^ ^ Bi'L?L50T5:
CRANFIELD
INSTITUTE OF TECHNOLOGY
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ADDITIONAL DAMPING FOR TALL STRUCTURES
by
CRANFIELD INSTITUTE OF TECHNOLOGY
ADDITIONAL DAMPING FOR TALL STRUCTURES
by K.C. Johns
Senior Visiting Fellow:
Assistant Professor:
Structural and Aerospace Dynamics Group,
Cranfield Institute of Technology, Department of Civil Engineering, Universite de Sherbrooke, Quebec Canada.
SUMMARY
A study is made of a method of vibration reduction in tall civil engineering structures in which flexibility is introduced artificially, but with passive dampers to absorb energy during any motion which results. In particular, an examination is carried out of the effect of introducing an articulation part-way up a tall tower, with passive dampers and springs to restrict the
rotation that results. The method involves assigning three degrees of freedom, and the use of Lagrangian
equations of motion. Power spectral density concepts are exploited using wind turbulence as a stationary random forcing function. In-wind vibrations only are studied. The optimal spring-stiffness/damper combination is sought for two specific tower designs.
provided by the Science Research Council, U.K., to whom the author is most indebted. Many useful discussions with Dr. C.L. Kirk and other members of the Structural and Aerospace Dynamics Group, Cranfield, are also
Page NOTATION
1. INTRODUCTION 1 2. TOWERS AND MODES OF VIBRATION 2
3. EQUATIONS OF MOTION 3 4. WIND EXCITATION 8 5. RESPONSE SPECTRAL DENSITY 13
6. PARAMETRIC STUDY 14 7. CONCLUSIONS 17 REFERENCES 19 APPENDIX I APPENDIX II FIGURES
all units are in ft., lb., and sec.
A,B,C functions which are included in self weight terms C„___ effective linear damping coefficient of
spring-damper system
C^ coefficient of drag
D function appearing in potential energy terms D ,D diameter of tower at base, and rate of change of
this with height
Do,D. diameters of lower and upper mast segments D. . coefficients in dissipative function, terms in
"^ damping matrix
D(x),D(z) general diameters, functions of height
E ,E moduli of elasticity of concrete and of steel c s
E.. complex matrix terms, equations of motion
J. J
F effective stiffness of spring system
Fy. Lagrangian dissipative function
F.,F. Lagrangian generalised forces
g acceleration of gravity
H height of spring-damper system above mast bearing
H.. complex frequency response functions
L' scale factor, wind expression
M. . coefficients in kinetic energy expression, terms i j
in mass matrix
M t , function forming part of M,,
P(x,t),T
P ( x ) , y total, steady and turbulent wind pressures
p(x,t) J
I^,H moments of inertia of lower and upper mast segments 1,1 ,Ii,'l moment of inertia of tower, and constant coefficients I2, I3 / in expression for this
i,j,k general subscripts
K surface roughness coefficient, wind velocity expression.
Li ,L2 ,L3 , Lit length of various components, see fig.l.
Qi.Q2 ,Q3 non-dimensional amplitudes of modes of deflection
S ' (zi,! cross-spectral density of wind at two heights
Z2 , W)
j-S (w) fully correlated wind spectral density
Sf.j^.Cd)) cross spectral density of generalised forces i and j
Sp.( w) power spectral density of generalised force i
t time
T kinetic energy
u,(x,t) turbulent wind velocity
V total potential energy
V. . coefficients in potential energy experession, terms in stiffness matrix
VÏ-, function forming part of V,,
V -, (x,t) ,1^ relative and steady velocities of wind
Vfxt j
V mean wind velocity at 10 metre height 10
Vp gradient wind at height Z^
W2 total weight of mast
W3,Wi» total weight of lower and upper mast segments
Wp weight of tower per unit foot at base
W external work by wind
xi height of tower points measured from the ground
X2 height of mast points measured from the tower top
X general measure of height from ground
yi,y2 el^tic deflection of tower and of mast (functions
of X i , X 2 )
yi total lateral movement of mast
Z rate of change of weight per unit foot of tower, with height
Zp gradient height, wind expression 3 structural damping constant
Y power exponent, wind lexpression
T) coefficient of exponent in coherence function
6 rotation of mast as rigid body, see fig.l
(|). displacement shape function in expression for F. Po density of air
T arbitrary time interval in correlation functions w,n angular velocity, reduced angular velocity
1. STRUCTURE AND MODES OF DEFLECTION 2. POSITION OF SPRING-DAMPER SYSTEM
3. SIMPLIFIED FLOW-CHART OF COMPUTER PROGRAM 4. FREQUENCY RESPONSE FUNCTIONS
5. FREQUENCY RESPONSE FUNCTIONS 6. P.S.D. CURVES
7. RESULTS, STRUCTURE I 8. RESULTS, STRUCTURE II.
1. INTRODUCTION
As materials and methods of design improve, civil
engineering structures become increasingly slender. Bridges have longer spans, buildings and towers are taller, cross sections and weights are smaller. This has meant that stability and vibration studies have to be performed in great depth as a standard part of the structural design procedure.
The problem of vibrations in tall buildings and some of their harmful effectswere emphasised by Waller (Ref.l). Fatigue life, architectural finish, comfort, even safety, all may be adversely affected by excessive vibrations. Various methods of vibration reduction in structures have been proposed in the past. The problem is generally seen as one of energy dissipation, since it is rarely possible in Civil Engineering structures to isolate the structure
from the source of excitation; usually the wind, earthquakes, or the loads themselves which the structure supports. A standard engineering undergraduate problem consists of the reduction of vibration of a dance floor by the suspension of a properly 'tuned' weight supported by springs and dashpots beneath the floor. Waller (Ref.l) proposes a rolling weight, with dampers, installed on an upper
service floor of a tall building as a variation on this. Some success is claimed, but efficiency will vary with
frequency of excitation, and the weight itself must be added to the dead weight the structure must support. To return
to the undergraduate problem, the beat of the music may change and the floor must be strengthened to support the weight!
Roorda (Ref.2) has proposed a more promising method of dynamic response reduction, involving the use of 'active' damping. This consists of a system of vibration sensors, electro-hydraulic servo-mechanisms, and forcers. The
reduction of dynamic response of a cantilever column, by the suspension of a feedback-controlled pendulum near the tip
is successfully demonstrated. It is suggested that the method might be applicable to more complex structures with several modes of vibration. In a practical case, however, the
question of electro-mechanical reliability must arise, and it must be admitted that common Civil Engineering
reliability figures (unity minus the probability of failure) are much higher than those generally obtained for electro-mechanical devices. One suspects for instance, that a malfunction causing badly phased feedback signals might cause a structure to self-destruct.
The investigation of yet another method of vibration
reduction in structures was thought to be warranted, especially in view of the shortcomings of existing proposals. Examination of a simple single degree of freedom bar and spring mechanism showed that it was actually possible, under certain
circumstances, to reduce dynamic response by the introduction of an additional degree of freedom, for example a hinge,
Additional flexibility is introduced, but the energy lost to the damper seems to outweigh this, and dynamic response is reduced. The idea is not obvious, but not unattractive either, since no extra weight is added, and passive viscous dampers, being unsophisticated, seem reliable enough. In the case of a tall building, for example, some of the upper floors might be carried on a flexible mounting which would allow passive dampers to be effective, and, should the principle hold, the structure might vibrate less in gusty wind or during an earthquake.
It was decided to investigate tower-type structures because of their relative simplicity, and because some recent designs consist of two obvious main components,
between which an articulation might logically be introduced. A common design involves a reinforced concrete tower of
some considerable height, supporting a steel-framed aerial mast. Total heights are of the order of 1000 feet and more.
The aim of the investigation was to compare performance of structures with and without the added
'flexibility'. Certain approximations in the analysis were therefore thought justified, since the relative magnitude of the output was considered most important. The structures were assigned three degrees of freedom, and Lagrange's energy method was used to get the equations of motion, which yield the frequency response functions. A mean-square spectral density approach was then used, with H9.rris' (Ref. 3) modified version of Davenport's (Ref. 4) spectrum of
stationary random wind excitation ag input. Earthquake excitation was not considered.
2. TOWERS AND VIBRATION MODES
A recent paper (Ref. 5) has discussed the details of the design of a specific television broadcasting tower and presented a comparison with eleven other towers in the world of comparable height (600 ft to 1600 ft approximately).
