ubrary copy - nasa

DOE/NASA/1900-1NASA CR-165589

l.

NASA-CR-16558919820022833

Vibration Analysis ofThree Guyed Tower Designs forIntermediate Size Wind Turbines

Robert J. ChristieW. L. Tanksley & Associates, Inc.

UBRARY COpyMarch 1982

. ,

~

LANGLEY RESEARCH CENTEftLi8RARY, NASA

HAMProN. VIRGINIA

Prepared forNATIONAL AERONAUTICS AND SPACE ADMINISTRATIONLewis Research CenterUnder Contract NAS3-21900

forU.S. DEPARTMENT OF ENERGYConservation and Renewable EnergyDivision of Wind Energy Systems

NOTICE

This report was prepared to document work sponsored by the United StatesGovernment Neither the United States nor ItS agent. the United States Department ofEnergy. nor any Federal employees. nor any of their contractors. subcontractors or theiremployees. makes any warranty. express or implied. or assumes any legal liability orresponsibility for the accuracy. completeness, or usefulness of any information.apparatus. product or prOCe!lS disclosed. or represents that its use would not Infringeprivately owned rights

•

,

Vibration Analysis ofThree Guyed Tower Designs forIntermediate Size Wind Turbines

Robert J. ChristieW. L. Tanksley & Associates, Inc.Brook Park, Ohio 44142

March 1982

Prepared forNational Aeronautics and Space AdministrationLewis Research CenterCleveland, Ohio 44135Under Contract NAS3-21900

forU.S. DEPARTMENT OF ENERGYConservation and Renewable EnergyWashington, D.C. 20545Under Interagency Agreement DE-AI01-76ET20320

DOE/NASA/1900-1NASA CR-1 65589

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TABLE OF CONTENTS

VIBRATION ANALYSIS OF THREE GUYED TOWER DESIGNSFOR INTERMEDIATE SIZE WIND TURBINES

SUMMARY OF VIBRATION ANALYSIS

INTRODUCTION

DESIGNS ANALYZED

Candidate Guyed Tower DesignsCase Studies . . . . . • . . .

ANALYTICAL PROCEDURE FOR VIBRATION ANALYSIS .

Inertia and Stiffness Coefficients of Wind TurbineEquations of Motion for Wind Turbine Systems .Natural Frequencies of Vibration . . . .Computer Program for Vibration Analysis

RESULTS OF VIBRATION ANALYSIS

DISCUSSION OF RESULTS . . . .

CONCLUSIONS OF VIBRATION ANALYSIS

TABLES

FIGURES

APPENDICES

PAGE

1

3

4

47

9

Systems 9131516

IB

19

22

23

29

. . 50

A FORTRAN COMPUTER PROGRAM, FREE VIBRATION APPROXIMATIONSFOR HORIZONTAL AXIS WIND TURBINES: NATURAL FREQUENCIESOF WIND TURBINE SYSTEMS . . . . . . . . . . . . . . . . . . 50

B COMPUTER PLOTS OF VIBRATION ANALYSIS, CONCEPTS I, II, ANDIII, CASES A THROUGH Q . 57

C SYMBOLS. . . . . . . . .LOB

iii

SUMMARY OF VIBRATION ANALYSIS

The objective of this study was to perform a vibration analysis of threeguyed tower designs for intermediate size wind turbines. These tower designsconsist of tubular steel sections totaling 28.6 m (94 ft) in length, mountedon a soft pad foundation and supported by guys. Each wind turbine design hasa 38.1 m (125 ft) diameter rotor and a rated power of 200 to 400 kw with arotor axis at an elevation of 30.5 m (100 ft) above ground.

This vibration analysis was conducted because there was some concern thatthese guyed towers might have natural frequencies of vibration close to theblade passing frequency of the rotor or some multiple thereof. This concernarose because of several facts.

The first is the knowledge that cantilever towers of propeller type windturbines will vibrate violently if excited by the rotor. Secondly, guy cablesare usually flexible, even with high pretension loads in them. As a result,there was concern that the tower frequencies might be too close to the rotorspeed, Finally it is known that Darrieus type wind turbines, whose centralcolumn is stabilized by guys attached at the top, have experienced seriousvibrations.

Another reason for conducting the vibration analysis was to determine howthe tower natural frequencies were affected by the guy dynamic characteristics,the location of the guy attachment point on the tower and other factors.

The method of analysis used is one developed by R. W. Thresher, et al.in their report titled "Modeling the Response of Wind Turbines to AtmosphericTurbulence", Oregon State University Report No. RLO/2227-8l/2, DC-60, August1981. Their report was prepared for the United States Department of Energy,Division of Solar Technology, Federal Wind Energy Program. Because of thelarge number of analyses to be performed, a computer program was written inFORTRAN which determines the natural frequencies of vibration and plots thesefrequencies as a function of rotor rotational speed. The frequencies ofvibration determined are for the first two modes of bending-pitch and thefirst two modes of bending-yaw. The program determines these frequencies ofvibration using the inertia and stiffness coefficients of the wind turbinesystem which are calculated using the calculator program, Inertia and Stiffness Coefficients and Wind Turbines. This calculator program is includedwith the Oregon State University report.

Two designs in the analysis have two-bladed teetering hub rotors. Sincethe analysis presented by Thresher, et al is limited to three-bladed fixedhub rotors, it was necessary to modify their method. The analysis was modifiedby eliminating the rotational inertia of the rotor about its axis and evaluatingthe natural frequencies of vibration at zero rotor speed. For the three-bladedfixed hub rotor concept the method was not moditied. Furthermore, the methodof determining the spring rate of the guys used to support the tower had to bemodified to account for the spring rate of the grouted anchors that the guysare attached to.

1

In the parametric study several of the system pfoperties were varied todetermine the sensitivity of the tower's natural frequencies of vibration tovariances of these properties. properties varied in9lude the mass of thenacelle and rotor, effective guy stiffness, the height of the guy attachmentpoint, the location of the nacelle and rotor center of gravity, the mass momentof inertias of the nacelle and rotor about the tower axes, and the rotor rotational speed. Then, by comparing the results of the different concepts, theeffects of different tower section mass and stiffness properties can be determined and the effects of different rotor designs can be evaluated.

The results showed that only the lowest two frequencies of vibration werein an area where they could be excited by the rotation of the rotor. Theanalysis also indicated that these same two modes could be tuned by varyingthe effective guy stiffness, the height of the guy attachment, the mass of thenacelle and rotor, or the mass and stiffness properties of the tower sections.It was also shown that these same two modes are very close in frequency andthat both change very little with the rotor's rotational speed.

In conclusion, the results of the analysis showed that the naturalfrequencies of vibration of all three tower designs are not in a range wherethey can be excited by the rotor. The effects on the natural frequencies ofvibration caused by changing some of the wind turbine system characteristicswere also shown.

2

INTRODUCTION

NASA is performing ~ conceptual design study to investigate the possibilityof reducing the cost of intermediate size wind turbines. These wind turbinesare 30.5 m (100 ft) high and have a rated power between 200 and 400 kw. Threeconcepts of tower designs for wind turbines are being considered. Each of thesetower concepts are made of tubular steel sections mounted on a "soft pad" foundation and supported by guys with grouted anchors. The "soft pad" foundationis a concrete footing with a flexible pad interface between the tower base andthe footing. This approximates a ball and socket type of joint and minimizesbending moments transferred to the footing. Two of the concepts have two-bladedteetering hub rotors; the other has a three-bladed fixed hub rotor. All threehave a rotor diameter of 38.1 m (125 ft) with the axis at 30.5 m (100 ft)elevation.

Cantilever towers of propeller type wind turbines are known to vibrate ifthe tower natural frequencies are close to the blade passing frequency or somemultiple thereof. Also, in recent years, some towers of Darrieus type wind turbines which use guys to stabilize the upper end of the tower have experiencedsevere vibrations. These two facts prompted the vibration analysis being reported here, namely, to determine the vibrational characteristics of the threeguyed tower concepts under study as mentioned in the previous paragraph. Thisanalysis should identify areas where the rotor might excite the tower's naturalmodes of vibration. Furthermore, since these tower designs are still in thepreliminary design stage, it is important. to determine the sensitivity of thenatural frequencies of vibration to the various system pxoperties. In this wayproblem areas can be identified and methods of rectification determined.

A method for vibration analysis of similar wind turbine configurationssystems has previously been developed by R. W. Thresher, et al of Oregon StateUniversity. A major part of this analysis is based on their draft reporttitled "Modeling the Response of Wind Turbines to Atmospheric Turbulence",Oregon State University Report No. RLO/2227-81/2, UC-60, August 1981.

In the analysis of the three tower designs, the various tower and systemp!Dperties of the preliminary designs were estimated and the natural frequencies determined. The system properties were then varied individually todetermine their effect on the natural frequencies of vibration. The systemproperties varied include: effective guy stiffness; the height of the guyattachment; the mass of the nacelle and rotor; the location of the center ofgravity of the nacelle and rotor; the mass moment of inertia of the nacelleand rotor about the tower's X, Y and Z axes; and the rotor's rotational speed.Due to the large number of analyses to be performed, and to expedite thecalculations, a computer program was written in FORTRAN to calculate and plotthe natural frequencies of vibration. The results of all the above and thecomputer plots are included in this report.

3

DESIGNS ANALYZED

Candidate Guyed Tower Designs

In this report three concepts of tower designs for wind turbines wereanalyzed. The three tower concepts have a hub height of 30.5 m (100 ft). Oneof the unique features of these tower designs is that the towers are not cantilevered but are supported by guy rods. These guy rods radiate out from thetower at three equally spaced positions. The guys are attached to the towerat 10.7 m (35 ft) above the ground and to concrete pads located 9.1 m (30 ft)from the tower's centerline. The concrete pads are anchored by grouted rockor soil anchors which extend below the surface to bedrock or to competent soil.This is shown in Figure 1.

To prevent the guys from becoming slack, the guys are pretensioned to avalue greater than the change in tension experienced under maximum operatingloads. Similarly, the anchor rods are pretensioned to prevent the anchor padsfrom unseating undev all conditions. Thus, the pretension on the anchor rodis set to be greater than the maximum tension on the guy rod.

As mentioned above, the preload prevents the anchor pads from unseating,therefore, there is negligible movement of the anchor pads as loads change.This causes the effective spring rate of the anchors to be relatively highwhen compared to the spring rate of the guys.

For the vibration analysis, the effective stiffness of the guys must bedetermined. The derivation of the effective guy stiffness is presented in theThresher, et al report. To include the spring rate of the grouted anchors andthe soil, the guy system can be modeled as three springs.

4

k ak '

V

Thus, the total effective spring rate of the guys and anchors becomes

where

kT

= Total effective guy and anchor spring rate

kR

Spring rate of a guy rod

k Spring rate of an anchor roda

kV

Vertical spring rate of soil due to compression

N Number of guys per position

e = Angle between horizontal guys

This assumes that the guys and anchors are located at three equally spacedpositions around the tower and that one guy is attached to one anchor.

The preliminary design of these towers specifies three guy rods and threeanchors per position for each concept. Using a guy rod spring rate of29.55E + 06 N/m (2.025E + 06 Ib/ft) and a combined anchor and soil spring rateof 218.9E + 06 N/m (15.0E + 06 lb/ft), the total effective guy spring ratebecomes 44.36E + 06 N/m (3.045E + 06 lb/ft).

Each tower is made of several tubular steel sections; Concept I is madeof three sections, Concept II has seven, and Concept III has eight. Each ofthese sections has a different wall thickness and, therefore, each have different inertia and stiffness properties. The lengths and wall thicknesses ofthese sections are listed in Table I and the inertia and stiffness values arelisted in Table II. The sections are numbered vertically starting at thetower base. Additional sections were added to account for the extra mass andstiffness of the flanges and the segment from the top of the tower sections tothe rotor's centerline.

Located at the top of the tower, as shown in Figures 2 and 3, are the gearbox, generator, pitch change mechanism and controls. These are all mounted ona rotatable platform. The sum of all these parts will be referred to as thenacelle throughout this report. Also located at the top of the tower is therotor, which consists of the rotor hub and blades. Concepts I and II bothhave two-bladed rotors with teetering hubs, whereas Concept III has a threebladed rotor with a fixed hub. Furthermore, Concept I has variable pitchrotor blades, whereas Concepts II and III have fixed pitch blades.

The vibration analysis requires the total mass and location of the centerof gravity for the nacelle and rotor to be calculated. For the various concepts,

5

the total mass and center of gravity locations were estimated as follows:

TOTAL MASS OFNACELLE AND ROTOR,

CENTER OF GRAVITYCONCEPT ~ + mR LOCATION, q

I 16.19E + 03 kg 2.35 m(1110 Ib sec2/ft) (7.7 ft)

II 14.59E + 03 kg 2.16 m(1000 Ib sec2/ft) (7.1 ft)

III 14.18E + 03 kg -91.43E - 03 m(972 lb sec2/ft) (-0.3 ft)

Since a fixed hub rotor blade can be located much closer to the towerthan can a teetering hub rotor blade, the rotor of Concept III is located only1.52 m (5 ft) from the outside of the tower wall while Concepts I and II havetheir rotor located 3.96 m (13 ft) from the tower. This accounts for the largedifference in center of gravity locations listed above. The center of gravitylocation is measured from the tower centerline and a negative value indicatesthat the center of gravity is located downwind of the tower centerline.

The mass moment of inertia of the nacelle and rotor about the tower axeswere also calculated, including the mass moment of inertia of the rotor aboutits spin axis. Concepts I and II have teetering hubs; this allows the rotorto remain parallel to its plane of rotation during small angular changes inpitch and yaw of the nacelle. Thus the rotational inertia about the X and Z

axes is greatly reduced on Concepts I and II.

The theory that this analysis is based on does not account for theflexibility of the baldes, therefore, it is most applicable to small systemswhere the mass of the blades is a small percentage of the total mass. In thisanalysis the mass of the blades was about 10\ of the total mass for ConceptsI and II, and 13% for Concept III. Also, approximately half the mass of theblade was located within the first third of its span. Thus, this simplification is justified. Furthermore, this theory is based on a three-bladed fixedhub rotor, therefore, care should be taken in interpreting the results forConcepts I and II since they have two-bladed teetering hub rotors.

