AER-5G-3

Some Studies on the F lu tte r of A irfoils an d P ropellers

By W. HAROLD TAYLOR,1 ANN ARBOR, MICH.

During the last few years there has been considerable

interest in the study of vibration of airfoils and propellers,

both in Europe and in this country, and most of the litera­

ture regarding it is of fairly recent date. A complete solu­

tion to the problem has not yet been attained, the com pli­

cated m athem atics having proved too involved. Again,

some of the factors are only now the subject of research.

This paper deals w ith a few of the m any interesting phases

of the problem. Because of the complexity of the m a the ­

matical expressions for the airfoils in com m on use, a flat

plate has been used. I t has also been assumed th a t the

angle of attack in the range to be covered is below the

burble po int and th a t the am plitude o f the vibrations is

considered small. The paper is divided in to three m a in

parts: (a) A theoretical development of the deflection

curve of a cantilever flat bar of un iform cross-section, by

air forces distributed according to an elliptic-load grading

curve, to show the effect of the warping produced by such

loading in the strain energy of the bar so loaded, (b)

An approximate solution of the free torsional vibrations of

a cantilever bar of th in rectangular cross-section, (c)

An analysis of the problem of self-induced torsional vibra­

tions which w ill apply for any airfoil.

1-

D- In t r o d u c t io n a n d S t a t e m e n t or

P r o b l e m

kURING the last few years con­siderable interest has been at­

tached to a study of vibration of airfoils and airplane propellers. Sev­eral severe accidents occurring prior to 1925 led such investigators as Younger in the United States, Fraser and Cox in England, and Blenk and Liebers and Kussner in Germany to attempt solutions.A general solution has not yet been de­

rived; so far the mathematics has proved too involved. This contribution deals with a few of the many interesting phases of the problem.

The work of Younger gives an approximate solution of vi­

brations of two degrees of freedom as applied to propellers. The differential equations have been solved by means of graphical integrations. His solution, however, does not take into account the aerodynamic couples which exist on the airfoil.

Fraser investigated the torsional and bending vibrations of a

1 Department of Aeronautical Engineering, University of Michigan. Mem. A.S.M.E. Mr. Taylor received the degrees of B.Sc. from Mc­Gill University in 1915 and Sc.D. from University of Michigan in1933. He was chief designer of the Chisholm Moore division of the Columbus-McKinnon Chain Company. He was instructor in mathematics in the University of Buffalo, and was assistant to the director of the 1933 summer school at the University of Buffalo.

Contributed by the Aeronautic Division and presented at the Semi- Annual Meeting, Chicago, 111., June 26 to July 1, 1933, of T h e

A m e r i c a n S o c ie t y o f M e c h a n i c a l E n g i n e e r s .

N o t e : Statements and opinions advanced in papers are to be understood as individual expressions of their authors, and not those of the Society.

wing with spars, and also with various settings of ailerons or flaps. Since that time it has been shown by Younger that ailerons or flaps are not completely responsible for the vibration of air­foils. Cox investigated the torsional properties of the stripped airplane wing.

Liebers2 calls attention to the recent propeller failures ascrib- able to vibration troubles. He attributes one of the main causes to the periodic changes in the loading curves due to interrup­tions and disturbances by such things as wings, other propellers, or disturbed inflow air. Again3 in another paper the same author states that the aerodynamic forces are of less importance for the bending vibrations of aircraft propellers than for the torsional vibrations. He mentions the great difference in frequency at which bending and torsional vibration exist and suggests that each can exist without involving the other. Blenk and Liebers4 in their papers show the equations set up for the torsion of the airfoil and give in more detail practically the same information as in the two papers by Liebers. Kussner calls attention to the vibration of wings from the point of view of the two-spar wing. He shows that the manner in which ailerons or flaps are attached determines the manner in which monoplane wings will vibrate.

It will be the endeavor of this paper to show (1) a theoretical development of the deflection curve of a cantilever flat bar of uniform cross-section by air forces distributed according to an elliptic load-grading curve and to show the effect of the warping produced by such loading in the expression for the strain energy of the bar so loaded; (2) to give an approximate solution of the torsional vibrations of such bar; (3) to show that induced torsional damping exists in an airfoil and to indicate the effect of self-induced torsional vibrations on the bending of the bar.

