Report No. 257

~

A study of the wind-indueed oscillations in the downwind direction has been performed for a taU

slender mast that ineorporates a unique form of damper. Thc damper consists of a hinged extension

to the main mast, rotation of which is opposed by springs that pro vide stiffness and by vlscous

dampers. In th at portion of the domain of hinge parameters (stiffness and damping) where

eonven-tional (approximate) analysis shows substantial beneficial effects of the damper, a more exact analysis shows th at the benefit is not in fact realized. The "exact" analysis is 50 tented

because it treats the natural modes of the linear vibration problem exactly, in contradistinction

to conventional vibration analysis, which approximates complex modes by real modes. 111e cross

vibrations associated with vortex-shedding are briefly discussed.

Available copies of ~his report: are limi~ed. Re~urn ~his card ~o UTIAS, if you require a copy. Available copies of this repor~ are limi~ed. Re~urn ~his card ~o UTIAS, if you require a. copy.

UTIAS Report No. 257

Institute for Aerospaee Studies, University of Toronto (UTlAS)

4925 Dufferin Street, Downsview, Ontario, Canada, M3H 5T6

EFFECT OF A DAMPER ON THE WIND-INDUCED OSCILLATIONS OF A TALL MAST

Et1c.in, "B., Ransen, J. S.

1. Wind engineering 2. Wind loads 3. Vibra,ion 4. Dampers, vibration

I. Etkin, B., Hansen, J. S. II. UTIAS Report No. 257

~

A study of the wind-indueed osciUations in the downwind direction has been performed for a taU slender mast that ineorporates a unique form of damper. The damper consists of a hinged extension

to the main mast, rotation of whieh is opposed by springs that provide stiffness and by viseous

dampers .. In that portion of the domain of hinge parameters (stiffness and damping) where con

ven-tional (approximate) analysis shows substantial beneficial effects of the damper, a more exact

analysis shows that the benefit is not in fact realized. 111e "exact" analysis is 50 termed

because it treats the natural modes of the linear vibration problem exactly, in contradistinction

to conventional vibration analysis, which approximates complex modes by rea 1 modes. 111e cross

vibrations associated with vortex-shedding are briefly discussed.

Available copies of ~his report: are limited: Re~urn ~his card to UTIAS, if you require a copy.

UTli\S Report No. 257

Institute for Aerospaee Studies, University of Toronto (UTlAS)

4925 Dufferin Street, Downsview. Ontario, Canada, Mlli 5T6

EFFECT OF A DAMPER ON '!'HE WIND-INDUCED OSCILLATIONS OF A TALL MAST

Etkin, 8., Hansen, J. S.

1. Wind engineering 2. Wind loads 3. Vibration 4. Dampers, vibration

I. Etkin, 8 .• Hansen, J. S. II. UTIAS Report No. 257

~

A study of the wind-indueed oscillations in the downwind direction has been performed for a taU

slender mast that ineorporates a unique form of damper. The damper consists of a hinged extension

to the main mast, rotation of which is opposed by springs that provide stiffness and by viseous

dampers. In th at portion of the domain of hinge parameters (stiffness and damping) wh_r_

eonven-tional (approximate) analysis shows substantial beneficia! effects of the damper, a more exact

analysis shows th at the benefit is not in fact realized. 111e "exact" analysis is 50 termed

because it treats the natura 1 modes of the 1 inear vibration problem exactly. in contradistinction

to conventional vibration analysis. which approximates complex modes by real modes. The cross

vibrations associated with vortex-shedding are briefly discussed.