The total mass moments of inertia were calculated for each concept andare as follows:

6

CONCEPT

I

I rkg rn2

(lb ft sec2)

113.8E + 03(83.9E + 03)

MASS MOMENTS OF INERTIA

I xxkg m2

(lb it sec2)

113.5E + 03(83.7E + 03)

l69.8E + 03(125.2E + 03)

CONCEPT

II

III

MASS MOMENTS OF INERTIA

I r I xx I zzkg m2 kg m2 kg m2

(lb ft sec2) (lb ft sec2) (lb ft sec2)

156.6E + 03 125.4E + 03 132.3E + 03(115.5E + 03) (92.5E + 03) (97.6E + 03)

164.1E + 03 133. OE + 03 139.9E + 03(121. OE + 03) (98.1E + 03) (103.2E + 03)

I r Rotational mass moment of inertia of the rotor about itsspin axis (Y)

I xx Mass moment of inertia of the nacelle and rotor about thetowel; 's X axis

I zZ Mass moment of inertia of the nacelle and rotor about thetower's Z axis

The calculator program for determining the inertia and stiffnesscoefficients also requires the number of intervals per tower section to beused for integration. The following number of intervals were chosen.

CONCEPT

I

II

III

NUMBER OF INTEGRATIONINTERVALS PER SECTION

10

4

4

NUMBER OFSECTIONS

3

7

8

TOTAL NUMBER OFINTEGRATION INTERVALS

30

28

32

This completes the required input data for determining the inertia andstiffness coefficients for the tower designs.

Case Studies

To determine the effects of varying the properties of,the tower, aparametric study was performed. In this study each of the tower propertieswere varied, one at a time, to determine their effect on the natural frequencyof vibration of the tower. First, the effective guy stiffness was varied from25% to 200\ of the design value. Next, the height of the guy attachment wasvaried from 2/3 to 4/3 of the design height. Similarly, the mass of thenacelle and rotor was increased and decreased by 1/3 and the center of gravitylocation was also changed. Finally, the mass moments of inertia about theX and Z axes were increased and decreased. The following table lists all ofthe cases examined. Each case was analyzed on each of the three concepts.

7

CASE

A

B

C

D

E

F

G

H

I

J

K

L

M

N

o

P

Q

PROPERTY VARIED

25\ of design effective guy stiffness

50% of design effective guy stiffness



200% of design effective guy stiffness

Height of guy attachment increased by 1/3

Height of guy attachment decreased by 1/3

Nacelle and rotor mass increased by 1/3

Nacelle and rotor mass decreased by 1/3

Distance to center of gravity increased by 1/3

Distance to center of gravity decreased by 1/3

Nacelle and rotor inertia about the tower's Zaxis increased by 1/3

Nacelle and rotor inertia about the tower's Zaxis decreased by 1/3

Nacelle and rotor inertia about the tower's Xaxis increased by 1/3

Nacelle and rotor inertia about the tower's Xaxis decreased by 1/3

Both of the nacelle and rotor inertias about theX and Z axes increased simultaneously by 1/3

Both of the nacelle and rotor inertias about theX and Z axes decreased simultaneously by 1/3

Case C is considered the norm in this analysis.

The rotor has a design operating rotational speed of 4.19 rad/sec (40 rpm).Since the natural frequency of vibration of a wind turbine system can vary withrotor's rotational speed, each concept was analyzed from 0 to 6.3 rad/sec (60rpm) at ten equally spaced points. Since Concept I and II have teetering hubs,the natural frequencies of vibration will not change with the rotational speedof the rotor. Thus, the frequencies of vibration should be evaluated at zerorotor speed. On the other hand, Concept III has a fixed hub which causes thenatural frequencies of vibration to change with rotor speed. Therefore,Concept III should be evaluated at operating speed.

8

ANALYTICAL PROCEDURE FOR VIBRATION ANALYSIS

Inertia and Stiffness Coefficients of Wind Turbine Systems

To determine the natural frequencies of vibration of wind turbine systems,the inertia and stiffness coefficients of the tower must be calculated. Thedetermination of the inertia and stiffness coefficients is explained in detailin the Thresher, et al report.

For displacement of the top of the tower in the y direction, the stiffnesscoefficient is (see Figures 4 and 5)

Lf EI(z) ('¥~(Z))2dZ + kc'¥~(a)o

where

V Translational displacement in y direction

L Length of tower

EI(z) Tower bending stiffness

'JIV

Displacement function

k Effective spring constant of guysc

a Distance from ground to guy connections

'JII, 'JI" Represent the first and second derivatives with respectto Z

The stiffness coefficient for displacement due to pitching about the X axis is

where

kXX

Lf EI (z)

o('¥" (z) ) 2 dz + k ,¥2 (a)

X c X

X Rotational displacement about x axis at the upper end ofthe tower

For rotational displacement about the tower's Z axis the stiffness coefficientis

Lf GJ(z)

o2

('JI ~ (z)) dz

9

where

GJ( z) Tower torsional stiffness

Rotational displacement about Z axis at the upper endof the tower

For displacement due to translation in the y direction (V) is influencedby the rotational displacement about the x axis (X), and vice versa. Thestiffness coefficient relating these two displacements is

L

kxv ~ EI(z) ~;(z)~~ (z) dz + k c ~v(a)~x(a)

Finally, the stiffness coefficient for translational displacement along thex axis is

k uu kW

- k2

/kVx XX

where

U = Translational displacement in the x direction

The displacement functions (~) are interpolating functions which relatethe displacements within the tower to the displacements at the top of thetower. For a tower with a ball joint at its base, these displacement functionsare

~v(z)1 22' (z/L) (3 - z/L) )

~ ( z) = z/2 (1 - (Z/L)2)X

~ <jJ (z) = z/L

~"( z) 3V

-3z/L

~"( z) 2-3z/L

X

~¢(z) = l/L

The equations for a tower with a ball joint at its base were chosenbecause the tower designs utilize bases mounted on flexible pads which reducebending moments transmitted to the foundation. Similar equations for acantilever based tower can be found in the Thresher, et al report.

10

The inertia coefficients of the tower are expressed in a similar fashion.They are

L

Mvv ; m(z) ~V2 (z) dz + (mR + ~)

where

m(z) Mass per unit length of the tower

Mass of nacelle and rotor

and

where

MXX

Ixx

LI m(z) ~ ( z) ~ (z) dz0

V X

L'¥ 2I m( z) (z) dz + I

0 X xx

L'¥ 2II (z) (z) dz + Im <P zz

0

Mass moment of inertia of nacelle and rotor about thex axis

also

where

Izz

I (z)m

Mass moment of inertia of nacelle and rotor about thez axis

Mass moment of inertia per unit length of tower aboutthe tower centerline

q The distance from the tower centerline to the center of gravityof the nacelle and rotor

Finally

MW

- 2(k /k ) M + (kv /k )2 (MVx XXVX X XX XX

I )xx

11

The displacement functions are the same as those used in determiningthe stiffness coefficients. These coordinate functions are used in a finiteelement technique and are explained in the Thresher, et al report.

The stiffness and inertia coefficients as outlined above can be calculatedusing the calculator program Inertia and Stiffness Coefficients of Wind Turbineswhich is included in the Thresher, et al report. The required input datainclude (see Figure 4) :

1) The type of tower base.

2) The height of the tower (L) •

3) The effective spring constant of the guys (k ) •c

4) The height of the guy connection (a) .

5) The length of the tower sections (L') •

6) The bending stiffness of the tower section (EI(z».

7) The torsional stiffness of the tower section (GJ(z» .

8) The mass per unit length of the tower section (m(z».

9) The mass moment of inertia per unit length about the towercenterline (I (z».

m

10) The number of intervals for integration (N).

11) The mass moment of inertia of the nacelle and rotor aboutthe x and z axes (I and I ).

xx zz

12) The mass of the nacelle and rotor (~ + mR

) .

13) The distance from the tower centerline to the center ofgravity of the nacelle and rotor (q).

The output from the above program is

and

12

Equations of Motion for Wind Turbine Systems

Once the above mass and stiffness coefficients are calculated they can beentered into the equations of motion for wind turbine systems.

The equations of motion for a wind turbine system, excluding aerodynamicforces, are

·~u 0 MU<p 0 U 0 0 0 U

·0 l\;v 0 MVX

V 0 a a V

+ ·~<p a M<p<p 0 <p 0 0 Sir <p

r

0~X

0 M X a -Sir 0 xXX r

k a 0 0 u 0uu

0 kw 0 kvX

V 0

+ =0 0 k<p<p 0 <p 0

0 kvX

0 k X 0xx

To obtain the natural frequencies of vibration, these equations of motionhave the solution

u(t) Uo

vet) Voiwt

e<p (t) <PO

x(t) Xo

13

Substituting this into the equations of motion yields

(kUU

2 2- M w) a (-MU</JW ) a

UU

2 a 20 (kvv - ~w ) (ku - M w)

X Ux

2 2(iI Itw)(-MU</JW ) a (k<P¢ - M</J</JW ) r

(kvx

2 (-iI Itw) (k w2

)0 - ~Xw ) - Mr XX XX

Uo 0

Vo 0

</Jo a

Xo 0

where

i = I=l

W Natural frequency of vibration

I = Mass moment of inertia of the rotor about its spin axisr

It Rotor's rotational speed

This requires that the determinate of the square matrix be equal to zero.Expanding the determinate yields

14

If we let

2 2 2 2W + (MuuM¢¢ - ~¢) (w )

The expanded determinate becomes

Natural Frequencies of Vibration

At zero rotor speed (Q = 0), the above becomes

o

thus the roots of G¢u and HXV

will give the natural frequencies of vibrationat zero speed.

The equation for Duv can be factored to yield

Rearranging the determinate2

H V(w )X

2 2w

At infinite rotor speed this becomes

The roots of these will give the natural frequency of vibration at infiniterotor speed.

The calculator program, Natural Frequencies of Wind Turbine Systems,included in the Thresher, et al report was written to determine the frequencies

15

of vibration at zero and infinite rotor speed. It can also be used tocalculate the rotor speed at which a given frequency of vibration will occur.

In this analysis it was desirable to determine the natural frequenciesof vibration as a function of rotor speed. Therefore, a computer program waswritten in FORTRAN that extracts the roots in w2 of the expanded determinate.

computer Program for Vibration Analysis

The FORTRAN program uses the inertia and stiffness coefficients calculatedon the calculator program, Inertia and Stiffness Coefficients of Wind Turbines,and determines the coefficients of (w 2)O, (w2)1, (w 2)2, (w 2)3 and (w2)4. Inthe program

AG kuJ<¢<j>

BG - (kutr¢<p + k¢<pMUU)

2CG MUUM¢¢ MU¢

k kvX

2AH k

XX W

BH -(k ~ +~ - 2kv ~ )XX XX X X

2CH

~Xx t-VX

AD 0

BO ~2I 2 k kr UU W

co _~2I 2 (k uMw + k~ )r U UU

DO ~2I 2 M Mwr UU

Substituting these into the expanded determinate and collecting terms yields

(AG) (AH) + [(AG) (BH) + (BG) (AH) - (BO) lw2 + [(AG) (CH) + (BG) (BH) +

(CG) (AH) - (CD)] (w2) 2 + [(BG) (CH) + (CG) (BH) _ (DO) 1(w

2) 3 +

(CG) (CH) (w2

) 4 = °

16

After summing like coeffici.ents, the subroutine GNEWTN is called. Thisroutine determines the roots of a polynomial using a modified Newton methodand is capable of locating complex roots. This routine was supplied by NASA Lewis Research Center and requires the subroutine SYND1V for synthetic division.The program is also capable of plotting the natural frequencies of vibration asa function of rotor speed and uses plotting routines also supplied by NASA Lewis Research Center.

The computer program requires the following input data:

1)

2)

3)

4)

5)

6)

7)

8)

9)

Number of data sets (J).

Number of roots to be plotted (JPO).

Ti tle.

Mass moment of inertia of the rotor about its spin axis (I ).r

The lowest rotor speed at which roots are to be found (R . ).m1.n

The highest rotor speed at which roots are to be found (R ) .max

The number of points between the lowest and highest rotorspeeds at which roots are to be found (NS).

MUU ' MU~' ~, MVx' M~~, and MXX

kuu ' kvv ' kvx ' k~~, and k xx

The output includes:

1)

2)

3)

4)

The natural frequencies of vibration at zero rotor speed(i.e., the roots of G~U and H

XV).

The natural frequencies of vibration at infinite rotor speed(i.e., the roots of D

UV).

The natural frequencies of vibration at R . I R I and thef . d b h'm1.n ma~ .number 0 p01.nts requeste etween t e m1.n1.mum and max1.mum

rotor speed.

A plot of the natural frequencies of vibration versus rotorspin rate.

A flow chart of this program is included in Figure 6 and a listing can befound in Appendix A.

17

RESULTS OF VIBRATION ANALYSIS

The appropriate tower properties for each case and concept were enteredinto the calculator program, Inertia and Stiffness Coefficients of Wind Turbines, presented in the Thresher, et al report. The output of the calculatorprogram provides the inertia and stiffness coefficients of the wind turbinesystem. The inertia and stiffness coefficients were entered into the FORTRANprogram, Natural Frequencies of Wind Turbine Systems, which was developed forthis analysis. The output of the computer program provides natural frequenciesof vibration as a function of rotor rotational speed. Listed in Tables III andIV are the first four modes of vibration for Concepts I and II evaluated atzero rotor speed. Table V lists the same modes of vibration for Concept IIIexcept they are evaluated at a rotor operating speed of 4.19 rad/sec (40 rpm).The results of the different cases are plotted and can be found in Appendix B.These four modes of vibration represent the first and second modes of vibrationin bending coupled with yaw and the first and second mode of vibration in bending coupled with pitch. In these tables the first bending-pitch mode is labeledwI (Mode I), the first bending-yaw mode is labeled w2 (Mode II), the secondbending-yaw mode is labeled w3 (Mode III), and the second bending-pitch mode islabeled w4 (Mode IV).

Cases A through E for each concept are plotted in Figures 7 and 8. Theseshow the effect of varying the effective guy stiffness on the frequencies of thefour modes of vibration. Lines labeled lP, 2P and 3P have also been plotted onFigure 7 and represent multiples of the rotor's rotational speed. Figures 9and 10 show the results of Cases F and G and indicate the effect of changingthe height of the guy attachment point. The effect of varying the mass of thenacelle and rotor, Cases Hand L, is shown in Figures 11 and 12 and the effectof moving the nacelle and rotor center of gravity is plotted in Figure 13.Finally, Cases L through Q, which show the effect of changing the nacelle androtor inertia about the X and Z axes, are plotted in Figures 14, 15 and 16. Onall of the above plots WI is the first bending-pitch mode, w2 is the firstbending-yaw mode, w3 is the second bending-yaw mode, and w4 is the secondbending-pitch mode. In all cases, WI and w2 were very close and are plottedas one line.

The computer plot of the natural frequencies of vibration versus rotorrotational speed for the design case, Case C, can be found in Figures 17, 18and 19. The computer plots of the other cases can be found in Appendix B.

The natural frequencies of vibration of w2 at infinite rotor rotationalspeed are listed in Table VI. Since Concepts I and II have teetering hubswhich eliminate the gyroscopic effects encountered with a fixed hub, theseresults are not applicable. Even though these results are apparently not ofany significance now, their importance will be pointed out later.