With respect to the first section of this paper, the method used is suggested by Liebers’ papers, but the evaluation of the warp­ing due to aerodynamic forces is original. The second point covered contains nothing new and is here inserted for com­parison purposes. The third point covered is original and, to the best of the author’s knowledge, has not appeared elsewhere.

Restrictions. The airfoil sections commonly used are of con­ventional type, such as Clark Y, N.A.C.A. M6, RAF 15, and Gottingen 387, or are modifications of these. A mathematical development of these sections is highly complicated; conse­quently a thin flat plate was chosen for investigation because of the known factors applying to such a plate. Further, we will investigate for vibrations of small amplitude only, and in order to keep the problem at or near some practical value, the aspect ratio of length of airfoil to chord has been chosen with 6 as its

value.

2— N o t a t io n s a n d S y m b o l s

The axis system as specified for N.A.C.A. reports and the symbols employed by Timoshenko for the study of vibration have been used. The following notations have been employed:

* F. Liebers, “Zur Theorie der Luftschraubenschwingungen,” Zeitschrift fiir technische Physik, vol. X , 1929, pp. 361—369.

3 F. Liebers, “Resonanzschwingungen von Luftschrauben,” Luftfahrtforschung, May 16, 1930, pp. 137-152.

4 See Bibliography.

57

58 TRANSACTIONS OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS

area of airfoil under consideration, sq ft constantconstant coefficients

angle of incidence, radianssmall change in angle of incidence, radiansangle of incidence at root, radians

angle of incidence due to vibration, 2 radiansangle of warp at point y — Ibreadth of plate, in.

constant coefficientvalue of Lc at y = 0slope of lift curve for infinite aspect ratio = 2ir

torsional rigidity factordistance from the fixed end to point of appli­cation of load P, in.

thickness of plate, in. chord, ftcoefficient of drag

coefficient of induced dragcoefficient of liftplan form ratio = l/b = l/cidrag, lb per sq in. of spaninduced drag factor = 0.096 fori? = 12Young’s modulus of elasticity, lb per sq in.frequency of vibration, cycles per secmodulus of rigidity in shear, lb per sq in.acceleration due to gravity = 386 in. per sec2moment of inertia of cross-sectionconstantlift coefficient at point y lift coefficient at point y at angle of incidence a change of Le accompanying change of in­

cidence Aa lift, lb per in. length of airfoil, in. torque or twisting moment, lb-in. torque or twisting moment, about the posi­

tion Ci/2, lb-in.■ slope of lift curve for aspect ratio being in­vestigated; in this paper R = 6

= concentrated load, lb 2 a-/

mass density of air, slugs per cu ft weight of bar per unit of length, lb per in. intensity of distributed loading at point c on the bar, lb per in.

aspect ratiodistance from center of rotation to center of pressure, in.

■■ kinetic energy, lb-in. time, seccoefficient of induced angular change rectangular wings strain energy, lb-in. velocity, ft per secvelocity at infinity parallel to x axis, ft per sec

or in. per secrelative velocity, ft per sec angular velocity, radians per sec

3— C o n c e p t o p F o e c e s A c t in g

Let us consider a flat bar ABCD, the center line of which lies in the 7/-axis of a right-handed system of axes. (See Fig. 1.) Its chord or width CD is Ci. Our bar has an angle of attack of a„ deg to a wind of velocity v. The lift developed is considered

for

to be quarter ellipse EFGH, with maximum lift at the built-in end and zero at the tip.

The center of pressure of any cross-section at angle of attack a0 is represented by the line EF. Due to the lift developed, we can think of the airfoil bending to some position ABC,DU but the eccentricity of the load produces a torque which twists the air­foil to, say, ABC2D2. Now there has been a change of angle of attack at the tip, reducing to zero at the root as indicated by the angle CJC2. This change increases the lift forces along the bar and deflects it to, say, A B C 3D 3 , where the torque so built up might be considered to have further twristed it. This process can be considered to continue until the bar finally takes a position of equilibrium of, say, ABC Jit, where the angle CKCt will repre­sent the total twist angle a2 at the tip. Now of course our lift curve has also encountered some change of shape, such that on the straight bar it might be represented by EFGJI. We must remember that the increment GGi is small because the angle CKCt is by hypothesis considered a small angle.