A schematic diagram of the type of structure examined here appears in figure 1. It consists of a circular
tapering, hollow concrete tower Li in height, surmounted by a steel mast L2 high. "Tower" will be used to refer only to the concrete supporting structure, and "mast" to the steel superstructure. The tower tapers according to a linear
relationship between diameter, D, and xi, measured from the ground;
D = D Q - Dg xi, ... (1)
and the moment of inertia, I, varies according to a third order polynomial in xi
The weight per unit length, W(xi), over the tower height varies according to a linear relationship;
W(xi) = Wj, - Z XI (3)
These relationships apply only over the interval 0 £ xi £ Li.
The mast would commonly be steel lattice with geometry dictated by electronic (aerial) requirements. It might be shrouded with a circular plastic resin
shield. The mast considered here presents two external diameters to the wind. Da over length Ls and diameter Di, over upper length Li». Moment of inertia is constant over L3, equal to I3, and again constant over length Li,, equal to I4. Weight per unit foot is also assumed constant over the two intervals. Total weight of the lower part of the mast is W3 and of thé upper part, Wi, . Total mast weight is W2•
The arbitrarily chosen modes of vibration also appear on figure 1. The tower deflects according to the shape
y i Q i L ] 1 - cos irxi
2 L i ( 4 )
where Qi is a nondimensional deflection amplitude. The deflection of the mast consists of two parts, measured from the tangent to the top of the supporting tower. One is the rigid body rotation allowed because of an artificial articulation introduced here. A rotation Q2 radians arises and at any point the deflection is Q2X2 provided Q2 is small. A further elastic deflection
72 = Q3L2 1 - cos ÏÏX2
2Lj (5)
arises, in which Q3 is a non-dimensional amplitude.
It was felt that these modes would be close enough to reality in form, and sufficient in number to achieve the aims of the investigation. Higher modes are
excluded both for simplicity, and because most of the energy from wind excitation is concentrated at the lower frequencies which correspond to these modes.
3. EQUATIONS OF MOTION
It is necessary to calculate the total potential energy stored due to the deflections allowed for, as well as the energy dissipated in the dampers and the kinetic energy developed.
Beginning with the total potential energy, the flexural energy in the tower is
dxi ... (6) which is 'E„7T' Qi
*j/c^(-^)ll^
,U 32Li2d o + IlXi + I2X12+ 13X1^)^°^ I^LL dXj 2Li ... (7)
where E is the modulus of elasticity of concrete. This integral and many which follow are evaluated numerically in the computer program developed. The coefficient
multiplying Qi^ in (7) is later referred to as iV,, . . The flexural energy in the mast is, similarly
Q
2 jEjTL* 32L221'
-•' c\l3-S^i&
dx-1.I
I It c o s 2TTX 2L5 dX; . (8) which is D 2 Q3' 64L2^Ï'
+ I^L^ 2^ ^-i., ''^^3 (J + = ^ sin . IT L23 - 10]
(9)
where E is the modulus of elasticity of steel and D is defined implicitly.
A sketch of a possible mounting for springs and dampers at the mounting of mast to tower appear as fig.2. The deflection at the spring-damper system, height H above the hinge is
Q2H + QaLzf 1 - cos||j)
If F is the spring stiffness, the energy absorbed in the springs is
^ ' Q2^ + FHL2(l - coslfjQ^Qs + ^ ' ( 1 - cos|g^) Q32
... (10)
In a structure which is to be made flexible, such as these are, it was thought prudent to include terms accounting for decreases in total potential energy due to lowering of the self-weight of the structure during deflections. In the event, these proved of small
significance, but were nonetheless retained. Their calculation appears in Appendix I.
It will also be necessary to have the kinetic energy as a function of the assigned coordinates, Qi, Q2, Q3.
If a dot over a variable signifies a time derivative, then, for the tower, velocity squared is
2 2 _ = Qi
•.<! - - - I E : )
and total kinetic energy of the tower is M
Qi^ 11 = Qi'
i-i]
2gJo
(Wp-Zxi)(^ - -^irrj
(11)
dx
whe of gr
re M?, is defined implicitly, and g is the accelerati
gravity. on
The displacement laterally of any point on the mast is the sum of
y2(x) = QiLi + ^ X2 + Q2X2 + QsLzfl - c o s g ^ ^ .
... (13)
For the mast, therefore,
( y 2 ( x ) ) ' = ^ L i ^ + TTLiXz + ^ X2'JQl' + 2^LiX2 + J x / J Q i Q z + 2 + 2 X 2 L 2 ( - c o s TTX2 " 2 L 2 1 - C O S TrX2\ 2L2>/ Q i Q -L i + T T -L I X 2 +
L.L2(l - c o s l ^ ) . Y K
) Q2Q3 + X2' Q2' + L2'(l - c o s | ^ J Q : . . . (14)The kinetic energy of the mast is
L3 L2
X ^ rf (^^(^))'^^^ ^X^r;^^^^'^))'^-^
(15)
where (24) must be substituted and the integrations carried out for any given structure.
The energy dissipated at the dampers depends on the velocity of movement of the mast at the point of attachment. This is
y2i.(H) = H(i2 + Q3L2(l - c o s | ^ ^ (16)
If the effective damping constant of the system of dampers arranged at this height is C^rroro then the dissipative
„ . . _^ . VIÖL/ function IS •'vise H 2 -Q2' + 2HL2(l - c o s ^ ) Q2Q3 + L2' (1 - c o s ^ ) Q: ... (17)
Considering the free vibrations of the system, Lagrange's equations appear as
3 F.
9 /9T \ ^ 2-^ + IV = 0 '3Q 9Q, 9Qi
where T is kinetic energy, t is time, F„ is dissipative function and V is total potential energy. These yield a set of matrix equations;
i, j = 1, 2, 3 ... (19)
These matrices are 3 by 3, and have the following elements where only 6 of the 9 are given, since they are symmetric:
% 1
and
^ I!;-!:X ^.^3.^.^1^')
l l ( LiL3^ + Ikl \+ I± (hlliL + HL^ _ LiL3^ _ TTLS^N
L 3 \ 2 6 / L , , \ 2 6 2 6 / | ( L . . I x 2 ) ( l - c o s g ^ ) d x £ ( L . . I f ^ X l - c o s l ^ ) dx
22 " 3g • 3 L , g ( ^ 2 ^ - ^^0
L 3 ^ J X 2 ( l - C O s l ^ ) d X 2%2 = i ^
La W 3 L 2 1 3 L 3 g J 1^1 "^ 0^211 -L - I^UÖTJY—JUA2 L? + W,L2 L ' ^ g ^ L 3 L 3 WgL: 2 3 g L . ^ ^ L 2 W^L2 •33 g L . ^ „ + .L3 WuL 2 "*-^2 gLi j ( 1 - c o s | ^ ) dx2 . . . ( 2 0 a )^11
= ni -
K«^i'
*
" ' ^
*
c)
W L 2 TT 1 2 2 ^ _ BTT 1 3 2 V22 = FH^ - WL2 V23 = F H L 2 ( l - c o s g ^ ) - B 2 V33 = D + ^ ( 1 - c o s ^ ) - 2A . . . ( 2 0 b )where A, B, C are defined in Appendix 1, and ' D 1 1 D 12 D 22 D 23 33 ,01 M = D 1 3 C . v i s e = Q V i s e
=
s.i
s c c o s COS-TTH 2Lj .TTH_ 2L5 ) ( 2 0 c ) The expression for D,-, is the result of an approximateand arbitrary attempt to account for some structural, or material damping in the structure, especially the concrete tower. Taking the tower as a single degree of freedom structure for a moment, its natural frequency (i)_ will be
n
and its equation of motion appear as D Qo 11 M 11 Q] 11 M 11 Q] = O or Qi + 23w^Qi + ui^ Qi = 0 (21)
where 3, the structural damping constant can be taken as, say 2% and
D 11 ,04 M 11
11 M 11
(22)
Assuming a solution for the vector Q. of the form of the real part of
iwt
... (23) where i is the usual imaginary number and o) is angular velocity, and substituting in equation (19), it Ijecomes, dividing through by e^'^^'^;
' Q I Q2 Q3 = [Q.» Q2° Q3° - 0 )
MW]
^^^[y h i ^Kj] M = °
or [ E , . ( . ) ] [Q.°] = 0
( 2 4 ) ( 2 5 )w h e r e
KJ(""J
is a complex matrix whose elements areThe inverse of this matrix is the matrix of complex frequency response functions, since for any forcing vector of the
form of the real part of [F.I e^^'^,
hMh"]^'""
= [Fje^-* ... (27,
or r « i r i r i
. (28) which is usually written
[ Q I ] =[Hij«o)][Fj ... (29)
where the H.. are complex frequency response functions giving response as a function of w in mode i to the periodically varying generalised force related to mode j.
