The free uncoupled vibrations of a uniformly rotating beam

Kluyverweg 1 2629 HS Delft

THE FREE UNCOUPLED VIBRATIONS OF A UNIFORMLY ROTATING BEAM

by

E. KOSKO

THE FREE UNCOUPLED VIBRATIONS OF A UNIFORMLY ROTATING BEAM

by

E.KOSKO

This review was originally prepared in connection with a rotating wing project at Avro Aircraft Limited. The author wishes to acknowledge the help received in the calculation of the numerical data and in the preparation of the graphs. and to thank the Company for per-mitting to publish the paper.

SUMMARY

The review derives and presents the equations which govern the free uncoupled flexural and torsional vibrations of an un-twisted beam rotating at uniform speed. Consideration is given to the effects of an elastic hinge at the root and of a concentrated mass at the tip. The variation due to these effects of the first three

natural frequencies of the root bending moments, torques and shear

TABLE OF CONTENTS

PAGE

NarATION iL

1. INTRODUCTION 1

2. GENERAL ASSUMPTIONS 1

3. DERIVATION OF THE EQUATIONS OF MOTION 2

3.1 Method of Investigation 3

3.2 Flapping Motion 3

3.3 In-P1ane Vibrations 8

3.4 F1exura1 Vibrations in a Plane Inclined to the P1ane

of Rotation 10

3.5 Torsiona1 Vibrations 11

4. APPLICATION TO SPECIAL CASES 13

4.1 Mass1ess Beam with Concentrated Mass at Tip 13 4.2 Rigid Beam with Free or Elastic Hinges 14

4.3 F1exib1e Chain 17

4.4 F1exura1 Vibrations of Uniform E1astic Beams 19 4.4. 1 F1exura1 Vibrations of Stationary B'eams 19

4.4. 2 Elastic Hinge at Root 20

4.4.3 Cantilever with Tip Mass 22

4.4.4 Beam Free1y Hinged at Root, with Tip Mass 23 4.4.5 Beam with Elastic Hinge at Root arid with

Tip Mass 23

4. 4.6 Effects of Rotation on Flexural Vibrations 24 4.5 Torsional Vibrations of a Uniform Elastic Beam 26

4. 5. 1 The Uniform Stationary Beam 27

4.5.2 Effects of Rotation on Torsional Vibrations 29

REFERENCES 31

A PPENDIX A 32

APPENDIX B 33

'."

I •

(ii) NOTATION

A rotating system of rectangular cartesian co-ordinates is introduced: the origin 0 is taken at the intersection of the elastic axis of the be am with the shaft axis; the shaft axis is taken as the z axis. positive upward; the x axis is directed along the elastic axis and performs a steady rotation; the y axis lies in the plane of rotation and points in the direction of the rota tion.

Symbols A (in. 2) e (in. ) E (lb. in. -2) G (lb. in. -2) I (in. 4) J (in. 4) k (in. ) K (lb.in./rad.) L (in. ) m (lb. in. -2sec . 2) M (lb.in.- lsec. 2) p (lb . .> P (lb. sec. 2) R (in. ) r q s (in. ) t (sec.) T

cross-sectional area of beam distance of hinge from shaft axis Young's modulus of beam material modulus of rigidity of beam material second moment of area of cross section torsional constant of cross section radius of gyration

hinge elastic stiffness length of beam

mass per unit length of beam, m = ~ A

total mass of beam centrifugal force

centrifugal force function,

f

=

pSl 2 tip radius

mass ratio (tip mass to be am mass),r = Mt/MB

Lagrangian co-ordinate

(in hinged beams ) co-ordinate counted from hinge. s = x - e

time

v

u (in. ) v (in. ) w (in.) d..

s-e

K

lt

l'

cv

(sec. -1) ~l. (sec. -1 ) Subscripts o 1 2 i p r t T prime dot • potential energy

dis placement in direction of x axis displacement in direction of y axis displacement in direction of z axis

frequency parameter (torsiona1 vibrations) coning angle (also, parameter in flexura1 vibrations).

lag angle

pitching angle of cross section; a1so flexural frequency parameter

ratio of flexibilities (hinge to beam) K = EI/KL Southwell's coefficient

torsional frequency parameter natura1 frequency of vibration

angular velocity of rotation

12

=

27f' x (rpm) 60

refers to stationary beam

(12

=

0)

refers to flexure in flapping (x - z) p1ane

refers to flexure in plane of rotation (x - y p1ane). refers to i-th normal mode of vibration

po1ar (i. e. with respect to x axis) refers to value at root

refers to va1ue at tip torsional

denotes derivative with respect to the x co-ordinate

denotes derivative with respect to time Other notations as::.defined in context.

.

w

'

.

( 1)

1. INTRODUCTION

The solution of the problem of free vibrations of a

rotating beam is of great importance for its applications to rotor dynamics, blade stressing. ground resonance, and similar questions. A survey of the literature has shown that much information is scattered through

various books and reports, with a surprising lack of uniformity in notation, methods of derivation, and presentation of results. Special effects, e. g. effects of elastic hinges, are either emphasized or ignored. depending on the purpose o~ the particular report.

The object of the present report is to provide a source of information on the subject and to enable the user to make his own calcu-lations where necessary. The basic equations governing flexural and torsional vibrations of uniformly rotating beams are derived from f~rst

principles. Several special cases among which uniform beams with tip mass and/ or root hinges are discussed at some length, and numerical results are presented.

The effects of interaction between flexure in the two planes and torsion have not been included; to cover these. and also the effects of aerodynamic or structural damping. would require much additional work.

The method of step-by- step tabular integration due to Myklestad and others, has been quite adequately described and discussed in the literature. and it did not seem useful to refer to it here.

2. GENERAL ASSUMPTIONS

The beam representing a rotor b~ade is assumed to be attached to a rigid shaft which revolves at a constant angular velocity

.Q .

In this analysis the following assumptions are made. subject to modifica-tion.

(1) No external forces act on the system; in particular, forces of gravity are neglected.

(2) The beam rotates in a plane normal to the shaft axis.

(3) Mass distribution and elasticity may vary along the beam length; in particular. aflapping and/ or a lag hinge may be included.

(4) One principal axis of inertia of the beam cross section coincides with

the plane of rotation - this excludes skewed hinges. (5) Elementary flexural theory is assumed valid.

(6) Effects of rotary flexural inertia and of shear deflections are

neglected (these effects may become important for the higher modes and for thin-walled box beams, respectively).

(7) For torsional deflections, the effect of longitudinal stresses is taken into account.

(8) No damping action is allowed for.

In a further report it is planned to discard assumption (2) by assuming that (due to either gravity or aerodynamic loads) a steady motion along a conical surface is possible; and assumption (4), allowing elastic coupling between in-plane and flapping motions.