The mode shapes of vibration for a typical wind turbine system can beseen in Figures 20 and 21. These figures are excerpts from the Thresher, et alreport and represent a non-guyed wind turbine tower.

18

DISCUSSION OF RESULTS

The results of the vibration analysis were analyzed to determine theimportant features and to determine the sensitivity of the wind turbine systemto the variance of individual parameters. Note that in this analysis, in mostcases, only one parameter at a time was varied, but in actuality when one parameter is changed so are several others. For example, if the mass of the rotorchanges, the center of gravity of the nacelle and rotor would move and the massmoment of inertia about the tower's X and Z axes would also be affected. Therefore, several properties of the wind turbine system would change instead ofjust one.

While analyzing the results of Cases A through E, it was determined thatall frequencies of vibration are affected by changing the effective guy stiffness (see Figures 7 and 8). This was expected because changing the effectiveguy stiffness changes the stiffness of the tower. As expected, as the effective guy stiffness decreased, so did the natural frequencies of vibration andas effective guy stiffness increased, so did the frequencies. Comparing Figures 7 and 8, we find that the lower two modes of vibration are affected themost.

Points of concern are where a multiple of a rotor's rotational speedcoincides with a natural frequency of vibration. At these points the rotorwould excite the mode of vibration and cause the tower to oscillate near resonance. Since concepts I and II have two-bladed rotors, the first and secondmultiple (IP and 2P) of the rotor's rotational speed are of importance. Sinceconcept III has a three-bladed rotor, the first and third multiples (lP and3P) are of interest. When the lowest modes of vibration occur at frequenciesabove the lower rotor orders, the tower is defined as being "stiff". If theselower modes of vibration occurred at a frequency below the first rotor order,the tower would be considered "soft". In this analysis, the lowest two modesof vibration, in most cases, occurred at frequencies well above the first tworotor orders for the two-bladed concepts and the first three rotor orders forthe three-bladed concept. Therefore, all three tower concepts in this analysisproved to be "stiff".

In analyzing the results of Cases A through E (see Appendix B), it wasfound that the effective guy stiffness had to be decreased considerably beforeany natural frequency of vibration coincided with a lower rotor order. Forconcepts I and II the effective guy stiffness had to be decreased to 25% ofthe design value before the lowest natural frequency of vibration would beexcited by the second rotor order. For Concept III the effective guy stiffnesshad to be decreased to 50% of the design value before the lowest natural frequency could coincide with the third rotor order. This indicates the towerdesigns are very stiff and that the effective guy stiffness could vary significantly before reaching resonance. The high stiffness can be greatly attributedto the use of prestressed anchors and anchor pads. Note that the two secondmodes of vibration, w3 and w4' occur at relatively high frequencies and,therefore, should not be of concern.

19

The results of Cases F and G show the effect of changing the height of theguy attaching point. Figures 9 and 10 show that as the height of the attachingpoint is raised, the natural frequencies of vibration increase. This impliesthat raising the guy attaching point increases the stiffness of the system andthus increases the frequency of vibration. On the other hand, if the height ofthe guy attaching point is lowered, the stiffness of the system decreases aswell as the natural frequencies of vibration. In both of these cases, thefirst two modes of vibration were affected the most. The first two naturalfrequencies of vibration of Concept III became excited by the third rotor order(3P) when the height of the guy attachment was lowered to 2/3 of the designvalue. The first two natural frequencies of vibration for Concepts I and IIdid not coincide with the first two rotor orders in either case.

In Cases H and I, the mass of the nacelle and rotor was varied. As themass was increased the frequency of vibration of the first two modes decreasedand as the mass was decreased, frequency increased. Figure 11 shows that,within the range of the analysis, none of the concepts have their first twomodes excited by the rotor orders of concern. Figure 12 shows that the secondmode of bending-yaw (w3) is greatly affected by the mass of the nacelle androtor but the second mode of bending-pitch (w4) is not.

Changing the distance from the tower centerline to the location of thenacelle and rotor center of gravity had little effect on Modes I, II and IV,but Mode III was greatly affected (see Figure 13). This plot shows that asthe distance to the center of gravity is increased, the frequency of vibrationof the second mode of bending-yaw also increases. The opposite happens whenthe distance is decreased.

The plots of Cases L through Q show the effect of changing the inertiaof the nacelle and rotor about the tower's X and Z axes. As the inertia aboutthe Z axis was varied, the second mode of bending-yaw (w2) was affected themost, but as the inertia about the X axis was varied, the second mode ofbending-pitch (w4) was influenced. These effects can be seen in Figures 14through 16. The lower modes showed little change and, therefore, were notplotted.

This analysis has shown that only the natural frequencies of the firstmodes of vibration are in a region of concern, i.e., in a region where theymight be excited by the rotor. These first modes of vibration, wl and w2'were shown to be most greatly affected by changing the effective guy stiffness,the height of the guy attachment point, the tower mass and stiffness properties,and the mass of the nacelle and rotor.

Once the basic wind turbine system is designed, the designer cannotpractically change the mass and stiffness of the tower nor the mass of thenacelle and rotor, but the effective guy stiffness can be varied without agreat deal of redesigning. The effective guy stiffness can be varied bychanging the size and/or the number of guy rods. For these tower designs,the minimum number and size of the guys were based on the cyclic loading undernormal operating conditions and the static loads experienced during hurricaneconditions. Thus, the practical design variables become, increasing the sizeor the number of guys. The effect of the guys can also be changed by varying

20

, .

the height of the guy attachment point, but this would require significantredesigning of the tower sections.

Finally, the results in Table VI, which lists the natural frequencies ofvibration at infinite rotor rotational speed, show that w2 changes very littlewith rotor speed, i.e., w2 at zero rotor speed is approximately the same frequency at infinite rotor speed. It was also shown earlier that at or near zerorotor speed, wI and w2 were also approximately of the same frequency. Therefore, the frequency of wI and w2 at zero rotor speed is approximately the sameas the frequency of w2 at infinite rotor speed. As discussed in the AnalyticalProcedure, w2 at infinite rotor speed can be easily found by calculating thesquare root of kuU/MUU' Thus, the natural frequencies of vibration of thefirst modes can be easily approximated with one stiffness and one inertiacoefficient.

21

CONCLUSIONS OF VIBRATION ANALYSIS

In conclusion, a vibration analysis was performed on three preliminarytower designs for intermediate size wind turbines. A parametric study was performed to determine the sensitivity of the wind turbine system to variances ininertia and stiffness properties. The following conclusions were made as aresult of the vibration analysis.

1) Under design conditions none of the tower concepts are excitedby the rotor at or below the operating speed of 40 rpm.

2) At present, only the lowest two natural frequencies of vibrationare in frequency range of concern. It was also found in theoperating range, that these first two modes are very clOse infrequency. Furthermore, the first bending-yaw mode frequency(w2) changes very little from zero to infinite rotor speed.Therefore, the frequency of vibration of most concern can befound by evaluating w2 at infinite rotor speed. This is simply:

w =2(k 1M )1/2

UU UU

where kuu and MUU are the stiffness and mass coefficients fortranslation in the x direction.

3) The frequencies of vibration of the lowest two modes aresignificantly affected by the effective guy and anchor stiffness, the height of the guy attachment, and the mass of thenacelle and rotor. These modes are also affected by thetower section properties, i.e., tower sections' mass andstiffness. Thus, the tower may be tuned to avoid excitationof the lower modes by changing the effective anchor and guystiffness, the guy attaching height, the mass of the nacelleand rotor, or the tower sections' mass and stiffness properties.The most practical being the size and the number of the guys.

4) The center of gravity location of the nacelle and rotor hadnegligible effects on the lower modes of vibration. The sameis true for the rotational inertia of the nacelle and rotorabout the tower's X, Y and Z axes.

5) The frequencies of vibration of the upper two modes wereaffected in all cases, but were of so high a frequency thatthey are of little concern. The effective guy and anchorstiffness mostly affected the second bending-pitch mode(Mode IV), whereas the nacelle and rotor mass and the location of the center of gravity affected mostly the secondbending-yaw mode (Mode III). The inertia about the Z axishad the greatest affect on Mode III, and the inertia aboutthe Y axis mostly affected Mode IV.

22

TABLE I

TOWER SECTIOO LENGTHS AND THICKNESSES

SECTION LENGTH WALL THICKNESSTOWER SECTION m rom

CONCEPT NUMBER (ft) ( in)

I 1 7.62 7.94(25.0) ( 5/16)

2 10.67 9.52(35. 0) ( 3/8)

3 10.36 7.94(34.0) (5/16)

II 1 3.66 7.94(12.0) ( 5/16)

2 1.83 9.52(6.0) ( 3/8)

3 3.05 12.70(10. 0) (1/2)

4 5.49 15.88(18.0) (5/8)

5 5.79 12.70(19.0) (1/2)

6 3.66 9.52(12.0) ( 3/8)

7 5.18 7.94(17.0) (5/16)

III 1 3.66 14.29(12.0) (9/16)

2 2.44 11.11(8. 0) (7/16)

3 2.44 14.29(8.0) (9/16)

4 4.57 19.05(15.0) (3/4)

5 4.57 15.88(15.0) (5/8)

6 4.88 12.70(16.0) (1/2)

7 2.74 9.52(9.0) ( 3/8)

8 3.35 7.94(11. 0) (5/16)

23

TABLE II

TOWER SECTION INERTIA AND STIFFNESS PROPERTIES

MASS MOMENTBENDING TORSIONAL MASS PER OF INERTIA PER

STIFFNESS STIFFNESS UNIT LENGTH UNIT LENGTHEI(Z) GJ(Z) M(Z) Im(Z)

TOWER SECTION Nm2 Nm2 kgjm kgm2

CONCEPT NUMBER (lb ft2) (lb ft2) (lb sec2jft2) (lb ft sec2 )

I 1 1. 861E+09 1. 489E+09 279.9 44.27(4.502E+09) (3.602E+09) (5.845) (32.65)

2 2.226E+09 1. 780E+09 335.4 54.23(5.385E+09) (4.308E+09) (7.006) (39.99)

3 1.861E+09 1. 489E+09 279.9 44.27(4.502E+09) (3.602E+09) (5.845) (32.65)

II 1 1. 861E+09 1. 489E+09 280.3 44.29(4.502E+09) (3.602E+09) (5.855) (32.66)

2 2.226E+09 1. 780E+09 336.0 52.99(5. 385E+09) (4.308E+09) (7.018) (39.08)

3 2.948E+09 2. 358E+09 447.1 70.21(7.133E+09) (5.706E+09) (9.337) (51. 78)

4 3.66lE+09 2.929E+09 557.6 87.11(8.857E+09) (7.086E+09) (11. 645) (64.24)

5 2.948E+09 2.358E+09 447.1 70.21(7.133E+09) (5.706E+09) (9.337) (51. 78)

6 2.226E+09 1. 780E+09 336.0 52.99(5. 385E+09) (4. 308E+0 9) (7.018) (39.08)

7 1. 861E+09 1.489E+09 280.3 44.29(4.502E+09) (3.602E+09) (5.855) (32.66)

III 1 1.861E+09 1.489E+09 247.4 39.,09(4.502E+09) (3.602E+09) (5.168) (28.83)

2 2.588E+09 2.070E+09 345.7 54.36(6.262E+09) (5.009E+09) (7.220) (40.09)

3 3.306E+09 2.644E+09 443.5 69.45(7.998E+09) (6.398E+09) (9.262) (51. 22)

4 4.364E+09 3.491E+09 589.4 91.69(10.56E+09) (8.447E+09) (12.309) (67.62)

5 3.661E+09 2. 929E+09 494.2 76.93(8.857E+09) (7.086E+09) (10.280) (56.73)

6 2.948E+09 2.358E+09 396.4 62.21(7.133E+09) (5.706E+09) (8.278) (45.88)

7 2.226E+09 1. 780E+09 296.6 46.78(5.38"5E+09) (4. 308E+09) (6.195) (34.50)

8 1.861E+09 1.489E+09 247.4 39.09(4.502E+09) (3.602E+09) (5.168) (28.83)

24

TABLE III

NATURAL FREQUENCIES OF VIBRATION OF WIND TURBINE TOWER DESIGNS

CONCEPT I AT Q = 0

CASE CONDITION FREQUENCY OF VIBRATION, HZ (CPM)

WI w2

w3

w4

A 0.25 kT

1.26 1.26 9.07 11.35(75.9) (75.9) (544.2) (681. 0)

B 0.50 kT

1.72 1.72 9.18 11. 77(103.2) (103.2) (550.9) (706.4)

C (NORM) 1.0 kT

2.26 2.28 9.37 12.60(135.5) (136.7) (562.1) (756.0)

D 1.5 kT

2.59 2.62 9.51 13.39(155.4) (157.6) (570.9) (803.8)

E 2.0 kT

2.82 2.86 9.63 14.15(169.1) (171.9) (578.0) (849.1)

F 1. 33 a 2.83 2.88 9.59 13.05(169.8) (173.1) (575.6) (783.2)

G 0.66 a 1.65 1.66 9.17 11.90(99.4) (99. 7) (550.5) (714.2)

H 1. 33 (fiR+ItN) 2.02 2.04 11.02 12.52(121.1) (122.8) (661.4) (751.4)

I 0.66(fiR+~) 2.61 2.61 8.30 12.73(156.7) (156.8) (498.3) (763.9)

J 1. 33 q 2.22 2.28 13.52 12.60(133.5) (136.7) (8llA) (756.0)

K 0.66 q 2.28 2.28 7.95 12.60(136.7) (137.1) (477.1) (756.0)

L 1. 33 IZZ

2.26 2.28 7.57 12.60(135.4) (136.7) (454.5) (756.0)

M 0.66 IZZ

2.26 2.28 13.68 12.60(135.6) (136.7) (821.1) (756.0)

N 1. 33 I 2.26 2.27 9.37 12.01XX (135.5) (136.3) (562.1) (720.7)

0 0.66 I 2.26 2.28 9.37 13.29xx (135.5) (137.2) (562.1) (797.8)

P 1. 33 (IZZ ' IXX

) 2.26 2.27 7.57 12.01(135.4) (136.3) (454.5) (720.7)

Q 0.66 (IZZ

I IXX

) 2.26 2.28 13.68 13.29(135.6) (137.2) (821.1) (797.8)

25

TABLE IV


CONCEPT II AT n = O·


wI w2

w3

W4

A 0.25 kT

1.27 1.27 12.04 13.04( 76.2) (76.4) (722.3) (782.3)

B 0.50 kT

1. 76 1. 76 12.12 13.31(105.6) (105.8) (727.7) (798.7)

C (NORM) 1.0 kT

2.39 2.39 12.29 13.86(143.5) (143.6) (737.4) (831. 5)