4. WIND EXCITATION
It is now necessary to derive expressions representing the generalised forces acting on the structure. The
horizontal motion of the tower is given by expression (4) and that of the mast by (13). Incremental deflections are therefore
6yi(x) = 6QiLi(l - -^fff^)
6y2(x) = 6Qi(Li+ ^ ) + ÖQ2X2 + SQsLzd - ^ ^ ^ ^ )
... (30) If D(x) represents diameter of the tower as a function of height, and p(x,t) pressure on the tower from the wind (net in the direction of the wind), then incremental external work, SW , is,
.Li D(xi)p(xi,t)6yi(x)dxi o 6W e \ D(x2)p(x2,t)6y2(x)dX2 ... (31)
and the generalised force, F., associated with mode of vibration i in Lagrange's equation is
3(6Wg)
^i " , 8(ÖQ^) , , ••• ^^^^
o o 'L2 o L2 D(xi)p(xi,t)Li(l - •^|5^)dx, D(X2)p(X2,t)(Li + 2 X2)dX2 D(X2)P(X2,t)X2dX2 D(x2)p(X2.t)L2(l - •^§p^)dx2 . ... (33)
Two comments are worthwhile at this point. These expressions are all of the form
F, =
length
D(x)p(x,t)(()^(x)dx ... (34)
where the ^. are the relevant displacement shapes and are
distinguishable in expressions (33).
Further, the integrals in (33) which are over length L2 must be performed in two steps, with D(x2) set constant at Da and Di».
The wind pressure variation with height is
P(x,t) = èPoCj5(V^g^(x,t)) .. (35)
where po is air density, C„ is coefficient of drag, taken here as constant over heiglit, and V^g]^ is relative velocity of structure and wind. The structures studied are
relatively tall, heavy and flexible, and will therefore have low natural frequencies and negligible velocities compared to the wind. V -, is then approximately
V^gl(x,t) = V(x,t)
= V(x) + u(x,t) (36)
where V(x) is the steady state component of wind superimposed on which is the turbulent, time-varying component u(x,t),
comparatively smaller, and with zero mean. Pressure may be written as approximately
P(x,t) = èPoCj5V(x)^ + PoCpV(x)u(x,t)
where a term in_u(x,t) squared has been deemed to be negligble, and P(x) is the steady pressure, while
p(x,t) is the turbulent, zero mean, time-variant pressure. Only the time-variant pressure will be of interest, since periodic vibrations of the structure are to be studied. A check on all structures examined showed that there
was negligble error in assuming this alternating force applied to the undeflected geometry. The vibrations with zero
mean are, of course, superimposed on the steady wind deflection.
The generalised forces now appear in the form Fj^ = J D(x)Pi,CpV(x)u(x,t)<|)^(x)dx ... (38)
length
after a substitution from (37) in (34). Unfortunately u(x,t) has been observed to be a random variable, with only certain statistical properties as measureable
characteristics. A key measure of intensity of turbulence is the mean-square value or its root, the r.m.s. value. The spectral distribution of this mean square value with frequency, or with angular velocity,w , has also been measured, and a number of functions, largely empirical, are available to express this. This distribution, or spectral density of mean squared value is also called power spectral density, p.s.d., from the electronic
terminology, in which field early use of the concept was made. The integral of p.s.d.over all frequencies gives
the mean square value. We are virtually obliged to accept these statistical averages of wind input as all that is usefully available, and to use these to obtain statistical averages describing the output vibrations.
To get the p.s.d. of generalised forces, the cross-correlation at different heights of any force must be
accounted for. the cross-correlation of F^ and Fj at
heights zi and Z2 at times t and (t+x) gives the expectation
<<SF^(zi,t).6Fj(z2,t+T)>
= (1)^(21 )^j(Z2)D(Zl)D(Z2)P0^Cjj2v(Zi)V(Z2) <U(Zi,t) U(Z2,t+T)> ... (39) the cross spectral density of F^^ and Fj will be the
Fourier transform of the double integral of this over the structure's height, that is, the cross-spectral density
(or p.s.d. if i = j), Spj^p-is,
Sp.j, (w) = P O ^ C Q 2 V ( Z I ) V ( Z 2 ) S'(zi,z2,u))dzidz2
1 i J J
•^ height
where ^ ^ S' (Zi ,Z2 ,W) = — U TT <U(Zi ,t)u(Z2 , t + T ) > e "'"'''^dT _ c o ... (41) The latter Fourier transform is the cross spectral
density of wind velocity between two points at differing heights zi and Z2. In fact only the real part of (41) will interest us.
Harris (Ref.3) has suggested an expression for this cross-spectral density which consists of a coherence
function multiplied by a spectral density independent of height. This is
where
s;(zi,Z2,a)) = e-"l^i- ^\(i^)
2KL'Yi (42) S (to) = ^^^ ^H . . . (43) ^ Tr(2+n^)^/6 8V n = - ^ /2 + m'"- ... (44) L'V(z)
and reduced angular velocity öó , is wL'
TIT = —
2-nV 1 0
In these K is a surface roughness coefficient, L' is a scale factor of 5900 feet, Vio is the mean wind at 10 metres and S^(4;)) is the spectral density of velocity. K varies from
.005 for open terrain to .015 for terrain with 30 to 50 foot obstacles, to_.050 for built-up central urban areas. An expression for Vio is given later. The cross spectral density of forces i and j can be written as
S (0)) = [ [ (t).(Zi)(l) (Z2)D(Zi)D(Z2)Po'C^
*^i*^J hiiiht ^ J ° V(zi)V(z2)e~^l^i-^2ls^(a3)dzadz2
... (45)
These will be onerous to evaluate unless a simplification is made, since diameter varies under three regimes
discontinuously with height. Even the simplest of the expressions, the p.s.d. of generalised force F2 has to be written as
Ls L; S^.(co) = Po'C^2S„(a))J D,2 ^1-2 u V(zi)V(z2)e^l^i'^2| dzidz2 + 2D3D L, L, 31^"» L3.L3 + D L ^ X > ZlZ2V(Zi)V(Z2)e ^\^^ ^2|(i2idZ; ZiZ2V(Zi)V(Z2)e~^l^l ^2|(j2idZ2! o (46)
and the expression for Sp (co) require of this type. The double^integrals,
.res six double integrals further, are not easy to evaluate numerically. It was decided at this stage to admit an approximation to the study, that of turbulence fully correlated with height. This is equivalent to saying that the structure will be buffeted by turbulence in the same sense at all heights at any given instant. The coherence function is taken as unity, and S' (zi ,Z2 ,(JJ)
becomes S^(Ü)) as defined in expression (43). A welcome simplification of (45) is available;
»F-F-^'^^ ^ Po^Cp^S^(<Jj) (}>^(zi)D(zi)V(zi)dzi
height
height
(j) .(Z2)D(Z2)V(Z2 )dZ2 (47)
where these integrals may be evaluated separately and simply multiplied together. Each integral will be evaluated in two or three stages, following the discontinuous variations of D(z). Again, if i = j , these expressions simply give p.s.d. functions. For those wishing to follow the
computer program in detail (Appendix II) a sample cross spectral density function would be
where RFl = Fl RF2 = Po^jj'S^((o)(RFl(RF2); low •" ^ W d •" ^^top (48) Fl low Fl Fl mid top F2 ., + F2^ , mid top' f-L"! (Do- Zx)Li(l-^gf!^ )V(x) dx ^ ^ i d " J o 'L1+L3 Li li, 1+L2 L 1 + L 3 li + L 3 2L. D3(^(x-Li)+ I4) V(x)dx D^(^(x-L, )+ L, )V(x)dx D3 (x-Li )V(x)dx F2 top Li+L: Ll+L; D^(x-Li)V(x)dx (49)
Y Y Y = = = .16 , Z Q .28, Z Q .40, Z Q = 900 ft = 1300 ft = 1700 ft The expression used for mean wind is
V(x) = VQ(x/Zg)Y ... (50) where y is a power exponent appropriate to Z„, V-, is the average gradient wind at the gradient height Z Q , and x throughout (49) and (50) is a uniform measure or height from the ground. The value of Vio used in expression (43) is therefore
- 10 ^
Vio = V Q ( | ^ ) ... (51) G
A list of these constants appropriate to different conditions is given by Davenport (Ref.4):
Open terrain, few obstacles Uniform 30-50 foot obstacles Large irregular obstacles
Expressions such as (48) and (49) are readily calculable on a digital computer, giving spectral density functions for the generalised forces acting on the structure.
5. RESPONSE SPECTRAL DENSITY
It is now necessary to see how the frequency response functions defined earlier may be used in conjunction with this spectral density of input force, to give spectral density of output vibration displacement.