The following limiting or special cases will be considered: - massless beam with concentrated mass at tip

- rigid beam with free or elastic hinges - perfectly flexible beam (chain)

- uniform cantilever beam

- uniform e lastic beam hinged at the root 3. DERIVATION OF THE EQUATIONS OF MOTION

In deriving the Lagrangian equations of motion of a rotating dynamic system, it must be borne in mind that the term "potential energy" loses its sense when the movement is referred to a moving system of

co-ordinates. (Ref. 1 - p. 38) This results from the fact that one of the co-ordinates, namely that which corresponds to a uniform rotation of the whole system, does not occur explicitly in the expression for the Lagrangian kinetic potential, and can therefore be eliminated from the equations,

thereby reducing the number of degrees of freedom. This reduction, however, is achieved at the cost of introducing a modified form of the kinetic potential, in which the separation between kinetic and potential energies has been lost. Such a system is called non-natural, meaning that the kinetic potential contains terms linear in the velocities, while for a natural system only quadratic terms are present.

The same basic difficulty is encountered when alternatively deriving the equations of motion by means of d'Alembert's principle from conditions of equilibrium. A rtifïcial forces, such as centrifugal and C oriolis forces must be introduced, in addition to the natural external forces. These additional forces are precisely the terms that result from differentiation of the linear parts of the kinetic potential.

In the vibration problem it is also necessary to distinguish between the more usual vibrations about an equilibrium configuration in which the system can remain permanently at rest, and vibrations about a steady motion. In the latter case the equations of motion contain so-called

'

'

..

.

"

(3)

gyroscopic terms, which make the coefficient matrix non-symmetrie; as aresult it is not possible to transform the system to normal co-ordinates. (Ref. 1, p. 83).

3. 1 Method of Investigation

For the various cases, the Lagrangian equations of motion are established by writing down expressions for the kinetic and potential energy of a beam element, and integrating over the beam length. The potential energy contains, in addition to elastic terms, expressions for the effect of centrifugal and coriolis forces. The assumption (approximately true) is made of separable normal modes of vibration, each having its

own natural frequency. The kinetic and potential energy expressions are then transformed in terms of normal co-ordinates qi and of their velocities

qi. The kinetic energy T is aquadratic homogeneous form in the

iii ,

and the potential energy V is quadratic homogeneous from in the qi ; (both positive definite). The Lagrangian equations of motion are then written as

( 1)

for each of the normal modes.

Assuming the tim~ variation to be harmonie, i. e. putting

~;

;. Ot

C-O:. w;

t

we obtain from (1) a differential equation whieh, together with appropriate end conditions, governs the shape of the normal modes. This shape is a so-called characteristic function of the differential equation, and each such function is associated with a characteristic number proportional to the square of the natural frequency.

The object of this report is to obtain, for the various modes of the rotating beam, the differential equations, the approximate shape of their normal modes, and the corresponding natural frequencies. 3.2 Flapping Motion

In the absence of external forces, we take the steady motion to be in a plane normal to the axis; the oscillatory motion takes place in the z direction, parallel to the axis. (Assumptions (.1) and (2) above). The effect of a steady coning ang1e, such as produced by the combination of gravity and aerodynamic lift on a rotor with vertical axis and flapping hinges will not be considered here.

Consider an element of length dx of a beam which rotates with angular velocity

S2

about the z axis, and at the same time

oscillates in the z direction. The elementary kinetic energy is

The total kinetic energy is obtained by integration

L l.

T :

i

Q

1

x.

lol11

cl x

+

~1

m

W

2

rJ

x •

D 0

(3)

In the course of displacing the beam from its stead-motion position to a deflected position w(:X), work must be done against elastic restraints

and against centrifugal forces. Both mean an increase in potential energy. The first part, the potential energy of bending, is expressed in the usual

way by .

cl

V

_rs

~

i

El

(~:~)Zdx.

The centrifugal force acting on the element is equal to:

-The displacement against which this force does work is equal to the shortening of the beam inboard portion in the x direc tion:

The second part of the potential energy is therefore

...IV:

--!.LldP

( A . < . - 2

=

~

rr

~YYL

x

dx{t~;

)l

dx.

The total potential energy is:

-The integral in the second term on the right-hand side of (4) can be integrated by parts; with mxdx = dp{x) ,

lar

if1frd~ ~

["

['(;;r

Jf{

T

!r('!i!:)'dx

J

( 4)

p(x) represents the resultant centrifugal force acting outboard of station x, divided by

SI..

2.)

'

..

rex)

(5) L ::.lmxd~. x. (5)

The first term vanishes at both limits, leaving only the second term. Thus,

(6)

(the sign

+

in the second term is due to the change of limits of integration in the expres sion (5) )

A ssume that the be am is vibrating in its i- th normal mode,

(7)

where qi is the motion of the tip, W i~) represents the shape of the mode, .

normalized (i. e. reduced) to unit tip displacement. In the following, only

one mode will be eonsidered, so that the subscript "i" may be dropped. The veloeity is

W

(x.,

#;-)

~

i·

w6:)

The kinetie energy becomes

L. 2

T ::.

~

12.1

YY\

[W(x)]

d~

Here q. sin (,IJ t is the veloeity appropriate to the generalized co-ordinate

qi·

We may write for

~-C

=

ti

l\n

[Wlx.>ydx.

.

drt

"0

The time derivative of this (the first term in Lagrange's equation). is

A.

()!

~

;;/\'.,

[W(x)]2. dx.

dl-

à

~

"

0

L

::. _

c~l~l

'VVl

[W

(x)

1

z.

dx

In harmonie motion, q

=

Q . cos (iJ t,

q

= -

fA) 2 Qeos fA) t

= -

co

2 q

where Q is the amplitude of the tip displacement, and (.t) the natural

The potential energy is written

(8)

The corresponding term in Lagrange IS equation is

The equation for the frequency thus becomes

L L ! fL

_w2.!

m

[W"(:~)l\jx

ti

E

I

[W"()t)l;\~x

+-nj Pc

x )

[W'lx)1d:x.

-=>0

o 0 ~

(9) If the function W(x) represents the true shape of the beam

in one of the normal modes, it is possible from Equation (9) to obtain the frequency of the vibration. It has been proved that if the W(x) is only an approximation to the true shape, the frequency obtained is accurate up to errors of the second order. (Rayleigh' IS principle). In these circumstances,

we may write

(Southwellls formula), where

tVw-

_{o is the frequency of the stationary}

beam, CU w that of the rotating beam. vibrating normal to the plane of rotation, and

A

is approximately a constant - (Southw~llls constant). In Ref. 4, bending frequencies of rotating beams have been estimated with sufficient accuracy, assuIIiing the shape of the stationary vibrating be am for the deflection function. If we want to determine that shape, a

differential equation may be preferable to the integral eiquation (9).

In order to derive such an equation, we sha11 have recourse to the minimal properties of the total (kinetic plus potential) energy (varia tiona l'~a,ppr0aGh) .