D 1.5 kT

2.81 2.82 12.42 14.39(168.9) (169.3) (745.7) (863.8)

E 2.0 kT

3.12 3.14 12.54 14.92(187.7) (188.5) (752.9) (895.5)

F 1. 33 a 3.07 3.08 12.47 14.12(184.0) (184.8) (748.7) (847.3)

G 0.66 a 1. 70 1. 70 12.12 13.40(102.1) (102.3) (727.7) (804 .4)

H 1. 33(mR+~) 2.16 2.17 13.90 13.65(129.9) (130.4) (834.2) (819.1)

I 0.66(mR+~) 2.69 2.71 11.18 14.20(161.6) (162.5) (670.8) (852.0)

J 1. 33 q 2.37 2.39 15.53 13.86(142.5) (143.5) (931. 9) (831. 5)

K 0.66 q 2.39 2.40 10.89 13.86(143.5) (114.3) (653.3) (831.5)

L 1. 33 IZZ

2.39 2.39 10.14 13.86(143.5) (143.6) (608.7) (831. 5)

M 0.66 IZZ

2.39 2.39 16.84 13.86(143.5) (143.6) (1010.8) (831. 5)

N 1. 33 I 2.39 2.39 12.29 13.37xx(143.3) (143.5) (737.4) (802.1)

0 0.66 IXX

2.39 2.39 12.29 14.41(143.5) (143.8) (737.4) (864.6)

p 1. 33 (IZZ

' IXX

) 2.39 2.39 10.14 13.36(143.3) (143.5) (608.7) (802.1)

Q 0.66 (IZZ ,IXX) 2.39 2.39 16.84 14.41(143.6) (143.8) (1010.8) (864.6)

26

TABLE V


CONCEPT III AT r.l = 4.19 RAD/SEC


w1

w2

w3

w4

A 0.25 kT

1.28 1.29 10.58 13.51(77.3) (77.6) (635.1) (811.0)

B 0.50 kT

1. 79 1. 80 10.58 13.75(107.4) (108.1) (635.2) (825.1)

C (NORM) 1.0 kT

2.44 2.47 10.58 14.22(146.8) (148.2) (635.3) (853.3)

D 1.5 kT

2.90 2.93 10.59 14.69(174.1) (176.1) (635.4) (881. 4)

E 2.0 kT

3.24 3.29 10.59 15.15(194.9) (197.5) (635.4) (909.1)

F 1. 33 a 3.16 3.20 10.58 14.44(189.6) (192.0) (635.3) (866.5)

G 0.66 a 1. 73 1. 74 10.58 13.83(103.9) (104.6) (635.2) (830.1)

H 1. 33 (mR+~) 2.22 2.24 10.58 13.99

(133.6) (134.6) (635.3) (839.8)

I 0.66(mR+~) 2.74 2.78 10.58 14.59(164.9) (167.0) (635.2) (875.6)

J q+O. 52m( +1. 7 ft) 2.44 2.47 10.65 ·14.22(146.8) (148.2) (638.9) (853.3)

K q-O. 52m( -1. 7 ft) 2.44 2.47 10.71 14.22(146.8) (148.1) (642.9) (853.4)

L 1. 33 IZZ

2.44 2.47 9.22 14.22(146.8) (148.2) (553.2) (853.0)

M 0.66 IZZ

2.44 2.47 12.81 14.25(146.8) (148.2) (768.6) (855.0)

N 1. 33 I 2.44 2.47 10.58 13.72xx (146.5) (148.2) (635.2) (823.4)

0 0.66 I 2.45 2.47 10.58 14.78xx (147.1) (148.2) (635.2) (886.9)

p 1. 33 (IZZ

' IXX

) 2.44 2.47 9.22 13.72(146.6) (148.2) (553.2) (823.3)

Q 0.66(IZZ

,IXX

) 2.45 2.47 12.82 14.80(146.6) (148.2) (767.6) (886.9)

27

TABLE VI

NATURAL FREQUENCIES OF VIBRATION, HZ (CPM)

AT INFINITE ROTOR ROTATIONAL VE~OCITY

CASE CONCEPT I CONCEPT II CONCEPT III

W2

W2

W2

A 1.27 1.28 1.29(76.2) (76.8) (77.4)

B 1. 74 1.77 1. 80(104.4) (106.2) (108.0)

C 2.31 2.41 2.47(138.6) (144.6) (148.2)

D 2.67 2.85 2.94(160.2) (171.0) (176.4)

E 2.92 3.18 3.29(175.2) (190.8) (197.4)

F 2.93 3.12 3.20(175.8) (187.2) (192.0)

G 1. 67 1.71 1. 74(100.2) (102.6) (104.4)

H 2.07 2.19 2.24(124.2) (131.4) (134.4)

I 2.66 2.73 2.78(159.6) (163.8) (166.8)

J 2.31 2.41 2.47(138.6) (144.6) (148.2)

K 2.31 2.41 2.47(138.6) (144.6) (148.2)

L 2.31 2.41 2.47(138.6) (144.6) (148.2)

M 2.31 2.41 2.47(138.6) (144.6) (148.2)

N 2.31 2.41 2.47(138.6) (144.6) (148.2)

0 2.31 2.41 2.47(138.6) (144.6) (148.2)

P 2.31 2.41 2.47(138.6) (144.6) (148.2)

Q 2.31 2.41 2.47(138.6) (144.6) (148.2)

28

NOTE: ALL GUYS ARE SHOWN BUT

ONLY ONE ANCHOR IS SHOWN

SOIL

ANCHOR

ROD

II-GROUT~ BULB

!

GUYATTACHMENT

RING

BED ROCK

WIND TURBINE TOWER GUY AND ANCHOR ROD SYSTEM

Figure 1

29

CABINETS

wo

/'

!

TOWER CENTERLINE

GENERATOR

PITCH CHANGE MECHANISM

TEETERING HUB

TOP VIEW OF AN INTERMEDIATE SIZE WIND TURBINE, CONCEPT I

Figure 2

r b ....---~

U U

~ If]-'-

~lHI TT

" - ttfj - j -fi-INE

II

-L...--- r--.....--

TEETERING HUB

WIND TURB

BLADE

GEAR BOX

o

PITCH CHANGEMECHANISM

SIDE AND FRONT VIEWS OF WIND TURBINE, CONCEPT I

WI-'

Figure 3

z

a

~v

L

32

WIND TURBINE TOWERSTRUCTURAL AND DIMENSIONAL CHARACTERISTICS

Figure 4

.,. Q I/i)

DIRECTION

y

v

X'~ND

" y

""\

z z,z

x,x

Q = Rotor rotation rate1> = Yaw angleX Pitch angleU = Tower top X

displacementV = Tower top Y

displacement

TOWER COORDINATE SYSTEM

Figure 5

33

YES

NS NS - I

CallGNEWTN

DetermineCoefficients

ofGepuHxv-Duv

il il - ilStep

Rmax

CallGNEWTN

CallGNEWTN


of Ow


of HXV

CallGNEWTN


of GepU

J J - I

DetermineInterval of

Steps, nStep

COMPUTER PROGRAM FLOW CHART

Figure 6

34

TOWER DESIGNS FOR INTERMEDIATE SIZE WIND TURBINES

III

Wl ' WzConcept 1

Wl

, WzConcept

,/ /' wl ' w2Concept II

concepts I,ll and IIIcases A through E

Natural Frequencies of VibrationVersus Effective GUy Stiffness

3.0

Z.5

~Z.O 3P

:z:

200".jJttlI-<.00" 1.5:>....0

>. 2Pul::<ll I::lgo

1.0 I10<r>..

'iilI-<::l.jJ

lP'"Z0.5

250\200%150\100\50\

0.0 '-- ~ -'- _'_ _..J. _'

o

Percent of Effective Guy Stiffness, \kT

The frequencies of vibration of the first two modes as theeffective guy stiffness is varied from 25\ to 200\ of the

design value. kT = 44.36E + 06 N/m

Figure 7

35

TOWER DES IGNS FOR INTERMEDIATE SIZE WIND TURB INES

Natural Frequencies of VibrationVersus Effective GUy Stiffness

concepts I, II and IIICases A through E

18.0

- - w3

, Concept II

6. 6. 6- w3

, Concept II!

-0-- - w3

, concept I

-0-- -0-- -0-0-

16.0 concept IIIw4

,

- ::: w4

, Concept I!-w4 , Concept I

14.0

10.0

8.0

250%200\150\50\

6. 0 L- J.- ...L- ....I- ~ ___J

o

Percent Effect GUy Stiffness, \kT

The frequencies of vibration of the second two modes asthe effective guy stiffness is varied from 25\ to 200%

of the design value. kT = 44.36E + 06 N/m

Figure 8

36

TOWER DESIGNS FOR INTERMEDIATE SIZE WIND TUre INES

WI' w2concept IIIWI' w2Concept IIWI' w2Concept I

III

/ II

Concepts I, II and IIIcases F and G

Natural Frequencies of VibrationVersus Height of Guy Attaching Point

3.0

2.5

N 2.0 3P:r:

c0

-rl...,ttl~

.Q·rl

1.5 ~:>

""' 1'10

2P>,uCaJ::>0'(\)

1.0~~

'iil~

::>...,IPtllz

0.5

0.0 '-- ..L- ........ -L. --'- ---'

o 1/3a 2/3a a 4/3a

GUy Attaching Height, a

The frequencies of vibration of the first two modes asthe height of the guy attaching point is varied.

a = 10.7m

Figure 9

37

18.0


Natural Frequencies of VibrationVersus Height of GUy Attaching Point

Concepts I, II and IIICases F and G

16.0

w4' Concept III

w4' Concept II

14.0t<l:I: ....2

~=w4' Concept I....

0 ....• ,..j..., w3

, Concept II<1l~

.a 12.0 :gr.,..j - -:> ....II-< ....0

><()l:: - - -6 6 6---Cll w

3, Concept III::l

0'Cll 10.0~

fLo -0- - - w3

, Concept IM -0-<1l - - -0-'"::l...,<1lZ

8.0

6.0o 1/3a 2/3a a 4/3a 5/3a

38

Guy Attaching Height, a

The frequency of vibration of the second two modes asthe height of the guy attaching point is varied.

a = 10.7m

Figure 10

f


Natural Frequencies of VibrationVersus Nacelle and Rotor Mass

concepts I, II and IIICases H and I

3.0

2.5

.....wl ' w2 ' Concept III......

2.0 w2 ' Concept IIN ..... 1:I:

wl ' w2 ' Concept IC0

• .-j

+JIII

'".0 1.5•.-j ->"-'0 2PGc:<l);:l0'UI 1.0'"0..

.....III

'";:l+J IPIIIZ

0.5

0.0 L- .L.- ....L- --'- --'- --'

o 1/3M 2/3M M 4/3M 5/3M

Nacelle and Rotor Mass, M

The frequencies of vibration of the first two modes asthe Nacelle and Rotor Mass is varied.

Concept I M 16.19E + 03 kgConcept II M 14.59E + 03 kgconcept III M 14.18E + 03 kg

Figure 11

39


Natural Frequencies of VibrationVersus Nacelle and ftOtor Mass

concepts I, II and IIICases H and I

18.0

16.0

/w

3, Concept II

/

14.0 /N w

4, Concept III

:xl

2 w4

, Concept II0

OM+l

'" w4 , concept Ik,QOM 12.0 / Concept I:> w

3,

.... /0 />,uc w

3, Concept IIIQ)

::l0'

10.0Q)kr...~..~+l

'"z8.0

5/3M4/3MM1/3M

6.0'--- ......... ....L- -'- ....... --'

o

Nacelle and Rotor Mass M = (~ + MR

)

The frequencies of vibration of the second two modes asthe Nacelle and ftOtor Mass is varied.

concept I M = l6.l9E + 03 kgConcept II M = l4.59E + 03 kgConcept III M = l4.l8E + 03 kg

Figure 12

40


Natural Frequencies of VibrationVersus Center of Gravity Location

Concepts I, II and IIICases J and K

18.0

/ w3 ' Concept II

16.0 /

Iw3 ' Concept I

, _ w4 ' Concept III

~14.0 - w4 ' Concept II

:I:

~00"+> - - - w4 ' Concept Ittl10<.0"., 12.0:>Ii-<0

:>,()

~ - -- w3

, concept IIIQ);:ltl'Q) 10.010<

r>.......ttl10<;:l+>ttlZ

8.0

6.0 L..- '-- --J --J -1 --J

o l/3q q 4/3q S/3q

Distance to Nacelle and Rotor Center ofGravity from Tower Center Line, q

The frequencies of vibration of the second two modesas the location of the center of gravity is varied.

Concept I q 2.3SmConcept II q 2.l6mConcept III q -.09m

Figure 13

TOWER DESIGNS FO~ INl'ERMEDIATE SIZE WIND TllRl5INES

Natural Frequencies of Vibration VersusMass Moment of Inertia of the Nacelle and

~otor About the Tower'. Z Ax! s

concepts I, II and IIIcases Land M

18.0 \\

w4 ' concept IIIw

4' COncept II

W3

' COncept III

W3

' Concept II.....

- - - w4 ' Concept I

8.0

I zZ • l69.8E 032

Concept I + kg m2Concept II I ZZ .. l32.3E + 03 kg m

2- w3 , COncept I

Concept III I ZZ .. l39.9E + 03 kg m

6.0

0 l/3IZZ 2/3IZZ IZZ 4/3IzZ 5/3IZZ

14.0

10.0

12.0

16.0

Mass Moment of Inertia of the Nacelleand Rotor About the Tower's Z Axis, I ZZ

The frequencies of vibration of the second two modesas the mass moment of inertia of the Nacelle and

Rotor about the tower I s Z axis is varied.

Figure 14

42



Rotor About the Tower's X Axis

concepts I, II and IIICases Nand 0

18.0

16.0

N 14.0:I:

20

0';

+JIIIl-<.0.'; 12.0:>....0>,()

l::(l)::J0'(l) 10.0l-<

'"'idl-<::J+JIIIZ

~-----8=w4 ' Concept III

w4 ' Concept II

w3

, Concept II

w4 ' Concept I

- - -6c------i6-r----,6- - - w3

, Concept III

- - -O)----~O)-----O- - - w3 , Concept I

4/3IXX

8.0

Concept I I XX = 113.5E 03 kg 2+ m2concept II I xx = 125.4E + 03 kg m2Concept III I

XX 133.0E + 03 kg m

6.0 L- ----:I- --..JL.- --..J----: --..J'-- --'

o

Mass Moment of Inertia of the Nacelleand Rotor About the Tower's X Axis, I XX

The frequencies of vibration of the second two modesas the mass moment of inertia of the Nacelle and

Rotor about the tower's X axis is varied.