Looking at equation (29) it is clear that, for the present three degree of freedom system, any of the output amplitudes, say Q , may be written
Qi = Hii Fl + Hi2 F2 + Hi3 F3 ... (52)
Unfortunately it has been necessary to settle for obtaining only the mean square densities of the generalised forces, Fi,F2, F3, and it will be necessary to accept this type of information regarding the displacements obtained. The mean square value of any output, say of Qi again, takes the following form, where H..(w) is written as H.. for
compactness: "^ "^ <Qi2 > = <(Hfi Fi+ Ht2 F2 + Hfj F3)(H„ Fj + H^^ F^ + E,, F^)>
= Hti Hn <Fi2 > + H* Hj2 <FiF2> + Hfj R,, <F^ F3> + Hf 2H11 <F2Fi> + H*2 H12 <F2^ > + H=J^, Hi3 <F2F3> + Hfs Hn <F3Fi> + H=f3 H^^ <F3F2> + Hfg E^^ <FÏ^ >
Now <Qi > <F.F •> = 1 3 S Q (ü3)dü) p ^^ ... (54) 00 0 ^FiFj (f^) ^^ • • • (55) where S Q J ( W ) is the p.s.d. of output Qi, and the asterisks denote complex conjugates.
Substituting (54) and (55) back into (53), removing the common integral signs from both sides, and taking the modulus of the complex function which results from the use of the H.., gives
SQ^(<.) = in* HjSp^(a,) + 2|H*, Hj, |Sp^F^(«) + 2|Ht, H,3 |Sj,^pJ(a))
+ |Ht2Hi2 18^^(0)) + 2|H*,H„ |Sp^p^((.)
+ |Ht3Hi3 ISp/o))
Similar expressions exist for S Q ((J) and S^ (to). These are easily evaluated for a sequence of values of w by digital computer. Again for those wishing to examine the program in Appendix II in detail, a convenient way to write the above appears as:
SQ^(a)) = Po'Cj^2S^(w)||Hti H „ | (RFl)^ + 2lH*jHj2 |(RF1)(RF2)
+ | H t 5 H i 2 | ( R F 2 ) 2 + 2|H*^H^3 | ( R F 2 ) ( R F 3 ) + 2 | H * J H J 3 | (RF1)(RF3) + lHt3,Hi3 | ( R F 3 ) 2 l
It must be remembered that each of the H.. are complex
functions of w. "^
An integration of S Q . ( W ) for all values of w will give the mean square value''"of Qj^. This is easily done at the same time by the computer. A simplified flow chart for the program developed appears as figure 3.
6. PARAMETRIC STUDY
Two particular practicable structural designs were studied. The first is similar to the specific structure mentioned earlier. This structure was considered in some detail. The second structure is quite hypothetical and was used to verify whether behaviour would be generally similar or not, i.e. whether anything "peculiar" had been chosen by chance in the first instance.
A list of dimensions and other properties follows in the table below. Reference may be made to figure 1 for the definitions of the various symbols. It will be seen that structure I is a 900 ft concrete tower supporting a mast nearly 200 ft high. The tower tapers as stated earlier. Structure II has a tower only 700 ft high, but a longer mast of 240 ft. The greater proportion of the
Structure I Structure II Lengths (ft) Weights (lb) Diameters (ft) Second Moments (ft ) Li L2 L3 L, W2 W3 Wu ^F z D3 Ti, Do I^s I'a i ; lo Ii I2 I3 900 199 115 84 157,000 102,000 55,000 65,973.5/ft 61.4492/ft2 12.0 5.0 80.0 .067/ft 8.380 .589 351858 -1364.49/ft +1.85476/ft2 -.000854/ft3 700 240 140 100 188,000 126,000 62,000 59.828/ft 61.4492/ft2 14.0 6.0 73.0 .067/ft 11.00 1.50 233103 -1019.16/ft +1.59856/ft2 -.000854/ft'
H for both structures was constant at 15 ft, and the value Ej., elastic modulus for concrete, was taken to be one tenth that for steel. The goal of the parameter study was to vary the spring stiffness, F, at the articulation and the viscous damping constant, to find an optimum combination of the two. The criteria of optimality were the minimisation of
displacement at the top of the mast and angular rotation. The first criterion is necessary in order to minimise
structural dynamic stresses and the second is required for the purpose of reducing signal scatter.
The wind parameters used were those appropriate to an open rural site with surface drag coefficient K = .005,
gradient wind V Q = 80 m.p.h., gradient height ZQ = 900 ft.,
A typical mean square spectral density (p.s.d.) curve for one of the system's three modes appears in figure 6A, and the p.s.d. curve for movement at the mast
tip appears in figure 6b. The integral of this curve over all (effective) frequencies is 18.2 ft^, giving an r.m.s.
amplitude of 4.26 ft.
This r.m.s. amplitude applies only for this spring stiffness, F, and value of damping, C . . What is required is a survey of all r.m.s. displacement values for the
complete series of combinations over a reasonable range of values of parameters F and C . . Figure 7 presents such a survey, and as such, is th^'^'^Sin result of this study. Each curve in the figure is for a specific value of damping constant. The extreme right-hand ordinates have stiffness values so high that they approach the 'rigid' case. There
appears to be a local 'optimum' region around F = 350,000 lb/ft, and an optimally bad region in the vicinity of F = 1,000,000 lb/ft. The local 'optimum' however, represents higher deflections
than the rigid case. This would appear to suggest that the proposed method of amplitude reduction is a failure, and that the rigid case is, in fact, the best solution. This is partially true. Some advantages accrue however to the local 'optimum.' Superposed on figure 7 is a dashed curve of concrete tower-top r.m.s. displacements, for the
60,000 lb/ft/sec damping values. They are lower in the region of the optimum and dynamic bending moments will be smaller. This is however an undergraduate dance-floor type of solution, since displacements of the suspended mass, the mast in our case, are quite large. However, in some designs, lower tower bending moments may be an advantage.
The local optimum solution may have a further advantage for some systems. The p.s.d. curves for this solution show much of the energy concentrated at lower frequencies than that of the rigid case, since it is a more'flexible' system. Much research remains to be done on the environmental issue of which vibrations are most tolerable to persons in a
structure but velocity and more particularly acceleration are both as important as amplitude. Thus, in the case of
a tall building, the 'optimal' system may well prove preferable, since it involves only slightly greater displacements with
much of the energy concentrated at an appreciable lower frequency.
The extrema of the curves in figure 7 are explained by an examination of the mean square value of the
individual modes, and of the orthogonal vibration modes. It seems that in the optimally 'good' systems the dominant mode of vibration (at 1.4 rad/sec) involves motion of the
tower to the right while the mast tilts in mode 2 to the left (an vice versa). This not only reduces total motion
energy at the damper. In the optimally bad case, the dominant mode is at 1.3 rad/sec, and in this mode, both mast and
tower move in the same direction in phase, with small Qz amplitudes. This exaggerates motion at the tip and the dampers are least effective.
Results for Structure II, which has characteristics listed above, are similar. The concrete tower in this case has a higher natural frequency, and the taller mast requires greater spring stiffnesses to give the same results. The overall behaviour is summarised in figure 8. Again, there is an optimally good region of spring stiffnesses, this time at about F = 900,000 lb/ft, and an optimally bad region near 3,400,000 lb/ft. R.m.s.
displacements are however, even less for values of F nearing 4,800,000 lb/ft., and decrease monotonically to even better values for higher F. The stiffest case is again the best, and the optimum at 900,000 is only a local one.
All the above deals only with dynamic vibratory response. The rigid case is therefore even more attractive, since
dynamic deflections will be super posed on much lower values of steady-state wind deflection.
7. CONCLUSIONS
The random vibrations of a tall un-guyed television-broadcasting-type structure, subject to random wind
excitation have been studied. Only the in-wind vibration caused by turbulent wind velocity fluctuations is considered. Vibrations due to vortex shedding and due to earthquake
motion are not considered.
The scheme of introducing an articulation, with added springs and dampers, as an attempt to reduce r.m.s. response proved unsuccessful, in the sense that response was never as small as for the rigid case, without the articulation. This conclusion is valid, at least for the range of spring stiffness and damping values studied. This range was felt to represent something of what was available commercially.