We assume that the deflected shape W(x) is varied by an arbitrary small amount

oW,

such as to satisfy the end condition at the root. The change in the kinetic energy wil! affect only the second term of (3):

.

'

,"

(7)

Variation in potentia1 energy (expression(6).)

(12)

Integrating the first term by parts:

jLEI

Wil.

d~(~w)

dx

=

[El

W'l

d(dw)IL -

[(EI~)',owJL

o

d~

cl";(,

1

0 , 0

l

+J

(E

I

w;täw

dx .

o

Integrating the second term of (12) by parts:

L

- fL

1

[p'ex)

'rf' -+-

reX) WH]

ow-

cl

z-o

The variationa1 equation of motion is obtained by minimizing the tota1 energy O(T

+

V) = 0:

I

Noting that, in view of the boundary conditions, all the terms in brackets vanish at both ends. the sum of the three integra ls must vanish. This can happen, for an arbitrary

d

w

,

on1y when the sum of the integrands vanishes, yie 1ding t'harequir.e-d differentia1 equation:

(13)

A ssuming again harmonie motion,

W'

(:x.,

t)

=-

Cl

coS

wt.

W"

(x) J

an ordinary diffe:rentia1 equation for the vibrating shape W(x) is obtained:

;;<L

[Elf

W"ex1 -

JL2

d~ [p~),W'c1.1- ~2.mW())

-=-0 ₍₁₄₎

3. 3 In- Plane Vibrations

When referred to fixed co-ordinates, in-plane vibrations differ from vibrations normal to the plane of rotation in the expression for the kinetic energy, resulting from the double product in the velocity squared:

(xf2.

-tv)

2. = ~~J22.

+2.xS2.V

+y'

v

being the velocity of vibration superimposed on the uniform rotation. In the uniformly rotating co-ordinate system adopted in this report the azimuth ang1e is considered as àn "ignorable" co-ordinate, and the resulting centrifugal term is added to the potential energy. The kinetic energy is thus written

°

fL

fL

2.

r=-~Q2

D

x

2

mdx

+1.

0

mv

d)(,

/

(15)

and the bending term in the potentia1 energy remains

(EI 2 shou1d be taken here for the bending stiffness).

The centrifuga1 force now is directed radially and has a y-component transverse to the beam axis:

in the same direction as the displacement v; the corresponding potentia1 energy term will be:

ei V' -:. -

!.

V"

elP.

0-: -lv2m.Sl.~rJ.)(..

( 2

'i

2.

The tota1 potentia1 energy can be now written (instead of (6) ):

(16)

Following the same argument as in the preceding section, we inay write the integra1 form of the frequency equation:

'.

(9)

This is the same relation as (9), with (

SL'

r

WZ )

substituted for the factor (

w2. )

which multiplies the first term. This being an exact relation, we see that assuming the same elasticity and the

same mode shape of in-plane vibration V(x) as for the normal vibration, . W(x), the frequencies of the two are related:

2.

2..n.

2

Ww = W_v

-+

₍₁₈₎

or, comparing with Southwell's relation (10) for normal vibrations,

W y 2.. =- W~ Z

+

(A _ , )

J2.

Z

(18a) Southwell's coefficient for in-plane vibrations is thus se en to be A-l, a re sult of great practical importance. The two mode shapes are not identical, however, but the error involved by that assumption has only a minor effect on the frequency.

The differential equation for the vibrating shape V(x) is obtained in the same way as previously:

there again, the difference with Eq. (13) is in the factor _(Q 2

+

CA) 2) replacing (-W 2) in the last term of the L. H. side.

(19)

It can be seen that for

.n.

=

0, both equations (13) and (18) reduce to the classical equation of the vibrating beam. On the other hand, for EI = 0, the equations reduce to those describing the vibrations of a chain rotating in a plane.

It should be noted that the centrifugal term p(x) being a function of the lower limit of an integral, the result of differentiating with ,respect to that liInit is obtained with a negative sign.

This case has been treated in Reference.8 by the

tabular method; the frequencies calculated there agree very closely with values obtained from formulae (10) and (18). The differences in mode shapes between stationary and rotating beam have been found to be negligible.

3.4 Flexural Vibrations in a Plane Inclined to the Plane of Rotation In many propeller blades, the major axis of inertia of the cross sections is inclined with respect to the plane of rotation. Assum-ing that the angle

1),,,

does not vary along the blade length (i. e. no twist in the blade), the flexural vibrations of lowest frequency will take place in the plane which contains the minor axis of inertia; this plane forms an

angle ['Tf/2.) - ~o with the plane of rotation.

Ref. 9 gives a derivation of an integro-differential equation which governs this type of vibration. This equation contains a non-linear term due to Coriolis accelerations. For small amplitudes of

oscillation. such as in the case of a propeller with blades encastré at the

hub. the effect of this term is negligible. The equation without the C oriolis

term can be reduced to the following differential equation (in our notation)

where Imin is the minor second moment of area of the cross section.

and

W

is the deflection measured in the plane of vibration.

This equation is identical with equation (19), except for

the second summand in the coefficient of the last term: . there.f2z- is

replaced by ..Q.z Si'fll~ ••

As aresult, the frequency j)J of vibration in the oblique

plane is given by

(18b)

where lI)w- is the frequency of vibration of the same rotating blade

mounted so that its major axis of inertia would lie in the plane of rotation

(and therefore vibrating in a plane normal to the plane of rotation). On

the other hand, for theblade mounted so that its major axis of inertia is normal

ta

the plane of rotation, we have

{h,

=

900, and equation (18b)

reduces to (18).

It is thus possible to generalize the concept of a South-well coefficient to vibrations in an inclined plane; this generalized

coefficient is given by

~18c)

Comparing Eqs. (19) and (l9a) we may further conclude that the mode shapes of the oblique vibration will be identical with those

of the blàdevibrating normally to the plane of rotation. as long as both

( 11)

3.5 Torsional Vibrations

When only torsional oscillations of small amplitude are considered, no coupling occurs between them and uniJorm rotation of the beam. Under these circumstances, the vibration modes and frequencies are not affected by the rotation. However, even with drcular cross

section, longihidinal displ~cements are associated with torsional deflections and may be responsible for an apparent increase in rigidity due to centri-fugal action. _ This effect is most pronounced in the case of a blade attached to the hub by means of flat straps which by.themselves havenegligible

torsional rigidity; yet when rotating, the joint is capable .of transmitting appreciable torques. An additional contribution may be due to the warping of cross sections out of the y-z plane, but this, together with bending-torsion coupling, is a non linear effect of higher order which will not be considered here.

Another effect is the so-called "propeller moment" which occurs when the major axis of the cross section includes an angle {). with the plane of rotation. It results from the fact that in the plane pf rotation, the centrifugal force has· a component nprmal to the beam axis.