The lift at any point is given by

Now let Lc receive a small increase ALc such that for practical purposes the dotted figure may still be considered an ellipse. This small increase accompanies a change of angle of incidence Aa, as explained in the opening paragraphs of this section. It is to be noted that at the built-in end the change of angle of incidence due to lift forces is zero and that the value of Aa will increase from zero at the built-in end to a maximum at the tip. The value of Lc will remain the same value at the built-in end. At the tip the value of ALc must also be zero. So that

The slope of the lift curve for angles of incidence below the burble6 point may theoretically be considered a constant of slope 2ir; nevertheless such value is not attained due to the aspect ratio being finite. Glauert6 gives the following coefficient as the correction of the slope of the lift curve for aspect ratio:

Again, because, in the wind tunnel, viscosity affects the circula­tion about the airfoil, we cannot hope to attain this value. We may introduce a factor k to allow for that portion of the slope of the lift curve which may be attained, so that

5 The burble point occurs at that angle of incidence where the vortices in the wake of the airfoil have approached the trailing edge of the airfoil and are about to start rolling on the upper surface of the airfoil, or at that angle of attack where streamlined flow over the top of the airfoil ceases. See Ewald Poschl and Prandtl, “Physics of Solids and Fluids,” Blackie, pp. 321-322.

6 “Airfoil and Airscrew Theory,” H. Glauert, Cambridge Press.

Since the coefficient CL contains all the variables for a given airfoil and velocity, wre may deal with the coefficient, and in order to avoid confusion let us call CL the lift coefficient at any point y.

Referring to Fig. 2, from analytic geometry we have:

AERONAUTICAL ENGINEERING AER-56-3 59

would represent the slope of the attainable lift curve at the particular aspect ratio under consideration. In this expression k would have a value of from 0.85 to 0.90. Then

4— T o r q u e

The torque on the element with reference to the center of gravity of the rectangular section is given by

The distance from the leading edge to the center of pressure has been tabulated by various writers, and Eiffel’s7 results for aspect ratio 6 have been used herein. Working with values belowr 10 deg incidence, an empirical8 equation has been found which fits the data wyith a very close approximation—in fact, well within the experimental error of the data—so that the equa-

It will be noticed that Equation [4] is of two parts; the first part may be interpreted as the torque producing statical torque, while the second part containing Ac* may be interpreted as the

torque produced and main­tained by warping.

As was noted previously, Aa is not constant, but varies along the length. As it is a function of y, it must satisfy the end conditions:

(Aa), = o = 0 and(Aa), = i = maximum

A suitable expression for

Aa would be:

where a is expressed in degrees. The torque may then be written:

7 G. Eiffel, “Nouvelles Recherches sur la Resistance de l ’air et L’Aviation,” 1914.

8 See methods shown in Prof T. R. Running’s book, “Empirical Equations,” John Wiley & Sons.

the end conditions being satis­fied thereby.

The torque due to warp is then given by the expression :9

5— T r a n s v e r s e B e n d i n g

In this place in our problem we propose to express the de­flection of a cantilever bar, by a series; by using the energy method to evaluate the constants of such series and by the super­position theorem find the deflection curve for a bar loaded with a distributed load of the type of a quarter ellipse; then determine

9 Liebers, in his article, “Zur Theorie der Luftschraubenschwing- ungen,” from Zeitschrift fur technische Physik, vol. X , 1929, quotes Reissner for authority for the approximate use of a quadratic dis­tribution of this factor. Whether a quadratic or (1 —• cos wy/ll) expression is used makes very little difference, as the error introduced is extremely small and hence negligible.

tion of the center pressure referred to the mid-point of the chord is given by

where a is expressed in radians, or

60 TRANSACTIONS OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS

the strain energy of bending for such a deflection. (See Fig. 3.) A suitable series to express the deflection of the bar is given by:

F ig . 3

Differentiating the series we get for the slope:

and in substituting for d*z/dy2 the value

By direct integration it may be shown that

We will consider one concentrated force and find the expression for the deflection curve.

For a single force P applied at a distance c from the built-in end, and considering now a slight change in one of the terms of the series, say, the nth term, with all the other terms remaining fixed, we get for the additional displacement of P:

The work done by the force P in moving through

the curve is horizontal at the fixed end, satisfying that condition.In calculation of the coefficients Oi, a2, etc. of this series, we

will use the energy method. The strain energy is given by

The corresponding change of strain energy is

Equating this to the work done \g]

from which

Then Equation [6] becomes

We note that even-numbered terms must be dropped in order to satisfy the boundary condition at y = I, where the curvature is zero. Equation [e] involves the square of this derivative, which contains terms of two kinds in which n is an odd integer:

where m ^ n. Hence in integral [e] all terms containing products of coefficients such as aman disappear and only terms with squares of these coefficients remain. Then

Checking Equation [10] for the accuracy of solution, it may be shown that if we consider a cantilever with a single force concentrated at the outer extremity, the equation gives an accuracy within 1.2 per cent when the first term only of the series is used, while if three terms are used, the error will be less than Vio of 1 per cent.

Having the solution for a concentrated load, we will now solve for the actual distributed load. From Equation [2] the lift on

any element is given by

From Equation [10], by substituting for P the value qidc:

10 For similar development of these equations, see Timoshenko, ‘Strength of Materials,” vol. II, Van Nostrand Pub. Co.

Such series satisfies the end conditions, for when y = 0, z = 0, and y = I, z has a finite value.

and substituting the value of Lc, we get the intensity of load at any cross-section:

AERONAUTICAL ENGINEERING AER-56-3 61

This equation has terms of two types:

The factor 12 in the denominator is here introduced so that the load intensity may be expressed per inch of span for length being taken in inches, the chord Ci in feet, while v is taken in feet per second. To this point there must be careful adherence. From Equation [13] the coefficients au a3, etc. in Equation [6] may be evaluated, and on squaring these coefficients and sub­stituting them in Equation [8], an expression for the strain energy becomes:

Dealing with expression [_/], choosing n = 1, 3, 5 in turn, evaluating the integral for each value of n, and then summing the integrals, will give the factor depending upon a0. In per­forming these operations, it will be readily seen that by sub­

stituting c/l = sin u, c = I sin u, and dc = I cos u du, there will come terms of the form:

The latter two of these are of course Bessels functions and readily evaluated.

Dealing now with the expression [fc], this may be handled by direct integration after removal of the Z sign and putting the expression in the form of a series, so that on reduction:

d2z d2zSince — = — , we have M = E l — , and we have now found

E l ay2 ay2the values of ah a3, etc. in Equation [/], so that the stresses maybe determined.

In the present problem the determination of the relative stress distribution was based on solving for the moment, stati­cally. The foregoing series is given so that the problem of com­bined bending and torsion may be approached in a series form. Such series may possibly be combined with the series evaluations for vortex circulation about the airfoil.

From consideration of the bending moment for an elliptic load on a cantilever beam, it may be shown that the moment at the root is given by

there will be as much stress produced by twisting forces as is produced by bending forces.

In adjustable metal propellers which are relatively thin, it is quite evident that this effect will contribute greatly to the stresses carried. This effect may account for some of the numerous propeller failures noted by Liebers.

If a wing is rigid and if possible does not have any torsional vibration, then <*2 = 0, and the remaining portion of Equation [14] will give an expression for the strain energy of a wing of constant moment of inertia.

6— V i b r a t i o n o f P l a t e A i r f o i l

(a) Torsional Vibrations. In this section it is proposed to calculate the frequency of free torsional vibrations of our canti­lever plate airfoil, so that the frequency of resonance may be known. A suitable expression for the mode of vibration is of the form:

We will use the energy method in this case, equating the maximum strain energy to the maximum kinetic energy, and solve for the frequency.

The strain energy is expressed by the equation:

from which we conclude that when

62 TRANSACTIONS OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS

F ig . 4

Referring to Fig. 4, we note that if we reckon time from the mid-position of the vibration, 7« ., occurs when sin pt = 1, or when t = y 4 cycle.

From consideration of the rectangular cross-section of our plate, and noting that the width is long compared with the thickness, the torsional rigidity may be expressed in the form C = G • 1/3 6c3, where c represents the thickness of the plate.