This touches however, on the major limitation of the study, the assumption of linear springs and dampers. The springs arranged radially about the mast, at the position shown in figure 2, will have an effective system stiffness which is non-linear and hardening. This is easy to show and is due to the 'tightrope' effect of the pair of
springs perpendicular to any motion. This will have a
beneficial effect on the local optimum r.m.s. displacements, almost certainly reducing them. By how much it is not
possible to say, without a much more complicated formulation. Again, a linear damping constant was used as in most
conventional dynamics calculations, mainly because this renders the differential equations of motion tractable. An added justification is present here however, in that low natural frequencies are involved with small displace-ments. This makes velocities small, and linear damping quite desirable. It would not be surprising however, if in fact, a commercially available damper were able to
develop some portion of 150,000 Ibs/ft/sec; if this damper were also to display some non-linearity. One would indeed almost expect this. Certainly a system of synthetic rubber
blocks in shear able to dissipate this much energy linearly for such a relatively low linear stiffness would seem difficult to develop.
Figures (4a) and (5a) show the modulii of the complex frequency response functions for the optimally 'good' case , and figures (4b) and (5b) refer to the optimally 'bad' case. In the good case the frequency of the articulated mast
system coincides with that of the tower without the damping system. The resonant peaks present in these figures
correspond to the following modal configurations.
Figure (4a). At 1.4 rad/sec motion of mast and tower are 180 degrees out of phase. At 0.95 rad/sec the mast and tower are in phase.
Figure (4b). At 1.3 rad/sec the tower and mast are in phase whereas at 1.8 rad/sec both are out of phase.
Clearly the system equations of motion are highly coupled and the response functions are very sensitive to changes in stiffness and damping at the mast articulation point,
In addition to the r.m.s. mast top deflection being greater in the optimally 'good' case than in the 'rigid' case, the r.m.s. rotation or slope of the transmitter mast was also greater. It is essential in T.V. transmission to minimise mast rotation in order to reduce signal scatter. Thus the 'rigid' system gives the least rotation as well as the least tip deflection.
A suggestion for further work in this area would appear to be toward a study of manufactured damping systems and their exact characteristics. Some of the other work in the Cranfield group is quite compatible with this direction, especially the forthcoming work on non-linear systems by Kirk (Ref.7). At the same time it is far from certain that the initial idea (of added flexibility with added dampers) would not in fact be a good idea for the reduction of earthquake response in some structures. This appears to be another possible subject for study.
REFERENCES 1. WAUER, R.A, 2. ROORDA, J. 3. HARRIS, R.I 4. DAVENPORT,A.G. 5. BARTAK,A.J.J and SHEARS, M 6. SHEARS, M. 7. KIRK, C.L.
The design of tall structures with particular reference to vibration. Institution of
Civil Engineers, 48, p303, 1971. Active Damping in Structures.
Cranfield Report Aero No.8, July 1971. The Nature of the Wind. Semiriar on the Modern Design of Wind Sensitive Structures held at the Inst,itution of Civil Engineers, London, 18 June, 1970. The response of slender line-like
structures to a gusty wind. Institution of Civil Engineers, 23, p389, 1962.
The new tower for the Independent Television Authority at Emley Moor, Yorkshire. The Structural Engineer, 50. p67, February 1972.
Report on the measurement of wind and vibration at the Emley Moor Television
Tower, Paper 122, International , Symposium'Vibration Problems in Industry,
UKAEA Windscale and NPL Teddington, Keswick, England, April 1973.
Application of the Fokker-Planck Equation to the Random Vibration of Non-Linear Systems. Cranfield Report Aero No.20, September 1973.
APPENDIX I
SELF-WEIGHT TERMS
The change in height of the top of the tower, due to the variable slope of the tower with respect to its original vertical centre line causes a lowering of the mast. The accompanying change in potential energy is
-Wc
i^]
dX]16 Qi^ (Al.l)
The mast is also lowered because of the rotation,6, at its mounting, equal to
(¥^«0-If WL2 is the first moment of the weight of the mast with respect to the mast base, then this change in potential energy is
- èWL2^Qi'|' + QiQzTT + Qz". )
provided the movements are small.
... (A1.2)
General deflections of the mast in its elastic mode will be from an initially sloped position, angle 6 from the vertical. Elastic deflections of the mast will therefore have a vertical component. If RM and RN are distances above the hinge of the centres of gravity of the lower and upper parts of the mast, then this change of potential energy is
WsQsLzfl - cosTT | M - ^ + W,Q3L2(l ^^
- s m t COSTT 2L2
(A1.3)
Further, the slope of the elastieally deflected mast
with respect to its position at angle 6 draws the centre of gravity back toward the hinge. The change in
potential energy here is
cos9 W3^^ 8^ QB^ RM o ,2 TTX2 2L2 WLTT-s i n " 7^^^ dx2 + " Q; RN s i n ^ - ï ^ dx2 ... (A1.4)
and
Approximately sin6 by 6, and cos6 by unity to eliminate fourth order terms and higher, and adding
(A1.3) and (A1.4) gives,
-(|BQIQ3 + BQ2Q3 + AQ3M ... (A1.5)