The resultant torque about the x axis has a magnitude

cJ.Mx

=-SLl.plyzax

= -

~

SLr(l2.-lf)sL'n2-9.á:(,

and tends to reduce the angle ~ If the angle is due to the vibrational deflection, ~

=

e

(small), we may write

The blade is assumed to posse;ss a straight elastic axis which coincides with the axis joining the centroids of the cross sections . The effect of the two axes being distinct is of some importance in the consideration of flutter.

The kinetic energy of torsional vibration is

where lp

=

1

₁

+

_{12 is the polar second moment of area of the cross section} with respect to its centroid. The potential energy of elastic displacement (strain energy) is

l

de)'

V

r

=-~!

GJ

(dx.

dJl-

.

o

The P. E. of the centrifugal forces is equal to the

negative work done by these forces on the displacements; or, in this case, we may consider the centrifugal force acting

on

a fibre as'. rea.cted bYe a-

which is at a distance r from the torsional axis, has a helical slope rdQ/ dx. The elementary centrifugal force

.E..

dA

has an unbalanced

compon-ent A

~

de

~JA'

',~

A

Q

acting in the plane of the cross section at right angles to the radius vector r. The resultant restoriIig torque is

elk"

= - -:-

t

r2

,jA

~!

d",

~-1!

f

lp

cl"

The centrifugal force was. expressed (5) by

D z

2.l

L

lex.)

=

SI.

F(~) ':

J2

«n,.rdx.

The potential energy of the centrifugal forces is therefore

the amount contributed by the "propeller moment" is similarly

I

2jL

~

Yr

::'"T Q

0

f

(1:L -

I. ) ()

dx.

(note that in the latter expres sion the torsional displacement e itself, rather than the rate of twist del dx is introduced).

The Lagrangian equation of motion in integral form leads for harmonie vibrations to the frequency equation

If the exact shape of a torsional mode f) (x) is known, the natural frequency can be expressed as

2. Z.

A

()~

w ;:

Wo

r

T..Ilo J

(20)

(21)

where Wo is th.e frequency of a stationary beam assumed vibrating in the

same mode, while

A

r is a Southwell coefficient consisting of two parts: one due to centrifugal action and the other resulting from the gyroscopic "propeller moment". In the case of slender blades, this latter contribu-tion is negligible. Due to the change in mode shape with rotacontribu-tion the relation (21), as well as (10) and (18») is only an approximation, albeit a fairlv close one.

..

•

(13)

The corresponding differential equation for the modal function 9(x) is derived as in the previous cases:

i[GJ

~J

+

n~tx[t

P(X)*J -

n~~(Iz - 1

1 ) B(x)

+

w'2.~

Ir

9(x)= 0

(22)

4. APPLICA TION TO SPECIAL CASES

4.1 Massless Beam with Concentrated Mass at Tip

The mass Mt attached at the tip (x = L) of a massless inextensible beam. is assumed to have its centroid on the elastic axis of the beam. Three degrees of freedom are considered: one. in a dir-ection parallel to the axis of rotation (w-displacement); a second. in the plane of rotation (v-displacement); and a torsional one about the x-axis. Radial osc illations are not discussed here. as their natural frequency is easily shown to be much higher than that of flexural or torsional modes. The case of the massless beam. although without practical importance in itself. affords easily calculated reference values for more complicated cases.

Flapping Motion

If the funetion w(x. t)

=

W(x) coswt represents the shape taken by the centre line of the deflected beam oscillating in harmonic motion, the value of this function at the tip. q = w(L. t) may be taken as the co-ordinate which characterizes the motion. The single La.grangian equation of motion will be derived explicitly for this case. The kinetic energy is expressed as

T

=

~

Mt [(Lfi)'l.

+

ë\]

while the potential energy (including the apparent energy of the centrifugal

force P = MtL

n

2 ) is .

V =

t

KI

c(

+

MtLn.~ u(L)

The flexural rigidity Kl of the beam is the reciprocal of its flexibility under concentrated transverse tip load.

L

2-F

=

S

CL-Xl

dx

I 0 El,

To evaluate the last term of V it is necessary to obtain an estimate of the radial displacement u(L) of the centroid of the tip mass

(13a)

in terms of the modal displace.ment q. The distance from the axis of rotation to the centroid is shortened by an amount

L 'l

u

Cl.)

=

t

ç

(dw/dx:)

dx.

o

which may be written

where

C

is a non-dimensional coefficient.

The expression for the potential energy then beco.mes

v=

1(1<,

+

C Mt

n2

)a;

and Lagrange's equation can be written

_W2Mt

+

~C

+CMt

J1

2.= 0

from which the frequency of the oscillations is obtained

(23) This relation indicates that in our case of a single tip .mass Southwell's coefficient

A

is equal to the constant

C

which is deter.mined by the flexural contraction of the beam. The value of

C •

as that of Kl> obviously depends on the shape assumed by the median line of the bea.m. within the limitations of small-deflection theory.

At slow speeds of rotation. when the effect of the centri-fugal force on the shape of the elastic line can be neglected. we simply have a cantilever subjected to a transverse load Q at the tip. In the case of a uniform beam. EI

=

const .• the elastic line is given by

W

(x)

=

.Q.. .

.L

x.

Z

(3L-X)-=

t't-x.2.(3L--X)/L

3

ËI 6

the rigidity is Kl = 3EI/L3 and the coefficient ofaxial contraction has the value

C

=

6/5

=

1. 20. A slightly different assumption.

W(~)

=

q(1-cos"TT; x/2L) yields the value

C

=

rr

2 /8

=

1. 23.

i

Irt the limiting c,ase of zero flexural rigidity (as for a .mass-less chain or string with tip mass). the elastic line is straight. with a slope q/L. and the value of is 1. O. This also corresponds to the case of a rigid massless beam with a hinge at the root. being a li.miting case of the configuration discussed in Section 4. 2. One would expect this case also to represent the massless elastic bea.m for extremely large speeds of rotation. when elastic bending .moments become insignificant in comparison with those due to the centrifugal force.

(13b)

The actual shape of the elastic line is not difficult to cal-culate. The differential equation governing the deflection is found by equating the elastic bending moment acting on a cross section,

Eld 2W/dx2, to the bending moment of the two loads applied at the tip, a transverse load, . Q, and a radial load, P

=

MtL 2; the required equation is

EI

lW/J-:i'+

lJ('t-

W)

=

Q

CL-X.)

The initial conditions being those applicable to the cantilever bearn,

WCo)

=

"/(0)=

0

This is similar to the equation of a beam -column, except for the sign of the axial force.

In the case of a uniform bearn, the solution which makes

where the axial-load parameter 0< is defined by 0( 2

=

piEl. The relation between the tip amplitude and the transverse load Q (i. e. the flexibility) is givenby

a transeendental function of the dimensionless parameter 0(

L,

and hence of the centrifugal force p, The slope is found by differentiation,

dW/Jx.

= Q.

r\ -

cash

ol, (L--x.)] / I ?