Figure 15

43



Roto r About the Tower •s Z and X Axe s

Concepts I, II and IIICases P and Q

18.0 \\\

w4

, Concept III

w4

, Concept II

- w4 , Concept I

w3 ' concept II

w3

, concept II!

Concept I I ZZ 169.8E + 03 kg2

8.0 m2I XX 113.5E + 03 kg m2Concept I! I zZ 132.3E + 03 kg m2I xx 125.4E + 03 kg m2 - w3 ' Concept IConcept II! I ZZ 139.9E + 03 kg m2I XX 133.0E + 03 kg m

10.0

16.0

14.0

12.0

6.0 '-- .L- ...I- --L.. ~ __J

o

Mass Moment of Inertia of Nacelle and RotorAbout the Tower's Z and X Axes, (Izz , Ixxl

The frequencies of vibration of the second two modesas the mass moment of inertia of the Nacelle and Rotor

about the Tower's Z and X Axes are varied simultaneously.

Figure 16

44

i I

W

I I [ I I ! I I I! I I I 4

, I1-~I-

i IiI ! i I I II I I I

! i \ ! II

I I I I ! I i :i

,I I I

1 i I I I I , II, I I ,i ! I i

!

w3

!I

I

:

ii I

i I I i I

! I. i i i

J_I II I I I i i I~-l~. i I I I

I , I-+--t~ , I i II : ; 1

iii i I I i II' I i I

J+ I I I,

I ! III i

I i I,

I I , i i !WI' W

....+-"H ,. -'! I

.-

I-

! Iiii I I I I I

: : i+.- ! I I II

I I i II ! ! I I , I I I-i-..J..I I

·t -! I i I i I , I II I i

! 1 I I i I,

12.5

10.0

""7.5

r

~uzW::l

'"Wa::"-..Jex0:: 5.0::l~

c:z.

2.5

O. 0

o. 0 0.5 1.0 1.5 2. 0 2.5 3. 0 3.5 4.0 O. P. 4.5 5. 0 5.5 6.0

2IP

2P

RaTeR SPEED. RRDI5

NATURRl FREQUENCIES OF WINO TURB!NE SYSTEMS

CONCEPT I, CASE C, 100~ EFFECTIVE GUY STIFFNESSFigure 17

10. C

'"r,.: 7.5uzW::Jawrr"-

5.0

2. S

C. r.

:;

:,

I ii I I

III

! I I I Iii iii i I !1--+--+-+1--+1-+-+-+-1 -+1 -+-f---+--+-+-I-f---+-+-+--il--+- --+-I---+-+--+-+--+-+--I---+-T ! I i I I I

i : I I 'tTl i"r i I :

iii I ! i [I I ! Ii' I Ii! i I! ,I i: ii I ! I I! I I I I I !Ii! ! ! I I !: ! ! i i !iii! Iii iii I' ; iii lit I 'i!' ,I iii"! i I ':' I ! i I I, ',..! Ti.,,'+-+-I -4 I I ll-r-t+

i- --I---~,r t-- '~II +i-- T" -1--1-... _L+-II-HI- --+--t-r'+- --1 --:-+- -i-;-f.

I-

i : i I: I I I : I ii, i: I I I' ,I I I : ! ,1 i I I! . iii II. _I,. -tt-l! ,.,_,i_l] I ,i I I! I I , I I ,j : TTl I I : I I I: i!:: I II! I I : I I I- , - '---+ ~- -.l-+-t--i---t --l.--,-t- +-+-+-: -+-I~..j- -, -H- ~++-+- -it-H- LTIT--f--l---!i,! iii I I !! : iii' I I ! I ': "I I "I: ! i! I I I I I ii' I ': w w

! , :; :' i I r' I , , +' i, i,! i ±: I! I 't I 1 " I, '1 i I :' '1 1'_2

l-l-J'~-l'J--r--1"ll'~"-Jt="-l+jt~-'~J=!~'''~'l---It----~' ~-'J~~~~~;~~-~-~~~~~fTt~~' ~.L~~-+~'--tt~-1~--~~'~-+t-t~:~~--~-~+~tj--f-f·f-f-~~1r==1 P; 'I I i I I ! ' ! I' I' _~ ~ i : I :' "I I II. : : I ; I ; I I L' i I lit i ' I ~Ii' !, , 1" , '.1 i! I J,'._ I,: Ii' '-+' . I , '_ =-;---: " I ._-' ' ' I J I : I 2P---rT tirr I' i'r- -ITill t' -! ; i' ii -1" -- I" +-- ~"I_ I II

: I I • I ! , i Ic. c o. S i. C l.S 2. 0 2. S 3. C 3. S -l. C O. P. 4. S 5. C 5.S 6. C

ABTeR SPErO. RRD/S

~ATURRL FREQUENCIES OF ~!ND TUR81NF SlSlfM5

CONCEPT II, CASE C, 100% EFFECTIVE GUY STIFFNESSFigure 18

---_...--~_._---~---_ .. _.._--_.._-_._-.. --<--- ------------ ---- --- -------~---------~~-~--- ---------~--- -------- ----------------------

3P

22P

IP

6. 05.55. 0~. 0 o. P.~."3.53. 02.52. 01.51.00.5

: ! W4-N ' ,+- f- t f-- 1--,- l-I-- I-t-

! iii

! I : ~rt++ iI

IIitt - -

, :I

. I

~! I I ,

I- ,

II ,! I ,

!I I

ii

I I! I iIi W

3

,

I

iI

iI I I I

II i !, ,, ;I I i I

: I I ! i ! i !I

I i ! I : i I

i I i-l--; I

! i :I Ii

i

I i , I iI !i , 7,·-1 I I i

r-ri I--I ....

~I I f ;WI' W

r-W-J- +I I T I

. ..,~ ..-l- I !I I I,' I- ,-l- t-- I-H T r I I' i ! ' I I i

.JJJI i i i i I,

I I I! : i I -! I ' ! '''-;-l-l- I I i I. , I, ,

I ! ! i I I : ! I, ,I

; I

2.5

0.0

o. 0

12.5

10.0

r>J:r

~ 7.5uzW::JC3We:II-

-'a:a:::JI-a: 5.0-z.

R~T~R SPEED. RAD/S

NATURAL fREQUENCIES ~f HIND TURBINE SYSTEMS

CONCEPT III, CASE C, 100% EFFECTIVE GUY STIFFNESSFigure 19

~l~-;Xl

\ //)~\ /~

x v \ /

\ /\ /

1\1 \I \I II I\ I\ I

1st V-X ModeBending-Pitch

~l/,---~ Xl

I \ ~{ \ ~

\ ~\ ~

\ ~

\ \\ \\ \\ \\ I

\ /\ /

\ /\ //

\ /\ / 2nd V-X Mode

\ #/

BENDING-PITCH VIBRATION MODES AT ZERO ROTOR SPEED

Figure 20(Excerpt from Thresher, et all

1 AX /~,

1 /

jJx I

<P ~ /

u \" I\ I\ I 1st U-<p Mode\ I Bending-Yaw

I \I \I II I\ I\ I

1~X /~

/ \

//

//

2nd U- <P Mode

I \I \I \I II ,, I\ I

BENDING-YAW VIBRATION MODES AT ZERO ROTOR SPEED

Figure 21(Excerpt from Thresher, et a1)

APPENDIX A

FORTRAN COMPUTER PROGRAM

C*FREE VIBRATION APPROXIMATIONS FOR HORIZONTAL AXIS WIND TURBINES*C NATURAL FREQUENCIES OF WIND TURBINE SYSTEMSC

C THIS PROGRAM EVALUATES THE FREE VIBRATION FREQUENCIES FOR A WINDC TURBINE SYSTEM. THE REQUIRED INPUTS ARE INERTIA AND STIFFNESSC COEFFICIENTS DERIVED FROM CALCULATOR PROGRAM 'INERTIA AND STIFFNESSC COEFFICIENTS OF WIND TURBINES'. ALSO REQUIRED ARE THE MASS MOMENTC OF INERTIA OF THE ROTOR ABOUT IT'S SPIN AXIS, THE MINIMUM ANDC MAXIMUM ROTOR SPEEDS AND THE NUMBER OF ROTOR SPEEDS BETWEEN MINIMUMC AND MAXIMUM THAT ROOTS ARE TO BE FOUND.CC THE OUTPUT WILL BE THE ROOTS IN FREQUENCY SQUARED OF THE NATURALC FREQUENCY OF VIBRATION AT ZERO ROTOR SPEED, INFINITE ROTOR SPEED,C AND THOSE SPEEDS REQUESTED.CC THIS PROGRAM WAS WRITTEN FOR NASA LEWIS RESEARCH BY ROBERT J. CHRISTIEC OF W. L. TANKSLEY BASED ON WORK OF R. W. THRESHER AND C.E. SMITH OFC OREGON STATE UNIVERSITY.CC TASK ORDER 152-01 FOR NASA CONTRACT NAS3-21900CC * * * * * * * * * * * *CC THIS PROGRAM REQUIRES SUBROUTINE GNEWTN AND SYNDIVCC R = ROTOR ROTATION SPEEDC W = NATURAL FREQUENCY OF VIBRATIONC PLOTTING ARRAYS

DIMENSION ZZ(200) ,P(400) ,X1(400)DIMENSION CX(3) ,CY(4),CT(8)DIMENSION TITLE(12)

CCOMMON /COEFS/ NDGREE,NC,NSF,EPS1,X,COEF(40) ,ROOT(40)COMPLEX COEF,X,ROOT

C PLOTTING DATADATA NX/18/DATA CX/'ROTOR SPEED, RAD/S'/DATA NY/21/DATA CY/' NATURAL FREQUENCY, HZ' /DATA NT/43/DATA CT/'NATURAL FREQUENCIES OF WIND TURBINE SYSTEMS'/

C NUMBER OF SIGNIFICANT FIGURES IN ROOTS DESIREDNSF=4

C EPSILON TESTEPS1=.OOl

50

C NUMBER OF SETS OF INPUT DATA AND PLOT OPTIONREAD(5,10)J,JPO

10 FORMAT (2I6)CALL MODESG(ZZ,12)

C TITLE120 READ(5,20) (TITLE(L) ,L=1,12)20 FORMAT (12A6)

C MOMENT OF INERTIA, MINIMUM AND MAXIMUM ROTOR SPEEDS, NUMBER OF STEPSC BETWEEN MINIMUM AND MAXIMUM ROTOR SPEED

READ(5,30)RI,RMIN,RMAX,NS30 FORMAT(3EI2.6,I6)

C MASS COEFFICIENTSREAD(5,40)RMUU,RMUO,RMVV,RMVX,RMOO,RMXX

40 FORMAT(6EI2.6)C STIFFNESS COEFFICIENTS

READ(5,50)RKUU,RKVV,RKVX,RKOO,RKXXNP=NS+2

50 FORMAT(5E12.6)C WRITE TITLE

WRITE(6,180) (TITLE(L) ,L=I,12)180 FORMAT(2Hl ,12A6)

C WRITE MASS MOMENT OF INERTIA AND ROTATIONAL SPEED LIMITSWRITE(6,130)RI,RMIN,RMAX

130 FORMAT(26H MASS MOMENT OF INERTIA = ,E12.6/126H MINIMUM ROTATION SPEED ,EI2.6/126H MAXIMUM ROTATION SPEED = ,E12.6///)

C WRITE MASS COEFFICIENTSWRITE (6,160)WRITE(6,140)RMUU,RMUO,RMVV,RMVX,RMOO,RMXX

140 FORMAT(7H MUU = ,E12.6/7H MUO = ,E12.6/7H MVV = ,EI2.6/17H MVX = ,E12.6/7H MOO = ,EI2.6/7H MXX = ,EI2.6///)

C WRITE STIFFNESS COEFFICIENTSWRITE(6,170)

170 FORMAT(23H STIFFNESS COEFFICIENTS//)WRITE (6,150)RKUU,RKVV,RKVX,RKOO,RKXX

150 FORMAT(7H KUU = ,E12.6/7H KVV = ,EI2.6/7H KVX = ,E12.6/17H KOO = ,EI2.6/7H KXX = ,EI2.6///)

160 FORMAT(18H MASS COEFFICIENTS//)C ROTOR INERTIA SQUARED

RISQ=RI**2C COEFFICIENTS OF POLYNOMIALSC *GOU*

AG=RKUU*RKOOBG=(RKUU*RMOO+RKOO*RMUU) *(-1.)CG=RMUU*RMOO-RMUO**2

C *HXV*AH=RKXX*RKVV-RKVX**2BH=-(RKXX*RMVV+RKVV*RMXX-2*RKVX*RMVX)CH=RMVV*RMXX-RMVX**2

51

C FOR ZERO ROTATION SPEEDCOEF(3)=AGCOEF(2)=BGCOEF(l)=CG

C DEGREE OF POLYNOMIALNDGREE=2

C NUMBER OF COEFFICIENTSNC=3

C SOLVE FOR ROOT SWRITE(6,190)

190 FORMAT(20(2H *»CALL GNEwrNWRITE (6,60)

60 FORMAT(SOH ROOTS OF GOU IN W SQUARED AT ZERO ROTATION SPEED)C FOR ZERO ROTATION

COEF(3)=AHCOEF(2)=BHCOEF(l) =CHWRI TE (6 , 190)CALL GNEWTNWRITE (6, 70)

70 FORMAT (SOH ROOTS OF HXV IN W SQUARED AT ZERO ROTATION SPEED)C FOR INFINITE SPEED

W2=RKUU/RMUUW3=RKVV/RMVVWRITE (6,190)WRITE(6,80) W2,W3

80 FORMAT (34H W SQUARED AT INFINITE ROTOR SPEED/ 2E12. 6)R=RMAXRS=(RMAX-RMIN)/(NS+1)GO TO 100

110 R=R-RSC COEFFICIENTS OF DUV

100 OMSQ=R**2BD=OMSQ*RISQ*RKUU*RKVVCD=-OMSQ*RISQ*(RKUU*RMVV+RKVV*RMUU)DD=OMSQ*RISQ*RMUU*RMVV

C COEFFICIENTS OF ENTIRE POLYNOMIAL AT RCOEF(S)=AG*AHCOEF(4)=AG*BH+BG*AH-BDCOEF(3)=AG*CH+BG*BH+CG*AH-CDCOEF(2)=BG*CH+CG*BH-DDCOEF(l) =CG*CH.NDGREE=4NC=5X=O.WRITE (6,190)CALL GNEWTNWRITE(6,90)R

90 FORMAT(27H ROOTS IN W SQUARED AT R ,E12.6)

52

C BUILD VECTORSM=NS+2P (M) =RXl(M)=REAL(CSQRT(ROOT(1»)/6.2832M=M+NPP(M)=RXl(M)=REAL(CSQRT(ROOT(2»)/6.2832M=M+NPP(M)=RXl(M)=REAL(CSQRT(ROOT(3»)/6.2832M=M+NPP(M)=RXl(M)=REAL(CSQRT(ROOT(4»)/6.2832