^ ' ^ ^ ^ ^ n w T / n T T R M \ ^ ™ , (^ T T R N N B = W 3 L 2 ( 1 - c o s 2 ^ - ) + W i t L 2 l l - COS2Y;- ) . . . ( A 1 . 6 ) A = ? i (W3RM . W.RN - ^ ^ i i ^ s i n - ^ - ^ s i n ^ ) 1 6 ^ TT L2 TT L2 . . . ( A 1 . 7 ) A more significant loss of potential energy is the
lowering of self-weight of the concrete tower during flexure. The weight above any point on the tower at height x ,
excluding the mast, is
(Wj,-Zxi)dxi = WpLi - ' ^ - Wj,Xo + ' ^
... (A1.8) the downward movement at any point is
2 2
Kdyi/dxi)^ = ^ V ~ ^^"'Sr ••• (A1.9)
so that the change in potential energy is
-CQ.' = -Q>' iV'CV- - ^ - V '
*
^ ' -"' Üt)
"^^
' O... (ALIO) where C is implicitly defined.
APPENDIX II COMPUTER PROGRAM
The following pages comprise the computer program used to obtain numerical results. Most of the notation is as it appears in the report text, with minor changes with which most readers will have no difficulty. The
following equivalences between text and program apply, as well as some others which are obvious.
Text ^ j M.J ^ij ^ij Ij' (Jj V* Program DAMP(I,J) EM(I,J) V(I,J) EQIJ EIJ W VllA % 1 EMllA
P « O G R A ^ ( D 1 1 A )
I I M P U T 1 = C R 0
UUTPUT 2 = L P Ü
COMPRESS I N T E G E R AND L O G I C A L ElMD
MASTER TOWER PROB COMPLETE
'^ U N I T S THROUGHOUT ARE F T . L B S . S E C . D I M E N S I O N E M ( 3 , 3 ) , 0 A M P ( 3 , 3 ) » V ( 3 , 3 ) R E A L L 1 . L 2 , L 3 » L 4 # I 3 # 1 4 R E A L L P R I H E C Ü M P L E X E 0 2 2 , E U 2 3 , E Q 3 2 , E 0 3 3 , D E L T A , H l 1 , H l 2 f H l 3 » H 2 1 » H 2 2 / H 2 3 » H 3 1 # H 3 2 C ü M P L E X H 3 3 , S r t 1 1 , S H l 2 , S H 1 3 , S H 2 1 . S H 2 2 » S H 2 3 , S H 3 1 , S H 3 2 , S H 3 3 C Ü M P L E X H T 1 , H T 2 , H T 3 , S H T 1 , S H T 2 , S H T 3 A F U Ü W ( X ) = ( I ) 0 - D S * X ) * L 1 * ( 1 . - C O S ( P I L * X ) ) * V G Z G * X * * ü A M M A A F I M I D C X ) = D 3 * ( . t > * P I * ( X - L 1 ) + L 1 ) * V G Z G * X * * G A M M A A F I T O P ( X ) = D 4 * ( . 5 * P I * ( X - L 1 ) * L 1 ) * V 6 Z 6 * X * * G A K M A A F 2 M I D ( X ) = D 3 * ( X - L 1 ) * V G Z 6 * X * * G A M M A A F 2 T 0 P ( X ) = D ' » * ( X - L 1 ) * V G Z G * X * * G A M M A A F 3 M I D ( X ) = L 2 * 0 3 * ( 1 . - C 0 S ( P I L * ( X - L 1 ) ) ) * V G Z G * X * * G A M N ! A A F 3 T 0 P ( X ) = L 2 * D 4 * < 1 . - C O S ( P I L * ( X - L l ) ) ) * V G Z G * X * * 6 A M M A C X ( X ) = ( P I S Q R / 8 . ) * ( W F * L 1 - Z * L 1 * L 1 * . 5 - W F * X + Z * X * X * . 5 ) * ( S I N ( P T L 0 * X ) ) 1 * * 2 V I I A X ( X ) = ( t C * P I * * 4 / ( 1 6 . * L l * L 1 ) ) * ( E I N 0 T + E I 1 * X + E l 2 * X * X * E l 3 * X * X * X ) 1 * ( C 0 S ( F T L 0 * X ) ) * * 2 E M 3 A X ( X ) = ( 1 . / G ) * ( ( W S / L 3 ) * ( L 1 * L 2 * ( 1 . - C ü S ( P T L T * X ) ) 1+ . b * L 2 * P I * X * ( 1 . - C 0 S ( P T L T * X ) ) ) ) E M 3 B X ( X ) = ( 1 . / 6 ) * ( ( W 4 / L 4 ) * ( L 1 * L 2 * ( 1 . - C 0 S < P T L T * X ) ) 1 + . ! J * L 2 * P I * X * ( 1 . - C 0 S ( P T L T * X ) ) ) ) E M 2 3 A X ( X ) = ( 1 . / G ) * U ( 3 * L 2 * X * ( 1 . - C 0 S < P T L T * X ) ) / L 3 E K 2 3 B X ( X ) = ( 1 . / G ) * W 4 * L 2 * X * ( 1 . - C O S ( P T L T * X ) ) / L 4 t M 3 3 A X ( X ) = < 1 . / G ) * W 3 * L 2 * L 2 * ( 1 . - C Ü S ( P T L T * X ) ) * * 2 / L 3 E M 3 3 B X ( X ) = ( 1 . / 6 ) * W 4 * L 2 * L 2 * ( 1 . - C 0 S ( P T L T * X ) ) * * 2 / L 4 5 5 F 0 R M A T ( 2 X , A 8 , l b ) 6 = 3 2 . 2 P I = 3 . 1 4 1 ' 5 9 P I S Q R = 9 . 0 6 9 6 0 P I F 0 Ü R = 9 7 . 4 ( ) y 1 EC = . 4 3 < r t + 0 9 E S T = . 4 3 2 Ë + 1 0 R t A D ( 1 , 1 ) L l , L 2 , L 3 , L 4 1 F Ü R M A T ( 4 F 0 . Ü ) R E A O d , 2 ) W 2 , U 3 , W 4 2 F U P M A T ( 3 F 0 . 0 ) R E A D d , 3 ) W F , Z , .i F Ü R M A T ( 2 F Ü . Ü ) R E A O d , 4 ) 1 3 , 1 4 4 F ü R M A T ( 2 F ü . ü ) R E A O d , 2 1 ) E I N U T , e i 1 , E I 2 , E I 3 2 1 F U R M A T ( 4 F 0 . 0 ) UL2BAR = W 3 * L 3 / 2 . + W 4 * ( L 3 + L 4 / 2 . ) R t i f O d , 1 4 ) D 0 , Ü S , D 3 , D 4 1^4 F O R M A T ( 4 F Ü . O ) R h A 0 d , 1 b ) R K , L P H I M E , V B A R G , Z G # G A M M A 15 F 0 R M A T ( 5 F 0 . 0 )
1 / , 2X,4')H*******************************************'**) WRITE(2,12)
12 FURMATC/«2X,42HSTEFL MAST ON CONCRETE lÜWER WIND RESPONSE, 1/,2X,33HKEN JUHNS-COLIN KIRK SUMMER 1973 ,/)
WrtlTE(^,'>6)
••)6 FORMAT (//,2X,16HT0WER PROPE RT I E S , / , 2 X , 1 6H ) WKlTE(2,32)Ll,L2,L3,L<f,W2,W3,W4,WL2BAR,WF,2,Ii,I4
52 FüRMAT(2X,3HL1 = ,F'^.i,3X,3HL2 = ,F9.3,3X,3HL3*,F9.3,3X,3HL4=,F9.3f
1//,2X,3H'J2 = ,F9.0,3X,3HW3 = ,F9.Ü»3X,3HW4=.F9.0,3X,7HWL2BAR = ,E12.6, 2//,2X,iHUF = ,F8.1,.5X,2H2 = ,FÖ.4,//,2X,3HI3 = ,F6.3,3X,3HI4 = #F6.3)
i/J« I T E ( 2 , 3 6 ) F I N O T , F 11 # E I 2 , E I 3
56 F Ü R M A T ( / / , 2 X , 6 H E I ^ 0 T = , F « . U , 2 X , 4 H E I 1 = , F 9 . 3 , 2 X , 4 H E I 2 = # F 9 . 6 , 2 X , 4 H E I 3 = 1,F9.6)
WRITE(2 ,3/)DU, 0S,I)3,D4
37 FÜRMAT(/.2X,3HDU=,F9.3,3X,3HDS=,F9.3,3X,3HD3=,F9.3,3X,3HD4=,F9.3) WKITE(2,3Ö)
5H F U R M A T ( / / , 2 X , 1 5 H W 1 N [ ) PROpE RT I E S , / , 2 X , 1 5 H — - ) WRITE(2, 39 )RK,LPRIM E,V13ARG,Z6,GAMMA
.Vi FÜR(-IAT(/,2X,24HSURFACt DRAG C0EFF*T RK=,F5.3#3X,
12UHSCALE LENGTH LPRIME = ,F6.Ü,7,2X,20HGRAD I ENT WIND VBARG = , 2 F Ö . Ü , 3 X , 1 9 H G « A D I E N T HtlGHT ZG = ,F6.ü , / , 2X,