L'

cos

h

0\ L

from which the flexural contraction is obtained as

u(

Ll

=

i(

*f

L

[~

(

I -

iL

ta

n

h

0(

0 -

~

ta

n

h

20(

Y

For smal! values of ~L, an approximation may be calcu-lated by expanding the function taV\h 0( L as a series of powers of the

argument, and retaining only terms of the lowest order. The first term alone yields a value of the flexibility

. and similarly for the axial contraction

L.l(L)

=

k.(~tL.

i%(O<L')4-

~

ct/L.

whence

C

=

6

I

5, Both these results agree with the expressions derived above for zero centrifugal force.

At the other end of the scale (very high speed of rotation, or very low rigidity), for large values of 0(

L ,

the function

'

t

a

1"\

h

0(,

L

(13c)

tends to unity. The rigidity becomes Kl

=

PiL = Mtn 2 , it is thus entirely due to the centrifugal force. The asymptotic value of the axial

contraction is then

2.

1. 1.

Ll(l')

=

~(~)

L(i-

k)=

t(-t1

L

=

i

~i'L

yielding in this limit for

C

the value 1. O. as anticipated. In-Plane Motion

As shown in Section 3 .. 3. Southwell's coefficient for in-plane vibrations is obtained from that for flapping vibrations by subtracting unity. The in-plane frequency is thus given by

W:=

k2./M

_t

+(c-l')n'l.

(23a)

where

C

is the constant which describes the axial contraction due to flexure. The effect of rotation on the frequency of in-plane vibrations is thus rather small. the value of (C - 1) lying between 0.2 and O. The rigidity K2 is defined as the reciprocal of the flexibility

F.

=

$

CL -

-xJ.

'2.

d

"X.

'2. 0 EI '2.

Torsional Osciallations

If the elasticity of the beam is such that torsional deflec-tion at the tip is not coupled with any displacement in the direcdeflec-tion of the beam axis. then the torsional frequency is not affected by the uniform rotation of the system. The relation

(23b) is then valid. independently of the value of

..n.

The symbol T prepresents the rnass moment of inertia of the tip body with respect to the x-axis.

The rigidity KT is the reciprocal of the torsional flexibility L

d-x.

F

T

=

S

~

Q

Usually. however. gyroscopic effects are responsible for a rotation-dependent frequency term. as in Equation (21). A discussion of some of these. such as "propeller moment" and "strap effect". is pre-sented in Sections 3.5 and 4.2. respectively, and is equally applicable here.

(14)

4.2 Rigid Beam with Free or Elastic Hinges

The dynamics of the rigid beam is a subject weU covered in the literature of rotorcraft, and the parameters governing its motion often serve as reference values for flexible blades. Thus, Coleman's analysis of the ground resonance phenomena is based on the assumption of hinged rigid blades. In theAppendix to this report, the values of Coleman's parameters wiU be derived for flexible blades.

L

Let MB be the total mass of the hinged blade =

f

m dx;

~

the hinge at station e is assumed to be rigidly conneèted with the rotating shaft. It is often convenient to measure distances from the hinge, rather

than fr om the shaft axis: S = x - e.

Let Sc be the distance of the blade center of mass C

from the hinge H; Ic

=

kc 2 MB the mass moment Of inertia with respect to a transverse axis through

€.

Then SH

=

Sc MB is the first moment of mass with respect to the hinge axis, and IH = Ic

+

sc2 MB = (kc2

+

sc2) MB

is the mass moment of inertia with respect to that axis. Further, let KH be the spring constant of the hinge, i. e. the ratio of hinge moment to angular deflection - the spring is assumed to have a linear charactevistic (KH independent of deflection magnitude).

Flapping Motion - The generalized co-ordinate

correspond-ing to this flapping motion is the flapping angle (3 • The kinetic energy can be written at once, for smaU deflections ~

T

=(

1/2) IH .

~

2

The potential energy of the spring deflection is

That of the centrifugal forces is calculated as foUows: elementary C. F.

its hinge moment

dM

=

5f3dP

=

.J2.2-mp

(ei's)sds

L-e .

total hinge moment

=/

(3nz-m.(e+s)sds

=f3.n.Ye.5J1+I,.,)

o

The resulting equation of motion is

For harmonie motion, this yields the frequency equation

2

Comparing this re sult, with Eq. (10), we see that KH/1-H = CV 0 ' and we

find for Southwell's coefficient

::. t

+

(25)

it exceeds unit by an amount which is proportional to the hinge offset e.

It is sometimes practical for quick calculations to refer

to two substitute masses which are dynamically equivalent to the given

blade. One mass is located at the hinge:

and has no effect on the blade dynamics; the other is located at the centre of percussion associated with the hinge:

SF ::.

IH

/S~ ~

Sc.

f-k:/s

c I and its magnitude is

Southwell's coefficient can be expressed as:

(26)

For a uniform mass distribution along the blade of length

sT

=

L-e, we have

Sc. ::.

'~J

L

-e)

I

kc

2

::.,i

(L-e.)',

s~=

:(L-e)j

hence (27)

A

=1+1

e

t+

3 _cf

2 ~-e; ï: 1-E

where E

=

e/L is the ratio of hinge offset to tip radius.

(16)

Uniform b1ade with tip mass Mt

=

'1-. MB' the mass of the b1ade proper being MB'

=

(l-t)MB; it is more c'onvenient here to use the parameter

We have

hence

and

r'

= ~ ::: ~_r __

Ma

l-T

s~

=Mt(L-e)

+iM;(L-e.) =

(T'+

~)M~ (L-e),

l~ :M~(L-e)1.+ ~ M~(L-e)1.=(T'+ ~) M~

CL-er,

s~

=

1

/5 -::.

r' + 1

f.l

(L -

e )

ti H T"-tt/2 '

'("+1/2 e.

A

= 1 -t Tt -t 1/3 ' L -

e

(28) The part dependent on the hinge offset is therefore

modified by the tip mass by the factor 2(ri

₊

_{1/2) /3(r'}

₊

_{1/3), tabu1ated}

be10w

TABLE 1

r'

=

0

I

0. 25 0.5 1₀. _{75 1.0 2.0 !4.0} CXJ

r

=

0 0. 20 0.333 0.428 0.5 0.667 0.75 1.0 factor

=

1.0 0.860 0.8 0.770 0.75 0.715 0.692 0.667

In- P1ane Oscillations

As se en in the general derivation (Sec. 3.3). the effect of the cehtrifuga1 forces is reduced when considered in the p1ane of rotation. The result is that the Southwell coefficient is reduced by unity (Eq. 18a). In the case of a hinged rigid beam, this means that the rotation-d~pendent

part of the squared frequency is a function of the hinge offset; we have

A :::

e

IS

F J (29)

and the data of the previous section concerning the uniform rigid beam with a tip mass are equally applicab1e here.

The physica1 meaning of this is fair1y obvious: if the lagging hinge is located on the axis of rotation, there is no means by which the b1ade cou1d participate in the shaft rotation, except for the spring in the hinge.