C CHECK NUMBER OF ROTATION FREQUENCY STEPSNS=NS-lIF(NS.GT.-2)GO TO 110WRITE (6,190)WRITE (6,220)

220 FORMAT(55H ROTOR,RAD/S W1 HZ W2 HZ W3 HZ W4 HZ)WRITE(6,210) (P (L) ,Xl (L) ,Xl (L+NP) ,Xl (L+2*NP) ,Xl (L+3*NP) ,L=l,NP)

210 FORMAT(5(E12.6»IF(JPO.EQ.O)GO TO 230IF(JPO.EQ.l)N=NPIF(JPO.EQ.2)N=2*NPIF(JPO.EQ.3)N=3*NPIF(JPO.EQ.4)N=4*NPN=N+lX1(N)=0.P(N)=O.CALL GRAPHG(ZZ,N,P,X1,NX,CX,NY,CY,NT,CT)CALL PAGEG(ZZ,O,O,l)

230 J=J-1IF(J.GT.O)GO TO 120CALL EXITG(ZZ)STOPSUBROUTINE GNEWTN

C

C POLYNOMIAL ROOT FINDER, USES A MODIFIED NEWTON METHODC INPUT TO THIS SUBROUTINE IS AS FOLLOWSC NDGREE DEGREE OF POLYNOMIAL INTEGERC NSF NO. OF SIGNIFICANT FIGURES ACCURACY DESIRED (INTEGER)C EPSl AN EPSILON TEST, USUALLY APPROX .0001C COEF(I) ,I=l,NC COEFFICIENTS (COMPLEX)C CALLING PROGRAM MUST HAVE THE FOLLOWING CARDSC COMMON /COEFFS/ NDGREE,NC,NSF,EPS1,X,COEF(40) ,ROOT(40)C COMPLEX COEF,X,ROOTC X INITIAL GUESS OF A ROOT (SUB USES X=O IF NOT SPECIFIED)C

53

COMMON /COEFS/ NDGREE,NC,NSF,EPSl,X,COEF(40) ,ROOT(40)COMMON /STORE/ C(40)COMPLEX COEFFSCOMPLEX PVALCOMPLEX COEF,C,F,DELTA,FPR,X,XO,ROOTCOMPLEX FODIMENSION PVAL(2D) ,NP(21) ,COEFFS(40)EQUIVALENCE (AVX,NAVX) , (AVXO,NAVXO)LOGICAL SKIPSTDATA MASK7/D777777777777/N=NDGREENC=N+lIF(EPS1 .LE. D.) EPSl=O.ODD1

2 NNSF=(9-NSF)*3MASK=DNZ=36-NNSFFLD(D,NZ,MASK)=FLD(O,NZ,MASK7)MTEST=MASKDO 79 I=I,NCNP(I)=NC-I

79 COEFFS(I)=COEF(I)WRITE(6,2DD) N,(NP(I) ,COEFFS(I) ,I=1,NC)

20D FORMAT(22H POLYNOMIAL OF DEGREE I3/17H WITH COEFICIENTS/1 (5H X** I2,1P2E18.7)}

WRITE(6,2D4} X2D4 FORMAT(31HK USING STARTING VALUE FOR X= 2F15.8//

1 4X,IHM,4X,IHK,3X,2HIT,6X,4HABSF,14X,6HABSFPR,12X,IHX,32X,2HXO)NR=OK=DIT=OMN=O

19 CALL SYNDIV(N,X)SKIPST=.FALSE.F=C (NC)AVF=CABS (F)IF(AVF .LT. 1.0E-10 } GO TO 50M=lDO 1D I=N,32767,32767FPR=C (I)AVFPR=CABS (FPR)IF(AVF .EQ. D.) GO TO 50IF(AVFPR .EQ. D.) GO TO IDDELTA=-F/FPRIF(M .EQ. 1) GO TO 9IF(CABS(DELTA) .LT. 1.DE-14} GO TO 9SM=MDELTA=CLOG(DELTA}/SMDELTA=CEXP(DELTA)

9 IF (CABS (DELTA) .LT. 2.*AVF} GO TO 111D M=M+111 IF(IT .EQ. D} GO TO 6

54

IF(AVF .LT. STORE) GO TO 666IF(M .NE. MS) GO TO 66SKIPST= .TRUE.GO TO 16

666 IF (REAL (F*FO) .GE. 0.) GO TO 6SKIPST= .TRUE.GO TO 16

6 XO=XAVXO=CABS (XO)STORE=AVFIT=IT+lIF(MOD(IT,15) .EQ. 0) MTEST=MTEST*2SKIPST= . FALSE.K=O

66 IF(M .EQ. 1) GO TO 15H=2**KDELTA=DELTA/H

15 X= X+2.*DELTA16 X=(X +XO)/2.

K=K+lWRITE(6,201) M,K,IT,AVF,AVFPR,X,XO

201 FORMAT(3I5,6G18.8)IF(SKIPST .AND. K .LT. 10) GO TO 19AVX=CABS (X)NXMXO=NAVX-NAVXOIF (CABS (X-XO) .LT. 1.0E-6 ) GO. TO 50IF(AND(NXMXO,MTEST) .EQ. 0.) GO TO 50MS=MGO TO 19

50 NR=NR+lWRITE(6,505) NR,X

505 FORMAT(10HL ROOT NO. I4,9H FOUND, = 2G15.7)ROOT (NR) =XIF(NR .EQ. NDGREE) GO TO 100MINUSN=-NCALL SYNDIV(MINUSN,X)N=N-lNC=NC-l00 51 I=l,NC

51 COEF(I)=C(I)IT=OMTEST=MASKX=CONJG(X)GO TO 19

100 CONTINUENC=NDGREE+1WRITE (6, 200) NDGREE,(NP(I),COEFFS(I) ,I=l,NC)WRITE (6,202)

202 FORMAT(lHL,20X,32HTHE ROOTS OF THIS POLYNOMIAL ARE /1 4HLNO.,8X,4HREAL,10X,9HIMAGINARY,10x,9HMAGNITUDE,9X,5HANGLE)

PROD=I.

55

00 101 I=1,NDGREEPVAL(I)=COEFFS(1)00 102 J=2,NC

102 PVAL(I)=PVAL(I)*ROOT(I)+COEFFS(J)XMAG=CABS(ROOT(I»PROD=PROD*XMAGXANG=57.29578*AIMAG(CLOG(ROOT(I»)

101 WRITE(6,203) I,ROOT(I) ,XMAG,XANG,PVAL(I)WRITE(6,205) PROD

203 FORMAT(I4,1P3G17.7,OPF11.1,5H DEG.,1P2G17.7)205 FORMAT (27HL THE PRODUCT OF THE ROOTS = G20.8)

9999 CONTINUERETURNSUBROUTINE SYNDIV(NDEG,D)

CC SYNTHETIC DIVISION SUBROUTINEC N= DEGREE OF POLY, AND IF N.L.T. 0, DO ONLY 1 LINE OF DIVISIONC D= DIVISORC C= ARRAY OF (N+l) COEFFICIENTS CIX**N+C2X**(N-l) --- CNC

COMPLEX D,C,COEF,X,ROOTCOMMON /COEFS/ NDGREE,NC,NSF,EPS1,X,COEF(40) ,ROOT(40)COMMON /STORE/ C(40)L=NDEGNN=IABS(L)IF(L .LE. 0) L=l00 2 I=l,NC

2 C(I) =COEF (I)00 1 I=l,LM=NN+2-I00 1 J=2,M

l C(J)=C(J-l)*D +C(J)RETURNEND

56

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7.5

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,:uzW:oJ<3W.,

5.0"--'a:.,::>f-a:z

ROTOR SPEEO. RAOIS

NATURAL FREQUENCIES OF WINO TURBINE SYSTEMS

CONCEPT It CASE At 25% EFFECTIVE GUY STIFFNESS

VI(Xl'

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2P

2

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~

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-rj S. 0a::=:J;--a::z:

R~TOR SPEED. RAD.S

NATURRL FREQUENCIES GF WIND TURB!NE SYSTEMS

CONCEPT I, CASE B, 50% EFFECTIVE GuY STIFFNESS

12.5

10.0

'"7.5

:r

,..:uZw~

'"wlC"-

-'cr:0: 5. 0~s:z

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NRTURRL FREQUENCIES eF WINO TURBINE S1STEMS

CONCEPT I, CASE C, 100% EFFECTIVE GUY STIFFNESS

0\0'

12.5

iO.O

~ 7.5

,.:uZILl:::>0ILla:u..

..Ja:a::::> 5.0>-a:-"

2.5

O. 0

I I 1 I W4! ,

I ! I I I i I i II

I I III

i I ! ! I ! I I !I I I I

I [ !I • ,

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ReTBR SPEED. RADIS

NATURAL FREQUENC!ES BF WINO iURBINE SYSTEMS

CONCEPT I, CASE D, 150% EFFECTIVE GUY STIFFNESS

I I I W4I I !I

I II I

,

I

W3

I

I

WI'

,I

I

I

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NT

~ 7.5uzw::>0wlCU-

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o. 0

e. 0 C. 5 \.0 \.5 2.0 2.5 3. 0 3.5 <ta.0 O. P. 4.5 5. 0 5.5 6. 0

2P

IP

ROTOR SPEED. RRDIS

NRTURRL FREQUENCIES OF ~IND TURBINE S~5TEM5

CONCEPT I, CASE E, 200~ EFFECTIVE GUY STIFFNESS

I I ! I I

I I ! IW4

! I I I I

i I II ,

I I ! I I I I I

!

i

W3

I

iI I I

,i

WI' W

I i

i I I i

I , \

12.5

10.0

'":r 7.5,.:uzW::JC3UJa:11.

..J<:rc::J 5. Cr-<:rz

2.5

C. 0

c. c D. 5 \.0 2. D 2. 5 3. 0 3.5 ~. 0 O. P. 4." 5. C 5.5 5. e

2

2P

IP

ReTeR SPEED. RRDiS

NRTu~nl ~'REQUENCIES l'lF WINO TURBINE SYSTEMS

CONCEPT I, CASE F, GUY ATTACHMENT POIN!' RAISED 12 FT

2P

2IP

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! i ! 1 ! I T W4

i I I j !!

I i ! i ;, I

i I, I ! \ i \,

! i II !I

, I I1 w

3I

I

I,

I I

I

I I I -I

wI' WI

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o. 0

2.5

10.0

7.5

""T

,.:u-z.w:>awcr:lL-

-' 5.0a:cr::>I-a:z.

R~T~R SPEEG. RRGiS

NRTURRL FREQUENCIES ~F WINO TURBINE S!5TEMS

CONCEPT I, CASE G, GUY ATTACHMENT POINT LOWERED 12 FT

I I I ! I I I J !

I T r ! I T Iw4

i I I I !I,

I i ! I I

T !I

I

II

!I w3

I

-

II

WI'~

r :

I I II I I

I

12.5

10. 0

2.5

c. 0

C. C o. ~ I.C 2. 0 2. " 3. 0 3. ~ 4..0 O.P. 4.." 5. 0 5. " 6. 0

22P

Ip

R~T~R SPEED. RRD/S

NRTURRL FREQUENCIES ~F WIND TURBINF ~TSTFMS

CONCEPT I, CASE H, NACELLE AND ROTOR MASS INCREASED BY 1/3

12.5

10. 0

N 7.5r

,;uzW:0C3Wa:!L-

a!a: 5. 0:0l-a::z:

2.5

0.0

I : : I I A W4''t'

, 'f

! I I I i ! ! . I I i ! i I I !I I I ,, II I i i i

I

I ! I , I I I ! I I I I I 1 I i II -L ;I ! I I I I I I ! ! I I i Ii I

! I ! I i I i:

I I !I

I I

i I !! i I

II

I! W

3II

I i !I i

I I

!I I

! i, I I I I II

I !I

I II W

1, W

I II I -i I I I I

I I I !

I I I I ! , ! jI

r+- i i II

I- h

I I I i ! : II I I I I

2

2P

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RClTClR SPEFO. RRO/S

~RTURAL FREQUENCIES OF WINO TURBINE SYSTEMS

CONCEPT I, CASE I, NACELLE AND ROTOR MASS DECREASED BY 1/3

I i i I i i ! i 1 I ! ,i , i A: , i I W

4! i i I ~!

,I i I

I I I,

i I I I i I I I Ii I I~ I ,i J i ! I ,

I I, i

I I i

i ,I I I ! ! I I :

W3, I

I

1 I ,! ! !I

I iI I I i !i

I I, ,

I

tI

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i, I I I I

I

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-I

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i i , , i .

i~

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I I,

0'10'1

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NT

~7.5

uzw::>C3w0:u..

c!a:::><- 5. 0a:z

2. 5

O. 0

C. 8 0. 5 1.8 1.5 2. C 2.5 3. 0 3.5 4.0 0 • P • 4.5 5. 0 5. 5 6. 0

22P

lP

ReJTeJR SPEEO. RRD/S

NRTURAL FREQUENCIES elF WI~D TURBI~E SYSTEMS

CONCEPT I, CASE J, DISTANCE TO CENTER OF GRAVITY INCREASED BY 1/3

I i 11 : 1I I ! i I i i

, I i IJ i i 1 Ii , , I

, : I) i I 1 i I i ! I I I II I I i 1 I !

w4i i I I I --l-J i i I : I I : 1 I i j

I ! i I ! II iii I i I I i I i i 1 i I I I I i I

i i i I I ! ii , I i -4 ' ,

! I I , i ! I I i ! i i I I i I ! t I ! !i I I,

I i i ! I , I I i I I I I II II I

I ! I i I ! i I i I II ! II I I I I i I 1 j II

iI !

I

W3

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I II

I I I

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I , ,I I I -i iI II I

! I i I I II :

,I :

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I

I !I

I

I I

WI' W

-1-+ t r I

l- f-

I I I IiI I i ! I

I ! i -T i i I i I1-+ I , 1--- 1I I ! i ,I I ! ,

I I !I I I I

12.5

10.0

N 7.5:I:

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S. C

2.5

O. 0

o. 0 O. 5 i. C 1.5 2. 0 2.5 3. 0 3. 5 4.0 O.P. 4.5 5. C 5. 5 6. 0

22P

IP

p,eT~R SPEED. RRDiS

NRTURRL FREQUENCIES OF WINO TURB!~~ 5lSTE~S

CONCEPT I, CASE K, DISTANCE TO CENTER OF GRAVITY DECREASED BY 1/3

enex>

Nr

,..uZllJ::JC3llJ0:u..