321HP0WER EXPONENT GAMhA=,F5.3) ALF= PI*L3/L2 BtT= ALF/2 SALF = SIN(ALF) C A L F = C O S ( A L K ) S3ET = SlN(BET) CrlET = C<»S(BET) Ri-^ =.b*Li RN =L3 +.b*L4 PTLU = PI/(2.*L1) PTLf = PI/(2.*L2) A =.0625*PlSyR*(W5*RM+W4*RN-W3*L2*SIN(Pl*RM/L2)/PI 1-W4*L2*SIN(PI*RN/L2)/PI) H = W3*L2*(1.-C()S(.b*PI*RM/L2))+W4*L2*d.-COS((.i)*PI/L2)*RN)) D=(EST*PlF0Ut?/(32.*L2*l2))*(I3*L3+I4*L4td3*L2/PI)*SIN(PI*L3/L2) 1-(I4*L2/P1)*SIN(PI*L3/L2)) HU = 0. H ü = O . h 1 = L 1 H ^ = L 1 + L 3 H i = L 1 + L 2 E = .01 IMD = 3 C A L L F 4 l N T S M P ( h O , H 1 , C X , E , I N D , C ) W K l T E C ^ , ^ ) 5 F U R M A T ( / / / , 1 X , 2 2 H h V A L U A T I N G U S E F U L A B C D ) W K I T E ( 2 , 'O 6 F u R M A T d 4 X , A H A I S , r i X , 4 H B IS,11X,<:.HC I S , 1 1 X , 4 H D I S ) W » I T E ( 2 , ^ ) A , i i , C , D 7 F 0 « M A T d ' J X , E l 2 . 6 / 3 X , E l 2 . 6 , 3 X , E l 2 . 6 , 3 X , Ë l 2 . 6 ) ' R F I A R F G F N * Z D F O R C E S D I V I D E D BY ü ( T ) * R H 0 * C D S Q U A R E D ' A L L V E L O C I T I E S A R t IN F T . / S E C . 'Z'jM IS Zt' IN r E T R E S K n O = .Ü/'6fi4/3f^,
V u Z G = V H A R G / ( 7 Ü * * b A M r 1 A ) V b A R i O = V B A K G * d U . / Z G M ) * * G A M M A P I L = ..'5*P1/L2 F = . 0 1 I N O = ,^ C A L L F 4 1 N T S M P ( H ( J , H 1 , A F 1 L O W , E , I N D , F 1 L O W ) l U D = i C A L L F 4 l N T S M P ( h 1 , H 2 , A F 1 M I D , E , I N D , F 1 M I D ) li^n = 3 C A L L F 4 1 N T S M P ( H 2 , h 3 , A F l T Ü P , E , I N D , F 1 T Ü P ) I N C = i C A L L F < » 1 N T S M P ( H 1 , H 2 , A F 2 M I D , E , I N D , F 2 M I D ) I.NiO = i C A L L F 4 l N T S M P ( H 2 , h 3 , A F 2 T O P , E , l N D , F 2 T 0 P ) I ,\ D = i C A L L F 4 1 N T S M P ( H 1 , H 2 , A F 3 M I 0 , E , I N D , F 5 M I D ) I N D = 3 C A L L F 4 l N T S M P ( h 2 , H / ) , A F i T Ü P , E , l N 0 , F 3 T 0 P ) W K l T E ( 2 , 3 1 ) F l L O U , n M l O , F 1 T O P , F 2 M l D , F 2 T O P , F 3 M I D , F 3 T O P
31 F v J R i l A K / / / , 2 X . ?riF1 LUW= , E l 2 . 6 , 2X , ? H F 1 M I D= , E 1 2 . 6 # 2X , 7H F 1 TOP= , 1 E 1 2 . 6 , / , 2 X , 7 H F 2 M I I ) = , E 1 2 . o , 2X , ?HF2T0P= , E 1 2 . 6 » / » 2X , 7H F 3M 1 D= , 2 b 1 2 . ó , 2 X , / H F 3 T O P = , E 1 2 . 6 ) K F l =F1 LOW + F l i ^ I D +F1 TOP RF2 = F 2 M ! ) + F2TÜP R F 5 = F 3 M D + F 3 T O P W K l T E ( 2 , ' ^ 3 ) R F 1 , R F 2 , R F i ^^ F ' J R i l A T ( 5 X , 4 H R F 1 = , E l 2 . 6 , b X , 4 H R F 2 = , E l 2 . 6 , 5 X , 4 H R F 3 = , E 1 2 . 6 » H A V I N G A , B , C , l ) FORh' M A T R I C E S UF E Q - N S OF M O T I O N M A T R I X N A M E S - EM IS I N E R T I A , D A M P O B V I O U S , V IS S T I F F N E S S H. = .01 lul) = j C f l L L F ' + l N T S M P ( H O , L . 5 , E M 1 3 A X , E , I N n , E M 1 3 A ) I WO = 3 C A L L . F 4 1 N T S M P ( L 3 , L 2 , E M 1 3 B X , E , I N D , E M 1 3 B ) U D = 3 C A L L F ^ I N T S M P ( H O , L i , E M 2 3 A X , E , I N D , E M 2 3 A ) Ir^D = 3 C ' \ L L F 4 1 N T S M P ( L 5 , L ? , E M 2 3 8 X , E , I N D , E M 2 3 B ) I.^D = 3 C A L L F 4 1 N T S M P ( H 0 , L 3 , E M 3 3 A X , E , I N D , E M 3 3 A ) I/.D = 3 C A L L F 4 1 N T S M P ( L 3 , L 2 , E M 3 3 B X , E , I N 0 , E M 3 3B) E i ' . 1 1 A = ( L 1 * * 2 / G ) * ( M F * L 1 * d . 3 - 4 . / P l ) - Z * L 1 * * 2 * ( . 7 5 - 4 . / P I + ? . / P l S Q R ) ) E r ' 1 1 B = ( W ^ / ( f i * L 4 ) ) * ( L l * L l * L 2 + ü . 5 * P I * L 1 * L 2 * L 2 1 + . 0 r f 3 i 3 3 3 * P I S u R * L 2 * * 3 ) + ( 1 . / G ) * < W 3 / L 3 - W 4 / L 4 ) * ( L l * L 1 * L 3 + . 5 * P I * L 1 * L 3 2 * L 3 + . I ; Ö 3 3 3 3 3 * P I S U R * L 3 * * 3 ) h i ' d , 2 ) = d . / G ) * ( ( W 3 / L 3 ) * ( . 3 * L 1 * L 3 * L 3 + . 1 6 Ó 6 6 7 * P I * L 3 * * 3 ) + ( W 4 / L 4 ) * 1 ( . S * L 1 * L 2 * L 2 + . 1 6 6 6 6 / * P I * L 2 * * 3 - . 5 * L 1 * L 3 * L 3 - . 1 6 6 6 6 7 * P I * L 3 * * 3 ) ) b r ( 2 , 2 ) = W 3 * L 3 * L 3 / ( 3 . * G ) + W 4 * ( L 2 * * 3 - L 3 * * 3 ) / ( 3 . * L 4 * G ) E f'. ( 1 , 1 ) = E M1 1 A + F M1 1 B E K ( 1 , 3 ) = EM13A + EM13B É r ' ( 2 , j ) = EM23A + ECi2iB E K , ( i , 3 ) = ÉMi3A • EM3ii3 EC ( 2 , 1 ) = E M d , 2 )
t V • V V V V F ij H I) V O C U $4 F 4 « 1 0 1 1 = 1 0 . A L L F 4 1 d , 1 ) d , 2 ) d , 3 ) ( 2 , 1 ) ( 3 , 1 ) I S flA V i s e I = 1 b . O 4 9 J bX = J J = I O C o ')9 K. V i s e = K l T E ( ^ U M M A T ( K I T E ( 2 O R M A T ( A L L T l f ' I T E ( 2 A r* p ( 1 , A M P d , A X P d , A e p ( 3 , A M P ( < i , A f. P ( 2 , A N ' . P ( 2 , A M P ( 3 , A M p ( 3 , ( 2 , 2 ) ( 2 , 3 ) ( 5 , 2 ) ( 3 , 3 ) R I T E ( 2 O R M A T ( R I T E ( 2 C R M A T ( K l T E ( 2 iJ R M A T ( K I T E ( 2 K I T E ( 2 U R M A T ( K I T E ( 2 N T S M P ( H 0 , H 1 , v n A X , E , I N D , V l 1 A ) = V 1 1 A - 2 . * ( . 1 2 b * W L 2 B A R * P I S « R + . 0 6 2 b * P I S U R * W 2 * L i + C ) = - . b * W L 2 B A R * P l = - . b * f < * P I = V d , 2 ) = V d , 3 ) ST S U P P O R T S P R I N G S T I F F * S , L B . / F T . S DAMPING AT MAST S U P P O R T ,LB. / F T . / S E C . j = 4 , 4 8 , 4 0 0 0 . *Vi X = 6 0 0 U O , 1 ' 5 0 0 0 0 , 9 0 0 0 0 K , 3 4 ) F , C V I S ( / / , 2 X , ' f H F = , F r t . 0 , 4 X , 8 H C V l S C = , F 8 . ( , ) ) / , 4 d ) 2 X , 2 9 H * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ) M E ( T I M ) , b 5 ) T I M , 1 N n 1 ) = . U 4 * ( E ^ ( 1 , 1 ) ) * S Q R T ( ( V ( 1 , 1 ) ) / ( E M d , 1 ) ) ) 2 ) = 0 . 3 ) = ' ) . 1 ) = 0 . 1 ) = o . 2 ) = C V I S C * H * H 3 ) = C V I S C * H * L 2 * d . - C 0 S ( P I * H / ( 2 . * L 2 ) ) ) 2 ) = ! ) A M P ( 2 , 3 ) 3 ) = C V I S C * L 2 * L 2 * d . - C O S ( P I * H / ( 2 . * L 2 ) ) ) = F * H * h - WL2BAR = F * H * L 2 * ( 1 . - C 0 S ( P I * H / ( 2 . * L 2 ) ) ) - B = V ( 2 , 3 ) = O + F * L 2 * L 2 * d . - C 0 S ( P I * H / ( 2 . * L 2 ) ) ) * * 2 - 2 . * A .(<) II ,bX,1 7HN'ATRIX EM F O L L O W S ) ,V) ((Ff- ( I , J ) ,J = 1,3) 1=1 ,3) / , 3 X , b 1 2 . 6 , ' ) X , E l 2 . 6 , 5 X , E 1 2 . 6 ) ,10) //,')X,1'^HMATRIX DAMP F O L L O W S ) , 9 ) ( ( D A M P ( 1 , J ) , J = 1 , 3 ) I = 1 , 3 ) ,11 ) //,'>X,2«HMATRIX V ( S T I F F N E S S ) F O L L O W S ) , 9 ) ( ( V ( I , J ) , J = 1 , 3 ) I = 1 , 3 ) TUT IS A S U B T O T A L TO I N T E G R A T E S P E C T . D E N S E . OF W I N D T Ü T 1 = 0 . T 0 T 2 = O . T C T i = ( . . T ( J T T O P = 0 . WKI T E ( ^ , ' ^ 2 ) V ? F U R M A T ( / / , 4 X , 1 H W , . ^ X , 3 H H 1 1 , 8 X , 3 H H 1 2 , e X , 3 H H 1 3 , 8 X , 3 H H 2 2 , 8 X , 3 H H 2 3 , 8 X , 1 3 r t t t i 3 , f e X , 3 H S U 1 , 8 X , 3 H S i J 2 , 8 X , 3 H S t | 3 , 8 X , 4 H S T 0 P ) OU 9 9 1 = 3 , 2 0 0 0 , b S t X = I
E*J11= F >J 1 2 = E »J 1 3 = -b 0 2 1 = • K L < J 2 2 = A F. U t: 2 = FU2 2 = R t (J 2 3 = A t « 2 3 = E u 2 i = E < J 3 2 = £ 9 3 1 = R fc O 3 3 = A t 9 3 3 = E 9 3 3 = DELTA = 1E'^13*(F H11 =(F M12 =(F hl 3 h2l H22 H23 H32 H31 H3 3 Shi 1 Sh12 Srt13 S H 2 2 S H 2 3 S M 3 3 S M 21 S r l 3 1 S H 3 2 ^ H 11 = ( L1 HT2 = ( H ï 3 = ( S H T1 = C C S r i T 2 = CC S rt T 3 = C C SU= 2 . * 1 PuUj) R i i C O = ( R S ' j 1 = R H C 1 + C A f i S ( M l ' ^ l . * C A d 3 * < ^ . * C A K S I J 2 = R H C 1 + t A B S ( H 2 * c: . * C A P 3 + 2 . * C A F S J 5 = K H C 1 + C A B S ( H è.'^<L. * C A f i 3 + / . . * C A li S T P = R r i C = ( r : = H1 = ( F = ( t = H2 = H1 = ( F = C = C = C = C = c = C = S = s = s • I J * * w * * w * W12 ' J * D ' P L w*W I.J * O l-.PL U2 3 9 1 3 U*W * D A l ' P L 9 1 1 2 1 * 1 3 * 1 2 * E II ( 1 , 1 ) + V ( 1 , 1 ) E M d , 2 ) + V ( 1 , 2 ) E M d , 3 ) + V ( 1 , 3 ) *^.\'(.l,^)* \IU.,1) A l ' i p ( 2 , 2 ) X ( H H Q 2 2 , A E 9 2 2 ) * E M ( 2 , 3 ) + V ( 2 » 3 ) A M P ( 2 , 3 ) X ( R E Q 2 3 , A E 9 2 3 ) * F f M 3 , 3 ) + V ( 3 , 3 ) M P ( 5 , 3 ) X ( R F: Q 3 3 , A E O 3 3 ) * ( E . i 2 2 * F 9 3 3 E « 2 3 * E 9 2 3 ) E ( J l 2 * ( E ü 2 1 * E C ) 3 3 -E 9 3 2 - F 9 2 2 * -E 0 3 1 ) E 9 i 5 - t « 2 3 * E Q 2 3 ) / D E L T A E ' J i 2 - t 9 1 2 * E ü 3 3 ) / 0 E L T A E 9 2 3 - F 9 1 3 * E 0 2 2 ) / D E L T A E Q 2 3 * E Q 3 1 ) + 1 1 * E ' V . i , ' i - F ' W 1 3 * E 9 1 3 ) / D É L T A 1 3 * E ' J 2 1 - I 9 1 1 * E 9 2 3 ) / D E L T A 1 1 NJ ro t . j 1 2 1 3 2 3 + 1 1 JG JG JG K* * F ' i 2 2 - h Q 1 2 * E Q 1 2 ) / D E L T A G ( H11 ) G ( ri 1 2 ) G ( ri 1 5 ) G ( H 2 2 ) G ( h 2 3 ) G ( H 3 5 ) .')*PI*L2)*H11 + L2*H21 +L2*H31 + . S * P I * L 2 ) * H 1 2 + L2*H22 + L2*H23 + . 3 * P I * L 2 ) * H 3 1 + L 2 * H 3 2 + L 2 * H 3 3 (HTl ) ( H T 2 ) ( H T 3 ) L P R I M E * V B A R 1 0 / ( P I * ( 2 . + < W * L P R I M E / ( 2 . * P I * V B A R 1 0 ) ) * * 2 ) * * H O * C 0 ) * * 2 D * S 0 * ( C A B S ( H 1 1 * S H 1 1 ) * R F 1 * * 2 + C A B S ( H 1 2 * S H 1 2 ) * R F 2 * * 2 1 'ü * S H1 3 ) * R F 5 * * 2 S ( H 1 1 * S H 1 2 ) * R F 1 * R F 2 + 2 . * C A B S ( H 1 1 * S H 1 3 ) * R F 1 * R F 3 S ( H l 2 * S h 1 3 ) * R F 2 * R F 3 ) D * S U * ( C A B S ( H 2 1 * S H 2 1 ) * R F 1 * * 2 + C A B S ( H 2 2 * S H 2 2 ) * R F 2 * * 2 2 i * S r i 2 3 ) * R F 5 * * 2 S ( M 2 1 * S H 2 2 ) * R F 1 * R F 2 + 2 . * C A B S ( H 2 1 * S H 2 3 ) * R F 1 * R F 3 S ( H 2 2 * S H 2 3 ) * H F 2 * R F 3 ) 0 * S O * ( C A B S ( H 3 1 * S H 3 1 ) * R F 1 * * 2 + C A B S ( H 3 2 * S H 3 2 ) * R F 2 * * 2 3 5 * S r t 3 3 ) * R F 3 * * 2 S ( h 3 1 * S H 3 2 ) * R F 1 * R F 2 + 2 . * C A B S ( H 3 l * S H 3 3 ) * R F 1 * R F 3 S(rt3 2 * S H 3 3 ) * K F 2 * R F 3 ) 0 * S U * ( C A B S ( H T 1 * S H T 1 ) * R F 1 * * 2 * C A B S ( H T 2 * S H T 2 ) * R F 2 * * 2
R H l 1 = C A H S ( H 1 1 ) R r l 1 2 = C A B S ( H l 2 ) Rri13 = C A 9 S ( H 1 3 ) R h 2 2 =CAtiS(rt22) Rri2 3 = C A « S ( H 2 3 ) R H 3 3 = C A a S ( H 3 3 ) W K l T E ( 2 , 9 i ) w , R H 1 1 , R H 1 2 , R H 1 3 , R H 2 2 , R H 2 3 , R H 3 3 , S 9 l , S Q 2 , S Q 3 , S T P V 3 F ü R H A T ( 2 X , F 4 . 1 , 2 X , E 9 . 3 , 2 X , E 9 . 3 , 2 X , E 9 . 3 , 2 X , E 9 . 3 , 2 X , E 9 . 3 , 2 X , E 9 . 3 f 1 2 x , t 9 . 3 , 2 X , E 9 . 3 , 2 X , E 9 . 3 , 2 X , E 9 . 3 ) Mbl A R F i ' . S . K E S P . N Ü N * D I M * S E D S U B T O T A L S W I L L G I V E M . S . R E S P O N S E <;o T T T T C M I Z Z I I w F 13 R R R W F 14 •5^ C 'V4 ivS ^9 0 T 1 0 T 2 0 T .5 U T T O u N T I . S . M S 1 = ;',S2 = ,' S 3 = r' S T 0 K I T E Ü R M A A ,/JH Ki S 1 = I'; S O T i'. S T 0 R I T E U R M A U H F E O N T I O N T I T O P NT I M S P = ML RF IS .0 .0 . ( J p (2 T( MS S(. = L P (<: T( ET NO NU TO T(i TO T E S P S .3* 5 * 3 * ,9 // TO RT 1 * = S ,'^ /, , E E Tl + S O I T 2 + S U 2 T 3 + S O i O T T O P + S T P O N S E I M P L T 0 T 1 T 0 T 2 T 0 T 3 0 5 * T 4 ) Z M //,*> P = ( Z M S R M S 1 Q R T ( 3 ) R M 5 X , 3 A N D MY / / L L O G R A M R U L E Y AN A L I A S F O R M S I OTTOP S I , Z I 1 S 2 , Z M S 3 , Z M S T Ü P X,6HMS1 = ,E9.3,3X,6HMS2 ,E9.3) 1) ,E9.3,3X,6HMS3 ,E9.3# Z M S T 0 P ) S D T , R M S T O P I H R . M . S . D E F L E C T I O N A T T O W E R T O P = , F 7 . 3 , R . M . S . D E F L E C T I O N A T M A S T T O P = ,F 7 . 3 , 1 2 H F E E T ( T O T A L ) ) h
artlculotlon Q,L| (I-COS n a [l-COS n a i j noil tongtnt to i tongtnt to towtr^ ' / / m o l t
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FIGURE 1: STRUCTURE AND MODES OF DEFLECTION
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^ (ft 2 j»c/rod) 1 "20 "lO L" 1 1 1. J 1 - j — r r H STRUCTURE I F = 3 0 0 0 0 0 Ib/ft CviSC " 60,0OO Ib/
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e 12 16 2 0 24 2 8 32 36 4 0 STIFFNESS OF RESTRAINING SPRING (lOO.OOO |bf/f t )
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