TorsionalOscjl\a,tions

sometimes in the form of a strap joint. If the joint is such that its length is unaffected by rotation about the x-axis, the natural frequency of torsional oscillations is the same for the rotating and for the stationary blade:

(30) L

where KT is the spring constant of the po.int, and IT = {

f

lp dx . is the mass moment of inertia of the blade With respect to x axis.

In the case of straps, the elastic restoring torque due to small deflections is neglig>ible;. however, the tensile loads due to

centri-fugal action produce arestoring couple which is proportional to

SJ..

2 and,

in first approximation, to the change G in pitch angle.

Assume a number of straps of length

t

.

with axes

initially parallel to the x axis at a distance a from it. When the rigid

blade is rotated by an angle G, the straps are deflected by an angle G . a/f, . The centrifugal force acting on the blade is P

=

MB Xc

.n..

2 , where Xc

=

distance of centre of mass of the blade from the axis of rotation; it has

unbalanced components (1/ n)P . O . a/! at each strap, which produce a

restoring torque about the x axis:

The squared frequency of torsional oscillations is obtained by dividing the apparent stiffness,

à

_{M x / () G / by the moment of inertia:}

W

r'

=

Msxc..

_Lr

G\.'

_l.1L,

('")2

or

( 31)

where

lvy

is the radius of gyration of the blade about the x axis. Usually, a and kT are of the same order of magnitude, while Xc is much larger

than

t ;

therefore, the frequency of torsional vibrations will be higher

than the rotor speed. 4.3 Flexible Chain

The problem of the vibrations of a rotating flexible uniform chain is treated in many textbooks on Dyp,amics or Differential Equations. Like most other limiting cases, its importance consists in yielding bounds for quantities which in actual cases are difficult to calculate.

'

,

(18)

For a uniform.mass distribution m, the function defining the centrifugal force is (see Eq. 5):

f

ex) -= [ L»'l X

d

x

~

T

yy\

(L

1.

-x}-)

d

x.

.

x

Vibrations Normal to Plane of Rotation

The equation of motion to consider is Eq. (14), from which the elastic term is removed:

..n.

z.

1x [

pl-x.)

W'tx.)]

+ wLyy\

WC-x)

-=

0

Substituting for p(x), introducing the ratios ~

=

x/L and

and di viding throughout by m, we obtain

(32)

This is Legendre's differential equation; solutions of the homogeneous

equation above (with zero right-hand side) exist only for certain values of the parameter À . These characteristic values determine the mode shapes which are described by polynomials in

S

•

the so-called Legendre polynomials. For our root condition W(a)

=

O. only odd degree poly-nomials are applicable. - The low·est characteristic values are

À = 1, 6. 15. 28 -- general expression À = n(2n-l)

These values are none other than Southwell coefficients for the rotating

chain; they give an estimate of the effect of rotation on the vibration frequencies of uniform beams.

Vibrations in the Plane of Rotation

The equation applicable here is Eq. (19), without the elastic term;

(33)

If the parameter

(lAl

/12.)2

+

1

=..t'

is used here, we obtain again a Legendre equation hav'ing characteristic va lues

.t'

= J, 6, etc. This confirms our earlier finding that the Southwell coefficient for in-plane

vibration is equal to that for normal vibration minus one. In this case, the mode shapes in the two planes are exactly the same.

Torsional Vibrations

In the çase of a chain, torsional vibrations do not appear to have a clear physical meaning.

(19)

4.4 Flexural Vibrationsof Uniform Elastic Beams 4.4.1 Flexural Vibrations of Stationary Beams

The equations of motion (14) and (19) fqr the rotating beam do not have any solutions in terms of known functions. It has been found practical (Ref. 4) to base approximate calculations on the assumption of the same mode shape as for a stationary beam. For that reason. we shall pass in review the results of calculations pertaining to stationary uniform beams, and we shall obtain some new results relating to such beams having an elastic hinge at the root and/or a concentrated mass at the tip.

The vibrations of a stationary uniform beam are governed by the differentia 1 equation

cl d'lW

-w

2

mW

₌₀

t.:

d

x.4 J (34)

which may be obtained from (14) or (19) by putting

12.

= 0, EI = consL The general solution is

w(:~) =A[os~x+B$[,~fSx.

+CCQSh(3:x.

+

DSiYlhf3X.,

(35)

where

p.,::

ti

In

w2./E [

,

and A J B, C, D are arbitrary constants

to be determined from the dis placement or load conditions at the ends of the beam. Non-zero solutions are possible only when

f3

assumes certain values, the characteristic values of the problem; these are the roots of a transcendental equation, the frequency equation, obtained by eliminating the arbitrary constants from the end conditions. The quantity

(3

.

always occurs in the form of the non-dimensio.nal parameter

f3L

=

B

,

the frequency parameter.

The frequency is then obtained as:

w

=

a

z

I

El

L2.Vh'l

(36)

2 ,

-H we substitute fA for m. k for I/A, and set

vElr

=c

(velocity of propagation of compression waves in the beam material)

we see that the frequency is

G4)

=

e~

t

L

(36a)

equal to the product of the square of the non-dimen&ional frequency parameter

e,

of a slenderess ratio (kiL) and of the velocity of sound

c

divided by the beam length L.

As in other instances of free vibrations, the amplitudes are indeterminate as to magnitude, only their ratios can be obtained. Usually,

(20)

in the case of beams having a free end, the reference value is taken at the tip, i. e. tip amplitude

=

1. The tabulation in Ref. 6 however, ,sets the tip amplitude- 2; some other conventions can also be encountered in the

literature ..

Two standard cases of interest here have been fully tabu~ated

as to mode shapes, and their derivatives for the first five modes (Ref. 6). These are the "c lamped-free" (i. e. cantilever) and the "supported-free"

(i. e. hinged at the root) beams. The main results are tabulated below for

easy reference. (first three modes only). · The table gives, apart from

the frequency parameter, the root bending moment and root shear force

for the cantilever , and the root slope and root shear force for the hinged beam, all in non-dimensional form. (Tabie 2).

4.4. 2 Elastic Hinge at Root

The conditions at the root (x

=

0) are:·

w

(0)

=

A

+

C ::

O

.

(a)

TABLE 2

FREE VIBRATIONS Of A UNIFORM BEAM ST ANDARD CASES

(A) CANTILEVER(Clamped-Free)

MODE FUNDAMENTAL 2ND 3RD

Frequency Parameter

e

=~L 1. 875 4. 694 7.855

Frequency

oo.L

'l../

m. Co

el.

t:l

3.516 22.0345 61. 697

*Bending Moment at Root 3.516 22.0345 61. 697

M(o).L 2./E[

*Shear Force at Root 4.839 105.35 484.30 _i

(21) (B) HINGED (Supported-Free) MODE PENDULUM 1ST 2ND 3RD' Frequency Parameter

e

=

(3

L 0 3.927 7.0685 10.210 , Frequency

WL2V

Wl

:Sz

ET

0 15.418 49.965 104.248 *S1ope at Root y' (0) . L 1 2.700 5.005 7.223

*

Shear at Root V(o)

--

L 3 0 44.013 249.52 752.5

EI

*

for unit deflection at tip.