, 1 i I i i i :, : I I I

,I -.i i, ~

I I I W4\2.5

11 T, il l II I I 1 Iii I I I I I I I ! i I I I

i,

; I I I I I I, I I I ;f-

I ! i I 1 1 I ITil i I ! I I I I ! I i I ! I I ! ! I I

! I ! I I !

I ! I I i I i I i,

I i . i I II;

i I I I : i I I iI I i, I

I i ! ; !! IiO. I}

!I I

1.0 1.5 2. 0 2. 5 3. 0 3.5 4.0 O.P.4.5 5 " 5. 5

I

G.O

I !! !

ROHIR SPEED. RRGiS

~qTURRL FREQUENCIES OF WING TURBINE S,STEMS

CONCEPT I I CASE L, NACELLE AND ROTOR INERTIA ABour Z AXIS INCREASED BY 1/3

1P

22P

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I I I i I I .-;. ,W

4H-'n 'r i II I ! I I ! i I ! i I I I i I i ! I ! :

-++!

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!,W

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+H+t+H I I i I , I

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, I I

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t-;

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: I I

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0.0

2. 5

.0.0

12.5

to<r

7.5,-uzUJ;:)

aUJtr:"-

5. 0

0',1.0' ReTeR SPEED. RRD/S

NR~URRL FREQUENCIE~ OF WING TURBINE SlST=MS

CONCEPT I, CASE M, NACELLE AND ROTOR INERTIA ABOUT Z AXIS DECREASED BY 1/3

-..Jo

IP

22P

! i I i i I : ! 1 i I I i I i i i ; I i, I iii 1 I i i ! i i !

I! !

J ! I i i Ui ! i i i I ill

I ! T ! I I ! I' I Ii ! i ! i ! ; ! I ! I I I I I

, ..,. 4

~t! I i . I ! i i j I I...1- i I I ; . ! I I I-+- T""-+-

I; I i ' ! I I I : ! I ! I l 1 I i I' I I ! I i \

; Ii ' ! I I I i i! i !

, I I i ! I I i I I i

H·I I i I I · ' I I i I i 1 I

I I ! ! i! II I ;I

I +t'-j- f-

I I! I I I I i I II I I ,

i I i I, I I I ,

I i iI ,

jw

3!

I

I II

I I i,

I I

!

! 1i

I i! I I

I I I I I I ! : II ! I ! ,i 1I I I

I I I i I : ! Ii,

t I iI ! ! I ! I I I i I,

I I, I

I I I i I iI IWI' w

T -i I I I I ! I

J,

I i....;..

I I !I i- .... I I I i II I i

,I I I i

i I i I i: ,

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i i ! II

! I I Ii , : : !I I I Io. ()

2.5

10.0

12.5

7.5

N:r

,.:uzw:;)

awIt:11.

-J5. 0a:

It::;)

;r:z:

D. C o. S l.0 l.5 2. C 2. 5 d. C 3.5 4. cO. P. 4.:' 5. a 5. S

ReTeR SPEED. flRD/S

~RTURfii ~REQUENCIES elF Wl'NO TURBHiE 5i5TEMS

CONCEPT I, CASE N, NACELLE AND ROTOR INERTIA ABOUT X AXIS INCREASED BY 1/3

12.!>

10.0

, 'I' 'f I 't', I I i 'f I I I I I! I I I

!I!, i III i I I ! i II I j I' Iii I I Ii: I I J ! I I ! I II! I!e-:+ ,_-Li' -1--l--'-+-~i-+++-J--l---1-W' ++1++-J;--+!-1-H-++--f-+-j--H1-+-i' 4,-+-;1.-++1~14-++-++-+-H-t--Hl-+i --ji-+-+-H+~I-+-t-+--JI Ii, i [ ! Iii: i ' i 1 i

I ' i! I I i I I I !! I ii, , !, ' I ! I

O. 0 O. 5

! I

! iI I I

1.0

I

I

I

I I I! i I i

i,

I I I! I i i II

3.!>

FHn~R SPEED. RRD/S

I I

i

I I

I Ii

, I;

!, I Wi' Wz

- 2P!

I I I I

: lP

l-

I i ! I I iI

'::. C O.P. 4. ~ 5. 0 5. !> G.O

NATURRL FREQUENCIES O~ WI~O TURBINE SYSTEMS

CONCEPT I, CASE 0, NACELLE AND ROTOR INERTIA ABOUT X AXIS DECREASED BY 1/3

12.5

iO. C

7.5

N:r:

,...uzw::>C3wcc"-..J 5. 0crcc::>...crz

2.5

D. D

I I i il ! ;1 ! ; I I I I i I i I I I Ii I

,Ii i , I I

! T I I ! I I I I i i"[ I !

w4i ,

I I i I

~1 I ,

I I, I ! i ! i ! I I I 1 I II I t i: t I

! i I i I i i I i,

I I I iI I

I I I,

II i

I W3

i

I

I I I

I I I

I I I I i I I ii I I i iI I i! i I I .' i

I I I I I I I I

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WI'

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I I 1 I

i i ! j Ii

i I ! I! ,

I 1 , i

! -1i I I ! i i I

i I I i I , , :

IP

D. Q O. 5 1.0 1.5 2. D 2.5 3. D 3. 5 ~. DO. P. 4.5 5. C 5. S 6. C

ROTOR SPEED. ARDIS

NRTURRL FREQUENCIES OF WI~D TURBINE 5YST=MS

CONCEPT I, CASE P I NACELLE AND ROTOR INERTIA ABOUT X AND Z AXES INCREASED BY 1/3

12.5

10.0

N:r

,.: 7.5uzUJ::JC34Ja:"-..Ja:a:::J.... 5. 0a:z.

2.5

o. c

: : W4I i I! !

I W3I i I I I i ! , I II

I I i I I ! i I ! II I II I

! i i I I ! ! \hI :

I I II

I

I

I

I II ,

I!

I I I,, i ! I [

I I II I I ! ! i I

,II

: I,

I

I I I

I I, I WI'

-I I I I

I I I iI

.

f---- I -I ! I i ! i I ! ! II I I

IP

O. 0 O. 5 1.0 1.5 2. 0 2. 5 3. 0 3. 5 4.0 O. P. 4." 5.0 5.5 6. a

-...JW

ReTeR SPEEO. RRG/S

NATURRL FREQUENCIES eF ~lNO TUR8!NE SlSTfMS

CONCEPT I, CASE Q, NACELLE AND ROTOR INERTIA ABOill' X AND Z AXES DECREASED BY 1/3

I ! I !, I I I ! I 1 i i , i i ,, ; i

I ! I ! I ! I I I I i I !, I 1 I i

W4

12. ~ i I I

I I i ' i ! I I I ! I , I I I I, I ,! ! I I

, W3; I

! ! I II

I i !

10. aI

,I

N:r:

>uZw::>owa:u..

..Ja:a:::>le:z

7.5

5. 0

i

Ii!I I

l-+-!H-!++-1,_+-[1+-+-+-++--il-+-+-+-+-+-+-f-+-+-+-t-+-+-H---jl-+-+-H-t-~IH-+-+,+--+I ++--iH--+--H++-H++--IH--+--H-+-+-H,

2.5

O. ~

i

1.0 1.5 2. 0 2. 5 3.0 3. 5 4. G O.P . .(. ~ S.C 5.5 6. C

RcrTcrR srEED. RRD/S

NRTURRL FREQUENCIES crF W!ND TURBINE SYSTEMS

CONCEPT II, CASE A, 25% EFFECTIVE GUY STIFFNESS

iP

2P

2

I I I,

i I IW

4,I I i ! ! i II I i ,I I

- W3I I I I I i

! I, ,

I! I I I

I II

tI

I ,

II

I

\I

tI I

I iI i I

II

-+-! i!I

I,I

II

I

I-

i i

I -I Wi' W

I i I II i !

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H-I

j ! i I i! I IO. 0

2.5

iO.O

12.5

....:r 7.5,:uzw::>

'"wa:u.

..Jcra:::> 5.0>-crz

Q. 0 o. :> 1.0 I.!> 2. 0 2.:> 3. 0 3.:> ~. 0 O.P. 4.!> 5. 0 5.5 6. 0-..JU1 RDTI'lR SPEED. RRD/5

NRTURRL FREQUENCIES OF iilNG TURBINE S~5TfMS

CONCEPT II, CASE B, 50% EFFECTIVE GUY STIFFNESS

_+-+'+-+'-+ I -i-~-+--+--f-:--+-+--,~-+I-+--+'-+--+-+---i'-+-+i+-+-'+-+---+---+-+-'-_+.--\'--++'+-+--1-+1-+-\1-+ ! i "I ! i I

! T! i I I T I I I I I I I Ii! , i -;-ti,-1_+--,If--+I-+--+'-+_If----C!-+-.-r-4~-+i -+_-+'-+--+1_+--1I-+-I--+-I--+-I--+-f---t-t-+--f--+--l--J-...-l---+-l---+-l-+-J--+-f--'-' -n---+-+---+--++++++

12. 5d>:l=+;:+::t:~4t::~li :t=H=t::t:ct=l=+:+~:::j:' ::j:;~1:i~'~l::t:'t:~!~I:<ti~it:j::::t:' ~!::t:'~':<t:'~; j!:t:tt:4=<t:t:t:t:tI=l:i~':j;!:t:t:t:~1=t:t31:t:j':t!:ttl:t:i ~I-+-+++-H-++-H-+-l-+-H-l...++-H-'r+---+-+-+-t--I--+-+-HH++-H-+++-H-++-~'---t-++-HH++-t-j-+-l-+-H-+-+-+-1I ' ! 'I I ! !' iii i I :

,.,:r

10. C

, ,, T

i

>uZILl~

C3ILlcr:4.

-'cr:cr:~

>-cr:z

7,5

5. 0

2.5

C. 0

I

-Ll I ' I !i i, I--+-l-+-+I--+-+-+-I-~+I-+--J-++-+-+-+-+-++-+-I-~+-t-++-+++-+-++-+--+-I-+-H~+-t-++-+I-+-+-+!-+-+-,I -+,-++-1'+-+++-+-,H

!ii' Iii i I I iii I

! I ! ++-+.+-+---14-+-H-++-t-+-l--+-l-++-t-+-+-+--l4-+-H-++-t-+I-1-H-+-+--+-1i +I_++I-l--+--l-+-ti-+--t-!_+-i'+-t!l-+1_~11-t1-+-+1+_

~--+--+-I:+-!,'~-+-+-+-+-+-+-+-+-+,-+-+-+-+-+-+-+-+-+--+-+-+-+-+--+-l-+-+-+-+-+++-+:-++-+-++--iI+1 +-+-+:-Hi:-+-+I---'I~-+!-+-+--+-+-+-+-+-i +,-l

O. 0 a. ;, LO i.5 2.0 2.5 3. 0 3.5 4.00. P. 4.5 5. C 5.5 G. 0

R~T~R SPEED. 'lAOiS

NATURAL FREQUENClES ~F wlND TURB!NE SYST"MS

CONCEPT II, CASE C, 100% EFFECTIVE GUY STIFFNESS

i I -t-L r i I if I I J-W- I I , i i i iW

4! ,

I ! I ! I II! Hi i i I ! i ! t ! ! T_t- Ht+J.l! [ i I :,[

II -+- I f-. , -+-

I ! I I I , i i i I +[ i i -tt I i II ,

iI I I I 1I , I ' I' I I I !

! I +- 'T-t i I

~._+-........... 'I I ii : i I I i i i I ; I iI i I I ! !

T I I' I I i I I ! ! I I I :W

3

! ! II ! ! I,-

II I I ! I,I

! I I I II I I i

I ,I i I !

I ,

I ,I I

II

, ,1

, ! I i I

I I I I I I

i i

! I

I,

I i I

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CONCEPT II, CASE D, 150% EFFECTIVE GUY STIFFNESS

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CONCEPT II, CASE E, 200% EFFECTIVE GUY STIFFNESS

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4.0 O.P. 4. 5 5. 0 5. 5: G. 0

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CONCEPT II, CASE F, GUY ATTACHMENT POINT RAISED 12 FT

I i ,i i ! ! i ! ! , ,I i ! I '-W I I

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~RTURRl FREQUENCIES DF WINO TURBINE S~STfMS

CONCEPT II, CASE G, GUY ATTACHMENT POINT LOWERED 12 FT

12.5

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ROTOR SPEED. RRD/S

~nnJf1qL FRfQilfl><CIES OF WINO TURBll><E SrSTEMS

CONCEPT II, CASE H, NACELLE AND ROTOR MASS INCREASED BY 1/3

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I;ATURAl fREQUEI;CIES ()f WINO TURBINE s;-sorEMS

CONCEPT II, CASE I, NACELLE AND ROTOR MASS DECREASED BY 1/3

15. ()

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NATURAL fREQUENCIES BF WINO TURB!NE S~STEMS

CONCEPT II, CASE K, DISTANCE TO CENTER OF GRAVITY DECREASED BY 1/3

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IItRTURRI FREQUENCIES OF WINO TURBIlltE SYSTEMS

CONCEPT II, CASE L, NACELLE AND roTOR INERTIA ABOUT Z AXIS INCREASED BY 1/3

_1 I I 't iii, I If, I II I 'I! i I I w4

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D. D O. ~ La 2. a 2.5 3.e 3. " 4..0 O.P. 4.. ~ 5. C 5. " 6. aROTOR SPEf~ RAO ~

CONCEPT II, CASE M, NACELLE AND ROTOR INERTIA ABOUT Z AXIS DECREASED BY 1/3

;[ i , I I 11' 1 i , 1 I I I ! i I 1W

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NRTURRL FREQUENCIES D~ WINO T~RBINE SYSTEMS

CONCEPT II, CASE N, NACELLE AND ROTOR INERTIA ABOUT X AXIS INCREASED BY l/~

00:00'

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~RTURAL "RfCUENCIES C~ MIND TURB!NE S1STEMS

CONCEPT II, CASE 0, NACELLE AND ROTOR INERTIA ABOUT X AXIS DECREASED BY 1/3

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CONCEPT II, CASE P", NACELLE AND roTOR INERTIA ABOUT X AND Z AXES INCREASED BY 1/3

1.0o

H-~!!-+-H-+-l-+-H-+-++'_IH-+-+I-+-I++-iH-++-H-+-+-f---'H-++--l-I!-+-+-+-+-+-++-f---'H,I, .il--!-+--Li,' -+,!i---+--+-+_~--+-+---H-+--l

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i-+

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RCTOR SPEE~ ~RO/S

~RTURRl. fREQUENCIES (:)" Wl~O TURBI~E S1STEMS

CONCEPT II, CASE Q, NACELLE AND ROTOR INERTIA ABOUT X AND Z AXES DECREASED BY 1/3

12.5

I! IT! I I I ! iii . !: ~;! !! i I!I -i-H-+---+-H-+---'-H-+-+-+-t-++-H-i--c-+--t,-+--+--H-+--+--H--+-+-h--+-+-hl-t--+-+-1--+-+-+---!I---t--+-H-i.--+---t-+-++--Hi : I I I 'ii i I I ! !! I

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W4

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-j--+-+-+-1-H-t--t--+-+-H--+-t--+-+-t-H---++--t-+-J---++--+-t-+-H--+-t--+-+-+-+-+---++--t-HJ---+++-l-t-H--+-t--t-+-+-+-t---++-+-H

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! ! I ! I I !, ! i_I-' ! I

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f------ - WI' w2

I-_++_-+f----t__ ~--+---_+--+ I U-i-"- i i ! I IP

0.0 ! ! i ! : "i !

o. 0

FlOTDP, SPEED. RRO/S

NATURRI fREQUENCIES 0" WI~O TURB1~E SYSTEMS

CONCEPT III, CASE A, 25% EFFECTIVE GUY STIFFNESS

..;. W4

Ii ! i

:

! I i I-11

W3

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- -+--f-+---+-+-~' +-, l!:! ~++-+,-l-t 'L~-- _11+:+-t--++-t-+:-+,- --r- ~+,---'r+ r-t-+- ++-I I: i I :Ii' i! ! I . i i : i I I!: i! Ii! 'i I! I

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I I I :!