Again we must note that a long rectangular prism, so con­strained that one section does not distort when a torsional moment is applied, has the effect of shortening the bar by an amount equal to 0.425 times the half width.11 If we apply a

correction to the coefficient of torsional rigidity C, to offset this reduction of length, the frequency may be approxi­mately expressed:

which, by differentiating Equa­tion [16] with respect to time, squaring, and integrating, re­

duces to

Tmox occurs at time I

so that= >-

The ratio of length of plate to width of plate may be called the plan-form ratio, and we may denote such ratio by

Assuming no loss of energy, we equate the maximum strain energy and kinetic energy, and solving for p2 we have:

Dealing with the expression l/l — 0.21256, we may by sub­stitution put this in the form y/y — 0.2125.

Again in Equation [21] we note in the denominator in front of the radical the factors lb, for which we may substitute l2/y, so that we may express this Equation [21] in the following form:

We note that the frequency is independent of the angular

amplitude.

11 S. Timoshenko, “On the Torsion of a Prism, One of the Crosa- Sections of Which Remains Plane,” Proc. London Math. Soc., 1921,

p. 389.

Differentiating Equation [16], squaring, and integrating, we have

The kinetic energy is given by

the equation

So that

AERONAUTICAL ENGINEERING AER-56-3 63

Thus, for example, a plate of these dimensions: Z = IS in., b = ci = 3 in., c = Vs in., y = 6, G = 11,650,000 lb per sq in., q = 0.10625 lb per sq in. of span, g = 386 in. per sec2. By direct substitution in Equation [21a ] and solving, we obtain for the frequency ‘16.7 cycles per sec.

(6) Induced Torsional Damping. Mr. J. P. DenHartog12

has called attention to the self-induced vibrations of sleet- covered wires at various wind velocities. He notes that the relative wind is inclined at an angle to the axis of the shape considered to simulate a sleet-covered circular wire, and there­from obtains a driving force to maintain vibrations in a vertical plane.

We will apply the method to another problem, but this time in erosion. Consider an airfoil to be loaded in such manner that the lift and drag forces may be considered as the only external forces acting, and allow these to act at the center of pressure of the airfoil. In general, the center of pressure and the axis of twist will not coincide, but will be at some distance apart. The relative wind due to the velocity of vibration will be in­clined to the translatory flow, so that the relative wind on the upswing of the vibration will be different from the relative wind on the downswing. Because the lift and drag act normal and parallel to the relative wind at any instant, they will both have components normal to the airfoil. Designating the sum of these components as the driving force Fd, we get therefore an induced torsional damping as a result.

Consider an airfoil to be represented by a flat plate and let it be at an angle of incidence a, such that

a = cko + «i sin pt............................[22]

as shown in Fig 5a. Let At; be a small change in the angle of incidence; assume the plate to be in torsional vibration about

its mid-point and that such vibration is small and simple har­monic; let r be the distance from the center of twist to the center of pressure, and va be the tangential velocity of the center of pressure at any instant during vibration. If the angular velocity of the plate is w, then the tangential velocity of the point P is

given by ur.Combining the velocity va taken in proper sense to allow for

the direction of representation of the translatory wind flow, we

can obtain the relative wind velocity vr. The lift and drag forces act normal and parallel to the relative velocity. The normal component of the lift will then be L cos 8, where 613 is the angle between the normal to the plate and the direction of the lift at the instant under consideration. Similarly D sin 6 will represent the drag component normal to the plate. The driving force Fn will then be the sum of these two components. It will be noted that the angle of attack 8 of the airfoil is greater at mid-position on the downswing than at the same position on

the upswing. Hence we may expect a greater value of FD at that time, and since this increase of force opposes the motion, damping will result.

Plotting the driving force FD against time for one cycle, we get a curve as in Fig. 6a, and plotting the angle of incidence against a time base we get a sinusoidal curve, Fig. 6b. Now plotting the product of the driving force and its moment arm r against the angle of incidence, we get a loop, Fig. 6c, which is a measure of the damping per cycle. This damping is small but positive.

12 J. P. DenHartog, “Transmission-Line Vibration Due to Sleet,” A.I.E.E., June, 1932.

13 See Appendix for mathematical derivation of 8.

F ig . 6a

Values of cc in Degrees

F ig . 6c

and substituting ci for b

64 TRANSACTIONS OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS

For self-induced torsional vibrations to exist there must be a moment acting on the airfoil greater than the damping moment. Such a moment may possibly be attributed to the impulsive moment connected with the shedding of the Karman-type vortices from the airfoil.

6 — C o n c l u s io n s a n d S u m m a r y

Self-induced torsional vibrations were observed at the natural frequency of the plate airfoil. The amplitude of the observed vibration being approximately 5 deg, there is little doubt that the forces producing such vibrations must appreciably exceed

the damping as indicated in Fig. 6c.If we denote the amplitude of the vibration as being a part of we see the effect on the lateral bending as expressed in the

equations for the stress distribution. From the expressions for the strain energy and moment we note that the vibration must play a major part in the energy and stresses in the airfoil.

Since a propeller is a rotating airfoil, it is probable that it also has a period of self-induced torsional vibration. Centrifugal force will have no effect on the torsional vibration so set up. This vibration is confirmed by the peculiar torsional fracture

with which propellers frequently fail.We cannot expect to remove these self-induced vibrations,

but we should endeavor to keep the periods of other vibrations such as lateral vibration of a wing or fore-and-aft vibration

of a propeller as far removed as possible from the self-induced vibration. In this manner we will remove the likelihood of resonance conditions being set up with damaging or critical

results.The method used in making observations in the wind tunnel

has indicated that great studies may be made in the vibration measurements by such means.

AppendixReferring to F ig. 5a, let xx be the line of reference, ao be the angle of

incidence a t the root, and a = ao + a i Bin pt.Assume d is tribu tion of load over the chord, represent the center of pres­

sure as c.p. and its distance from the axis of ro ta tion by r, a nd consider the c.p. to be m oving in the direction of a increasing.

For the sake of representing m otion of the airfo il in true sense, le t us refer again to Fig. 5&.

In the complex form vr = va + v. Now

for flat p late airfoil.W e can now solve for the value of 0 for any in stan t in one cycle. I t will

be noted for ci = 3 in ., ao = 4 deg, a i = 1 deg = 0.0175 radian, th a t 0 has a different value on the downswing th an on the upsw ing. C om puting the lift per inch of a irfo il and resolving i t norm al to the chord gives the d riv ing force due to the lift.

I t is a well-known aerodynam ic law that

and th a t for R = 12

and G laue rt14 in his book on airfo il theory gives 8 = 0.096

and hence its norm al component m ay be found, so th a t the driv ing force due to the drag m ay be found , and hence the to ta l driv ing force F m ay be determ ined a t any instan t.

B i b l i o g r a p h y

“Schwingungen mehrfach gestutzter Stabe mit Axial kraften,” H. G. Kussner, D.V.L., Sept., 1928. Jahrbuch D.V.L., 1929, S. 335.

“Schwingungen von Flugzeugflugeln,” H. G. Kussner, D.V.L., Feb., 1929. Jahrbuch D.Y.L., 1929, S. 313.

“Gekoppelte Biegungs- Torsions- und Querruderschwingungen von freitragenden und halbfreitragenden Flugeln,” H. Blenk and Fritz Liebers. Jahrbuch D.V.L., 1929.

“Gekoppelte Torsions— Biegungschwingungen von Fragflugeln,” Blenk and Liebers, Z.F.M., 1925, S. 479 und Z.F.M., 1926, S. 286 (errata sheet for 1925 articles).

Fraser, “An Investigation of Wing Flutter,” Great Britain, R. & M., 1042 (1926).

C. F. Greene, “An Introduction to the Problem of Wing Flutter,” Trans. A.S.M.E., 1928.

“Transmission-Line Vibration Due to Sleet,” J. P. DenHartog, A.I.E.E., June, 1932, Cleveland meeting.

“Self-Induced Vibrations,” J. G. Baker, A.S.M.E., June, 1932, Jan., 1933, Bigwin Inn meeting.

“On Relaxation Oscillations,” B. Van Der Pol, Phil. Mag., vol. 2, 1926, p. 978.

J. B. Younger, “Dynamics of Airplanes,” and B. M . Woods, John Wiley & Sons.

14 H. Glauert, “Airfoil and Airscrew Theory,” Cambridge Uni­versity Press.