If K is the spring constant of the hinge. the bending moment. EI • W" (0) must be equa1 to the s10pe, W' (o)j multiplied by K:

EI· W"(

0) =

k·

W'

(o) ,

or

(b)

We shall de fine a non-dimensiona1 root constraint coefficient. K = EI/KL

which becomes zero for a built-in cantilever, and infinite for a free hinge. The condition (b) then beca.'mes:'

(b ')

The conditions at the tip are those of zero bending moment and zero shear force.

W"(L)

=

-

A cos

e -

B sin G

+

C cosh G

+

'

D sinh

e

=

0 Wil! (L) = A sin G - B cos G

+

C sinh

e

+

D cosh

e

= 0

(c)

(d)

Elimination of the constants A.

:a,

C. D from the above

four re1ations yie1ds the frequency equation

1

+

cos

e

cosh

e

=

KG

(cosh

e

sin

e -

sinh

e

cos ~-) (37)

The direct s01ution ç'f this transcendenta1 equation is tedious; it is

preferab1e to assume a series of values for

e

and determine

(22)

!fhe corresponding plots are shown in Fig. 1, 2 &3.

The bending moment at the hinge is given, for unit tip deflection amplitude .. by

M

(0)

=

E"L

cJ"LW

ol.

)CL

x ... O The shear force at the hinge is

v

(D)

=

f

1 ei

3

~

=

EL~

El

olx

) ( : 0

S ~l'l

h

G

ft).,

e

St"t,/tt

e

+

Sl\'

e

4.4.3 Cantilever with Tip Mass

To determine constants in Eq. (35)';

Conditions at root: W(o) = W'(o} = 0, or A

+

C = B

+

D = 0 Conditions at tip: Wil (L)

=

0 (zero bending moment) .or

-A cos G - Bsin G

+

C cosh G

+

D sinh e = 0 (a, b) (c)

Shear force just inboard of tip mass is equal to the inertia force of the tip mass:

or, putting r

=

MtJmL

=

ratio of tip mass to mass of beam. and remembering that W 2m/EI

=

(34 , thiscondition may be written: A sin e -Bcos G

+

C sinh e

+

D cosh G

+

re (A cos e

+

B sin e

+

C cosh e

+

D sinh e) = 0 (d) Elimination of the constants from the four conditions (a) - (d) yields the frequency equation

1

+

cos G . cosh e

=

re (cosh e . sin e - sinh e . cos e)

This is the same as the frequency equation (37), for the beam witb elastic hinge and therefore, for the same numerical values, "-:.K, the roots of the equation, Le. the characteristic !lalues. wiU be the same.

Ther~ is a close relationship between the deflection functions and their derivatives for these two problems, when the characteristic values are the same: counting the x abscissa from the

" \

root in 0lle case, and from the tip in the other. the deflec'tion line in one problem repre.sents the bending moment distribution in the other, and

vice-versa; and the slope variation in one problem is the same as the shear force diagram in the other, and vice-versa (up to a multiplying factor).

The bending moment at the root is, for unit tip deflection, (see Fig. 4).

The shear force at the root is

'-J

(0) ==-

g

e~

L9

o-.-.l9

+~~

~l..9~f)- ~~Q

uy::>f)

Fig. 5

~ke-+ ~()_

~--::J,,-

()

~

() -

Uh()

~~

()

4. 4. 4 Beam Freely Hinged at Root, with Tip Mass

The root conditions are, for x

=

0, zero deflection and zero bending moment, i. e.

W(o)

=

W"(o)

=

0, hence A

=

C

=

0 (a, b)

Conditions at the tip are the same as for the cantilever with tip mass, i. e. -A cos 9 - B sin E>

+

C cosh 9

+

D sinh 9

=

0 (c) A sin G - B cos 9

+

C sinh 9

+

D cosh 9

+

rg (A cos 9

+

B sin 9

+

C cosh 9

+

D sinh 9 ) = 0 The frequency equation which results from eliminating the arbitrary constants, is

2 r G

=

cot 9 - coth 9

(d)

For zero tip mass, this is seen to reduce to the equation for the standard "supported-free" case: cot Q = coth g.

Characteristic values, with corresponding she·ar forces at the root are plotted on Figs. 6 and 7.

The shear force at the hinge, for unit tip deflection, is obtained from

4. 4. 5 Beam with Elastic Hinge at Root and with Tip Mass In this case the root conditions are

(24)

EI W" (0)

=

K . W' (0), i. e. (-A

+

C) h~Y

;;;;

(B

+

D). ₎ (b )

those at the tip (x = L) are

W"(L) = 0, i. e. -A cos G -B sin 8

+

C cosh G

+

D sinh 8

=

0 I (c) A sin 8 = B cos 8

+

C sinh 8

+

D cosh G

+

r8 (A cos 8

+

B sin 8

+

C cosh 8

+

D sinh 0) = O. (d) Elimination of the·arbitrary constants yields the frequency equation:

2 \<r 8 2 sin 8 . sinh G

+

(K

+

r) G (cosh 8 sin 8 -sinh G cos G)

= cosh G . cos G + 1 (39) This equation cannot be solved as easily as the previous. ones, as it

involves two parameters, K and r, in addition to the unknown G.

4.4.6 Effects of Rotation on Flexural Vibrations

For practical applications, the effect of centrifugal forces on the naturalfrequencies of beams is contained in the value of Southwell's coefficient,A According to Eq. (9) and (10) this coefficient may be

written as:

. (40)

for viJ:>rations normal to the plane of rotation; and according to Eq. (18a), the coefficient for in-plane vibrations is equal to

A -

1.

Herein, W (x) is to bE: taken as a mode shape which satisfies the differential equation (14), together with appropriate end

conditions. Usually, however, this mode shape is asmuch of an unknown as the frequency and aS the coefficient

A.

Fortunately a small error in assuming the mode shape still yields a fairly good estimate for tl1e

frequency. On this basis, charts have been presented in Ref. 4. for the rapid estimation of bençling frequencies of Fotating heams with various linear mass and stiffness distributions. also of uniform be'ams wHh tip

mass. •

For a uniform beam, and for the mode shape of the non-rotating beam defined by Eq. (35), the integrals of (40) can be calculated exactly. Some results are quoted below for the first three modes.

Results for higher modes can be expectetl to be much less accurate, due to neglect of rotational inertia and of shear deformations which for those modes become relatively more important.

parameter appropriate to the mode.

11

Ol

=

6.38;

/l

02

=

17.90;

A

₀₃

=

36.00 (in the "pendu1um mode", . we have

Aoo

=

1. 0).