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tta::::Jfa:.z

>uzu.;:::J(3u.;cr:"-

ROTOR SPEf~ RRDiS

NATLIARL FAEr:UE~C:FS 0" WING TURI3!NE S'rSTfMS

CONCEPT III,.CASE B, 50\ EFFECTIVE GUY STIFFNESS

i I . _~ !:! Ii: I ! I !I ) I . -++--j--++H-++H----l-+-+-if-i--+-+-ir-+-t--+-H-t:---+----t~I--+_+-:-+-+---t-I-+-+---+-t_+-+--+-H-T-j--+--t-TTTT, i 'I t Ii, ,

12. "

10.0

I

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I II 1

1

I

1I

i

J

it

I

I

i

~IW

1-+-t-hH--t-H-trH--r-t-t-t-r-t-1i'--tt'-itH-Tti'-r1

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i ....i T+r T-

i! ! I

! ,

2. 0 2. " 3. [) 3. 5 ~. 0 O.P. 4. 5 5.0 5.5

ROTOR SPEED. %O/S

NRTURAI rAEQUENClf~, tJF WIND TJRRlt>:E SYSTEMS

CONCEPT III, CASE C, 100\ EFFECTIVE GUY STIFFNESS

'5.C

12.5

iO. C

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003

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"OTDR S~EED. RRD/S

NRTU"RL FREQUENCIES DF HIND TUR8INE SYSTEMS

CONCEPT III, CASE D, 150t EFFECTIVE GUY STIFFNESS

i5.0

12.5

10.0

N:c

r:uzIII 7.S::l0ILl.,\l.

-'a:.,::l0-a:z

5.C

2.5

C.C

W4

W3

W1

,

-1-1-... ,.....

,

2P

lP

2.0C.D 0.5 1.0 1.5 2.5 3.C 3.5

ROTOR SPEEO. RAO/S

~ATURAL fREQUENCIES O~ HI~O TUR~INE SYSTEMS

~. CO. P. ~. 5 S.O 5.S 6.0

CONCEPT III, CASE E, 200~ EFFECTIVE GUY STIFFNESS

1.00\.

12.5

10.0

...;r

~ 7.5uz!!!aIIJ.,......Ja:.,::lf-a: 5.0z

2.5

W4

w3

WI'

--,----- ------

-'-..--

2P

IP

0.0

0.0 0.5 1.0 1.5 2. 0 2.5 3. 0 3. 5 <to 0 O. P . ~. 5

ROTOR SPEED. ~AD/S

NATURAL F~EQUENCIES OF HIND TU"~!NE SYSTEMS

5.0 5.5 6.0

CONCEPT III, CASE F, GUY ATTACHMENT POINT RAISED 12 FT

12.5

10.0

N:r

,.: ·7.5u..,w::>aOJc.....Ja:c::>f- 5.0a:

'"

2.5

W4

W3

,

~

_10-1-----_....

~ WI'

10-i--t-.-

....~~-~

3P

0.0

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R~T~R SPEED. RRD/S

NRTURRL FREQUENCIES OF HIND TUReINE SYSTEMS

CONCEPT III, CASE G, GUY ATTACHMENT POINT LOWERED 12 FT

W4

W3

-W

1,

-...I-

-~....1-

\0.(Xl

12.5

10.0

""or

,.: 7.5uzw:::>C3wa:II-

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3P

1P

~~T~A SPEEO. ~RO/S

~ATURRL FREQUENCIES ~F WINO TURBINE 51SIEMS

CONCEPT III, CASE H. NACELLE AND ROTOR MASS INCREASED BY 1/3

15.C

12.5

10. a

'":r,:uz 7.5w::>13wa::u....Jcrt:::>....crz 5.C

2.5

W4

W~.j

-W1 '

3P

1P

wiWi

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C.O 0.5 LO 2. 0 2.5 3.0 3.5 •. 0 O. P .4.5 5. 0 5.5 6.0

ROT~R SPEED. ~AD/S

NATURAL FREQUENCIES OF WINO TURBINE SYSTEMS

CONCEPI' III, CASE I, NACELLE AND ROTOR MASS DECREASED BY 1/3

...... 'oo

3P

IP

w2

2P

W4

W3

I

--........WI'

I-~r-- -.... - ....

f- ...._l- f----

2.5

10.0

12.5

'":r~ 7.5uz'":J"'"0::IL.

.Ja:a:::J...a: 5. 0z

0.0

0.0 O. 5 1.0 1.5 2.0 2.5 3. 0 3. 5 4.0 O. P. 4.:' 5. 0 5.5 6. 0

ROTOR SPEED. RAO/S

NATURAL fREQUENCIES Of WINO TURBINE 5~5TEMS

CONCEPT III I CASE J I DISTANCE TO CENTER OF GRAVITY INCREASED BY 1.7 FI'

12.5

10.0

NT

,.: 7.5uz..,::Ja..,c:"--'cra::::J....cr 5.0z

2.5

W4

W3

---- WI'

------

--_......---~~I-i

_......~

IP

ReTcrR SPEED. RRD/S

NATURAL FREQUENCIES crF WIND iUR~INE SlSiEMS

I-'oI-'

o. ao. c 0.5 1.0 1.5 2. a 2.5 3. 0 3.5 4. a o. P. 4." 5.0 5. 5 6. a

CONCEPl' III, CASE K, DISTANCE TO CENTER OF GRAVITY DECREASED BY 1. 7 FT

I-'0'f\),

12.5

ID.I)

N:r

,.:7.5u

z...,::JC3...,0:"-..Ja:0:::JI-a: 5. C:z.

2.5

W4

w3

Wi'

--- lP

D. C

o. I) O. 5 1.0 1.5 2. 0 2.5 3.0 3.5 4.0 O. P. 4. 5 5.0 5.5 6. 0

~BTBR SPEEO. ~AO/S

NATURAL FREQUENC1ES BF W1NO TURB1NE SrSTEMS

CONCEPT III, CASE L, NACELLE AND ROTOR INERTIA ABOUT Z AXIS INCREASED BY 1/3

12.5

NJ:

,:7.5u

zw:>C3wc....--'e:

'":>....e: 5.Dz

2.5

W4

W3

--- WI'---

-J,..-,l-I--

3P

IP

6. 05.55. 04. DO. P .4.53.52. 0 2.5 3.D

ROTOR SPEEO. RAO/S

NATURAL FREQUENCIES OF WIND TURelNE SYSTEMS

1.5l.DO. 5

D. D

D. D.....ow

CONCEPI' III, CASE M, NACELLE AND ROTOR INERTIA ABOUT Z AXIS DECREASED BY 1/3

12.5

10.0

N:I:

r: 7.5uz....:::>C3

""'C......Jcra:::::><- 5.0crz:

2.5

W4

W3,

-.-----...."..- WI'

i--..- -------_.....

---i------

3P

IP

O. 0

o. 0 O.~ 1.0 l.~ 2. 0 2.5 3.0 3.~ 4.0 O. P .4. ~ 5.0 5.5 6.0

R6TOR SPEED. RADIS

NATURAL FREQUENCIES OF WINO TURBINE SlSTEMS

CONCEPT III, CASE N, NACELLE AND ROTOR INERTIA ABOUT X AXIS INCREASED BY 1/3

15. C

12.5

10.0

N·r

~u

7.5z

'"=>CJ

'"cr:lL

..J<rcr:=>f-<r:z 5.0

2.5

..

w4

w3

-w1 '

--

3P

1P

ROTOR SPEEO. RRD/S

NRTURRL FREQUENCIES OF WINO ,URBI~E SYSTEMS

I-'o\Jl

C. 0

G. G a. " La 1.5 2. 0 2.5 3. 0 3.5 4.00.P.4.5 5. a 5.~ B.G

CONCEPT III, CASE 0, NACELLE AND ROTOR INERTIA ABOUT X AXIS DECREASED BY 1/3

1-',o0'1

12.5

10.0

N:r:

~7.5

uzbJ::>

'"bJc:ll..

--Ja:a:::>>-- 5. aa:

"

2.5

W4

W3

~I----0-

Wl'- -..... - ---

-----

3P

IP

0.0

0.0 0." 1.0 1.5 2. a 2.5 3. 0 3. " 4.. a O.P. 4." 5. a 5.5 6. 0

~~T~R SPEED. RADIS

NATURAL FREQUENCIES ~F WINO iURBINE S1STEMS

CONCEPT III, CASE P, NACELLE AND ROTOR INERTIA ABOUT X AND Z AXES INCREASED BY 1/3

.. ..

15.0

12.5

10. a

NX

,.:u

7.5z

'"'"C3

'"a:u.....Ja:a:

'"....a:z: 5.0

2.5

W4

W3

--I"""

Wi'

.....lP

6. 05. 55. a-i. a O.P.-i.53.52.0 2.5 3.0

ReTeR SPEED. RAD/S

NRTURAL FREQUENCIES eF WIND TURBINE SYSTEMS

1.5l.oO. 5

o. ao. aI-'

o-..J

I

CONCEPT III, CASE Q, NACELLE AND ROTOR INERTIA ABour X AND Z AXES DECREASED BY 1/3

a

A

E

f. ,~J

Fu' Fv

G

h

I

Im

I r

I xx' I zz

J

k a

k c

kR

k T

kv

k, .~J

1

L

I'

m

m. ,~J

108

APPENDIX C

SYMBOLS

Distance from ground to guy connections

Cross sectional area

Modulus of elasticity

Flexibility influence coefficients

Loads in u, v directions

Shear modulus of elasticity

Hub radius

Area moment of inertia

Mass moment of inertia per unit length about the centerline of tower

Mass moment of inertia about the rotor spin axis

Mass moment of inertia of nacelle-rotor around x, z axes

Polar moment of inertia

Anchor stiffness

Effective spring constant of guys

Spring rate of a guy rod

Effective guy and anchor stiffness

vertical spring rate of soil due to compression

Stiffness influence coefficients

GUy cable length

Beam, tower length

Tower section length

Mass per unit length

Tower inertia coefficients

..

"~

mR

/I

m..1.J

Mt

N

O.P.

P.1.

P

q

Q

s

T

U, V

w

B

e

A

<p, x

'¥.1.

W.1.

n

n

•

Mass of nacelle

Mass of rotor

System inertia coefficients

Total mass of tower

Number of integration intervals, also number of guys per position

Operating point

Generalized load at ith location

Rotor's rotational velocity

Distance to nacelle-rotor C.G.

Thermal energy

Strain energy

Kinetic energy

Displacements at tower upper end

Work

Approximate function for blade model deflections

Angle between guys and ground

Dimensionless blade natural frequencies

Rotational displacements at tower upper end

Displacement functions

Natural vibration frequencies

Rotor speed

Dimensionless rotor speed frequency

109

1. ~nA~'R-165589 I2. Government Accasion No. 3. Recipi.nt's Catalog No.

4. Title and Subtitl. 5. ".port Olt.

VIBRATION ANALYSIS OF THREE GUYED TOWER DESIGNS March IGS2

FOR INTERMEDIATE SIZE WIND TmBINES •• I'trforml!'l Or..nizltion Code778-33-41

7. Author(s) I. I'trforming Or..nizltion ".port No.

Robert J. Christie WLT T.O. "03-110. Work Unit No.

9. 'erformin, Organizltion N.m••nd Addr..

W. L. Tanksley and Associates, Inc. 11. Contract or Grant No.16106 Snow Road NAS3-21900Brook Park, Ohio 4414213. Type of Report .nd .....iod Covered

12. Sponsoring Agency N.me and Address Contra.ctor R~portU.S. Department of EnergyDivision of Wind Ener~ Systems . ,- 14. Sponsorl", A,ency -!odrReport No.Washington, D. C. 205 5 DOE/NASA/1900-1

15. ~ai"I't.ry ~t.'pepa • repared under Interagency Agreement DE-AIOl-7t1ET20320. Project Manager,J. saVino~Wind and Sta.tionary Power Division, NASA Lewis Resea.rch Center, Cleveland,Ohio 441 5.

16. AbstractA method fpr analyzing the vibrations of horizontal axis wind turbines, developed by

R.W. Thresher, W.E. Holley, C.E. Smith, N.Ja.fa.rey, andS.R. LinofOregonSta.teUniversity, was used to analyze three guyed tower designs for intermediate size windturbines. The method, which uses a simple wind turbine structural model, was used toestimate the four lowest natural frequencies of vibra.tion of the three towers concepts.A parametric study was performed on ea.ch tower to determine the effect of varying

various tower properties such a.s the inertia'and .tufness properties of the towerand guys, the inertia values of the nacelle and rotor, and the rota.tional speed ofthe rotor. The results showed tha.t only the two lowest frequencies were in a rangewhere they could be excited by the rotor blade passing frequencies. The analysisalso showed tha.t these two frequencies could be tuned by va.rying the guy stiffnesses,the guy attachment point on the tower, the tower mass and stiffness, and the nacelle/rotor/power trainma.sses.

-

17. K.y Words (SuggtSttd by Author(s)) 11. Distribution Stat.m.ntWind turbine Unclassified - unlimitedVibration analysis STAR Category 44Tower designs DOE Category UC-60Guyed tower dynamics

19. Security a••if. (of this r.port) 20. Security CIa.if. (of this pa,,1 21. No. of 'ages I 22. ~ice·

Unclassified Unclassified

• For sale by the National Technical Information Service, Sprinifield, Viriinia 22161

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UNITED STATES DE'AIITMENT OF ENEIlGVP.O. BOX 62OAK RIDGE. TENNESSEE 37130

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