The hinge offset correction factor also varies with the mode; it is defined by

( 41)

The resulting Southwell coefficient is th en

.. (42) (b) Cantilever beam: Calculations are somewhat lengthy, the results being:

.Jl

1 = 1. 193 ;

Jl

2 = 6. 4 78;./h = 17·. 86

(c) Hinged Beam with tip mass: The effect of a tip mass on Southwe11f's coefficient is very marked. According to Ref. 4, Fig. 20, the va1ues for mass ratios r = Mt/mL are approximately

r

=

0 0.5 1 2

110i 6.4 10.3 15.0 24.8

11

02 17.8 35 55 94

A03 36 75 119 208

(d) Cantilever Beam with tip mass (from Ref. 4, Fig .. 13)

l' :. _{0 O.$"}

_t

Al

1. 193 1. 196 1. 1975 /\2 6.5 11. 2 16.8

(26)

(er Effect of E lastic Root Hinge: C omparing results for free1y-hinged and cantilever beams (the fundamental cantilever mo.de corresponding to the pendu1um mode of the hinged beam, the 2nd cantilever mode to the lst mode of the hinged beam, etc.

L

we see that the elastic condition at the

root has on1y a very small effect on the Southwell coeff~cient. If necessary, va1ues for e1astic root hinge cou1d be interpo1ated between the two extreme cases above.

A lternative1y, it has been suggested in ReL 4 to de fine a new Southwell coefficient by referring its stationary frequency in the i-th mode, ~ io' to the fundamental of the same beam with its root fixed

wIF~

(43)

The usua1 Southwell coefficient,

Ai

'

and W io are for the beam with its actua1 rootcondition. The new coefficient,

A

i'. is insensitive to mass distribution, and is usefu1 in estimating bending frequencies for beams with similar stiffness distribution.

It can a1so be seen that the va1ues of the Southwell

coefficients above do not differ too muchfromthosé fur the flexib1e chain, Section 4. 3.

The assumption which takes for the rotating beam the mode shape of the stationary beam, although giving satisfactory results

in most cases, should not be applied without discrimination. A

comparison of the mode shapes has been carried out in Ref. 4, and it appears that for cantilever beams, the shape may vary quite appreciab1y .

with rotationa1 speed. The effect of a tip mass is a1so more pronounced at higher rotor speeds.

A consequence of the variation in mode shape with

rotation is that Southwell's equation (10) does not ho1d exactly, or in other words. the coefficient

A

is not a constant" but depends slightly on the

angu1ar velocity of the beam. .

4.5 Torsiona1 Vibrations of a Uniform E1astic Beam

- For a uniform beam. we may note that in Eq. (22) p(x) = (1/2)m (L2 .- x 2); the differentïal equation becomes

The second-order derivative is seen to have a variab1e coefficient which makes it probab1y impossible to obtain a solution in terms of known functions. It will be easier to obtain approximations to the natural frequency by assuming the same mode shapes for the rotating beams as for the stationary beam, and by app1ying Ray1eigh's method (Eq. 20).

4.5.1 The Uniform Stationary Heam

The differential equation reduces to:

419"

+w

'

oz

_f

lf9

=0.

.

(4'5)

It is convenient to set GJ

I

(f

lp)= cT2, where cT is the velocity of propagation of a torsiona1 impu1se a10ng the beam. The

general solution of (45) is of the form

e

=

A

StY\ <xx

+

B

Cos o{~ (46)

where (X, -:..

wo/cr ,

and A, Bare arbitrary constants to be determined

from the end conditions; The torque is given by T = GJ de / dx.

(a) Fixed-free beam: At the root x = 0. no torsional displacement can take p1ace, i. e. e (0) = 0; we must have B = O. At the free tip, the

torque must vanish; the condition is therefore A cos ex

L

,

=

O. Non-zero va1ues of A are possib1e only when cos ~ L

=

0; introducing the

non-dimensional frequency parameter

rp

=

odJ,

we obtain the characteristic va1ues

31i ST,

2.

'-2-'

The natura!' frequencies corresponding to, these are:

, CT ~

I

~J

'

w=Y;T=-t-Vfl

_p • ( 47)

We may note that the velocity CT can be represented as product of the velocity of propagation of 10ngit~dina1 'wa ves c

=

IE'

lp

(a true material constant), a numerical factor 1

1";2

(I..,.~)

=

0.62

( Y being Poisson's ratio. taken equal to 0.3). and a form factor F

--:. J

J

I

lp

I which depends on the shape of the cross section. For fuU

circu1ar or annular shapes. F = 1; for more e10ngated shapes, it is usually 1ess than unity. The torsiona1 frequency thus becomes

CA)

=

r'

O.62P

l.

(47a)

an expression to be compared with that for the. flexura1 frequency, Eq. (36). The slenderness kiL which occurs as a factor in the

expression for the flexural frequency is missing here. This is the main reason why torsiona1 frequencies are considerably'lmver than flexura1 frequencies of uniform beams.

The torsiona~ vibration modes of the non-rotating be am are described by the functions

e

'

7TX

. =: St'Yl - - I

, 2.L S~Y1

.

31ïx

2.L

I

(28)

(b) Elastic hinge at root: Let Ko (in. lb. Irad.) be the stiffness constant for the hinge. The condition at the root is: torque

=

Ko x rotation, or GJG' (0)

=

Ko . 0(0). Setting the relative hinge flexibility GJ/KoL

=

K , this yields a relation between the constants .

At the free tip the condition Gi(L» = 0 yields a second relation,

Non-zero values for A and B can exist only if the following frequency equation (obtained by eliminating A and B from these two relations) is satisfied

(b)

The roots of this equation are easy to obtain; they determine the~atural

frequency by means of Eq. (47).

Fot' zero flexibility, K

=

0, we have the case of the fixed root, just discussed. For a very feeble constraint, K - 00 , the beam

approximates a free-free condition, the roots of (c) being lf::. 0, 'JI, 2;;,---; the value

<;

= 0 represents a rigid-body motion with zero frequency.

The deflection amplitude at the tip is 0(L)

=

AI

sin" for unit tip amplitude, the root torque is

Frequency parameters and non-dimensional root torques have been plotted against the flexibility K in Fig. 8 &9 \

(c) Tip inertia : The effect of a concentrated mass at 'the tip having a mass moment of inertia ~

Ir

with respect to the x axis, when oscilla~.ing with an amplitude G(L),' is to produce ju st inboard a torque 'Of amplitude

l4J~IT

On)

I which is balanced by the élastic torque GJG'(L') .

. Setting the ratio of moments o~ inertia of tip mass to remaining blade equal to

and noting that 0<. » the tip condition may be written

rr

(A

S{yt

r

+

8

WH!) =

A

cos

r -

B

S1V,

j'

Combining this with the condition for a fixed root, B = 0, yields the frequency equation: