Deflection, stress, and vibration analysis of rotating propellers using holography

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NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER

Bethesda, Md. 20084

DEFLECTION, STRESS, AND VIBRATION ANALYSIS OF ROTATING PROPELLERS USING HOLOGRAPHY

November 1974

by

J. P. Sikora H. A. Peterson F. T. Mendenhall, Jr.

APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED

STRUCTURES DEPARTMENT

RESEARCH AND DEVELOPMENT REPORT

Lab. V.

Scheepsbouwkunde

T'-h14

tJi

-1n,

(.14

Report 4507

The Naval Ship Research and Development Center is a U. S. Navy center for laboratory

effort thrected at achieving improved sea and air vehicles It was formed in March 1967 by merging the David Taylor Model Basin at Carderock, Maryland with the Marine Engineering

Laboratory at Annapolis, Maryland.

Naval Ship Research and Development Center Bethesda, Md. 20084

*REPORT ORIGINATOR

MAJOR NSRDC ORGANIZATIONAL COMPONENTS

OFFICER-IN-CHARGE CAR DE RO CK 05 SHIP PERFORMANCE DEPARTMENT

*

STRUCTURES DEPARTMENT 17 SHIP ACOUSTICS DEPARTMENT 19 MATERIALS DEPARTMENT 28 N SR DC COMMANDER 00 TECHNICAL DIRECTOR 01 OFFICER-IN-CHARGE ANNAPOLIS 04 AVIATION AND SURFACE EFFECTS DEPARTMENT 16 COMPUTATION AND MATHEMATICS DEPARTMENT 18 PROPULSION AND AUXILIARY SYSTEMS DEPARTMENT 27 CENTRAL INSTRUMENTATION DEPARTMENT 29 NDW-NSRDC 3960/43b (Rev. 3-7 GPO 928-IO SYSTEMS DEVELOPMENT DEPARTMENT 11

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4. TITLE(and Subtitle)

DEFLECTION, STRESS, AND VIBRATION ANALYSIS OF ROTATING PROPELLERS

USING HOLOGRAPHY

5. TYPE OF REPORT & PERIOD COVERED

6. PERFORMING ORG. REPORT NUMBER 7. AUTHOR(e)

Jerome P. Sikora Herbert A. Peterson

Frederick T. Men denhall, Jr.

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Task 16874, Task Area S4622 1-173 0-150

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Naval Ship Research and Development Center

Bethesda, Maryland 20084

12. REPORT DATE November 1974

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Naval Sea Systems Command

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Propellers Holography Lasers Rotating

20. ABSTRACT(Continu,an r.v.r.. eid. if n.c..usy end identify by block numb.r)

A novel holographic technique has been developed which for the first time allows the measurement of the deflection of propeller blades while rotating. The technique requires that an axially-symmetric wavefront be directed onto both the propeller blade and a holographic plate rotating with the propeller. Some deflection results for a propeller rotating in air are presented and the technique has been shown to be feasible for

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Stresses for the rotating propeller have been obtained from the deflections by using pure bending-plate theory. The method was verified by strain gages for a propeller blade subjected to a uniform static pressure.

The technique provides a method for obtaining deflections, strains, stresses, and vibration modes for rotating objects such as marine and aircraft propellers, helicopter rotors, generator and turbine blades.

UNCLASSIFI ED

ABSTRACT TABLE OF CONTENTS ADMINISTRATIVE INFORMATION 1 INTRODUCTION 1 EXPERIMENTAL SETUP 2 EXPERIMENTAL PROCEDURES 3 DISPLACEMENT ANALYSIS 4 STRESS ANALYSIS 5 VIBRATION ANALYSIS 8

UNDERWATER FEASIBILITY STUDY 8

SUMMARY ₉

CONCLUSIONS AND RECOMMENDATIONS ₉

LIST OF FIGURES

i Schematic of Optical Arrangements 10

2 Vacuum Chamber 11

3 Fringe Pattern of a Beam Rotating at 32 Revolutions per Minute 12

4 Fringe Pattern of a Beam Rotating at 46 Revolutions per Minute 12

S Deflection of a Beam Subjected to Inertial Loads ₁₃

6 Propeller P4498 14

7 Fringe Pattern of Propeller P4498 Rotating at

197 Revolutions per Minute 15

8 Fringe Pattern of Propeller P449 8 Rotating at

237 Revolutions per Minute 15

9 - Fringe Pattern of Propeller P4498 Rotating at

301 Revolutions per Minute 15

10 - Deflections for Propeller P4498 at 60 Percent Radius

Rotating in Air 16

11 - Deflections for Propeller P4498 at 80 Percent Radius

RotatinginAir 16

12 - Deflections for Propeller P4498 at 90 Percent Radius

Rotating in Air 17

lu

Page 13 - Midchord Deflections for Propeller P4498 Rotating in Air 17 14 - Stress Distribution for Propeller P4383 Subjected to

Uniform Pressure 18

15 - Stress Distribution for Propeller P449 8 Rotating at

301 Revolutions per Minute 19

16 - Primary Bending Mode Pattern of Propeller P43 83

Rotating in Air 20

17 - Fringe Pattern of Propeller P4498 Rotating Underwater at

11 Revolutions per Minute 21

Table 1 - Change in Angle of Attack (ii Radians) 5

NOTATION

a, h, c Arbitrary curvature directions

C _{Chord length}

D Stiffness of a plate

d Time delay

E Modulus of elasticity

r Radius of curvature of surface

s _{Incremental distance for measuring curvature}

x,y Cartesian coordinate system

z _{Distance of surface from neutral axis}

Deflection normal to surface

e Strain

Poisson's Ratio

u Stress

w Rotational velocity

w' Equivalent rotational velocity

ABSTRACT

A novel holographic technique has been developed which for the first time allows the measurement of the deflection of propeller blades while rotating. The technique requires that _{an axially-symmetric wavefront be} directed onto both the propeller blade and a holographic plate rotating with the propeller. Some deflection results for a propeller rotating in air are presented and the technique has been shown to be feasible for propellers rotating in water.

Stresses for the rotating propeller have been obtained from the deflec-tions by using pure bending-plate theory. The method was verified by strain gages for a propeller blade subjected to a uniform static pressure.

The technique provides a method for _{obtaining deflections, strains,} stresses, and vibration modes for rotating objects such as marine and

aircraft propellers, helicopter rotors, generator and turbine blades.

ADMINISTRATIVE INFORMATION

The work reported in this report _{was sponsored by the Naval Sea Systems Command}

(NAVSEA), Code 0331G under Task 16874, _{Task Area S4622. This work was performed}

in the Structures Department under Work _{Unit 1-1 730-150 during Fiscal Year 1974.}

INTRODUCTION

The Center has for sometime been _{conducting research and development to originate} new propeller blade shapes for high performance ships. Among other problems, the deflec-tions of propellers under operating condideflec-tions are known to affect their performance by

causing changes in the surface pressures. The propeller blade designer should have a way of

estimating changes in the angle of attack for both forward and reverse directions of opera-tion. Stresses and vibration characteristics must also be determined for advanced blade shapes.

Holographic interferometry has already been _{shown to be a reliable method for}

deter-mining displacements for statically loaded _{propeller blades.1 The determination of}

deflec-tions and stresses on rotating blades is considerably more difficult. Strain gages may be used on propeller blades under operating conditions, but the gage installation may affect the flow patterns on the blade, and the number of_{gages is usually limited due to difficulties in} trans-ferring the gage signals from the blade. Finite element analysis might be used provided an

adequate load distribution is available. Unfortunately loads on rotating propellers _{are not}

well known.

Dhir, S. K. and J. P. Sikora, "Holographic

Displacement Measurements on a Highly Skewed Propeller Blade," NSRDC Report 3680 (Aug 1971).

The technique described in this report permits deflection analyses and, subsequently, stress analyses, to be performed on propeller blades while the blade is rotating in air. The holographic technique is full-field and non-contacting, thus it does not alter the deflection or mode pattern by touching the blade or affecting the flow pattern around it. The technique was applied to a rotating cantilever beam to check the reliability of the results. The deflec-tions of a propeller blade rotating in air and in a vacuum were determined. The results were analyzed to determine changes in angle of attack as a function of speed and direction of ro-tation, as well as bending stresses in the blade. The primary bending mode of a propeller rotating in air was obtained holographically. Preliminary experiments to obtain deflections

of propeller blades rotating in water demonstrated that the holographic technique can be used to obtain deflection data under actual operating conditions.

EXPERIMENTAL SETUP

Any attempt to make a double-exposure hologram of a rotating object requires that the object be repositioned in space within wavelengths of light for both exposures. This

preci-sion would require extremely accurate triggering of the pulsed laser. However, by using an

on-axis optical system and fixing the holographic plate to the hub but perpendicular to the axis of the rotating object, the triggering requirements are less stringent. A previous attempt by other researchers with an on-axis system reported inconclusive results.2

A schematic of the optical arrangement is shown in Figure 1. Light from a one-joule

ruby-pulsed laser is divided into two beams. The object beam is expanded by a lens, scatter-ed off the propeller blade back to a front surface mirror normal to the light axis, and then reflected to the holographic plate. The reference beam is brought off-axis at the

beam-splitter and returned on-axis just beyond the object beam lens. The reference beam is then expanded by a lens and illuminates the holographic plate. The distance the reference beam is brought off-axis is such that the reference-beam path length equals the object-beam path

length. This routing of the rQference beam is necessary to maintainthe difference in the

length of the two beams to less than one meter, the coherence length of the ruby laser, without requiring the mirror to be too close to the propeller.

Alignment of the elements was accomplished with the use of a 5 milliwatt He-Ne laser fixed to the pulsed laser rail. Standard burn spot tests were made to insure that the ruby laser was properly aligned. The front surface mirror waspositioned by reflecting the align-ment laser beam back down upon itself. Guide rails were attached to the table and the mirror was then translated far enough off-axis to allow the object to be illuminated.

2Tsuruta, T. and Y. Itoh, "Holographic Interferometry for Rotating Subject," Applied Physics Letters, Vol. 17, No. 2 (Jul 1970).

The end of the propeller shaft was centered in the unexpanded laser beam. A mirror was then placed in the holographic plate holder fastened normal to the shaft axis. The shaft was aligned by rotating the shaft and observing the reflected trace of the alignment laser. Alignment was considered satisfactory when the axis of rotation varied no more than 0.7 degree from the axis of the laser beam.

Ideally, if all the optical elements were exactly aligned, the exposures could be taken at any time that the blade was in front of the mirror. Then the blade and plate would be in perfect registry and no fringes due to rotation would appear. Since perfect alignment was not obtained, it was found necessary to trigger both laser pulses when the blade was positioned to within a tolerance of one minute of rotational angle.

The pulsed laser was triggered by an electrical signal initiated by reflecting another laser beam from a mirror on the rotating shaft via a fast photodetector arid a controllable time-delay circuit. The time-delay time is the interval between reception of the signal at the photo-detector and the firing of the pulsed laser. Electronic counters were used to monitor the delay time and shaft speed. It was possible with this arrangement to superimpose exposures to within 30 seconds of arc.

EXPERIMENTAL PROCEDURES

The deflections of the propeller due to rotation were determined from a double-exposure hologram. The first double-exposure was made while the blade rotated at its initial speed. The initial speed and initial delay time were monitored and recorded. The blade was then slowed to its final speed for the second exposure. To ensure that the propeller would be in

the same angular position for the second exposure as for the first, the time delay was ad-justed according to the equation d = w2 d2. Thus, if the second exposure is to occur at half the speed of the first exposure, the delay time must be doubled so that blade has

time to reach the angular position at which the first exposure was made. A non-zero veloc-ity for w2 was used for convenience of the triggering mechanism.

The equivalent speed that the blade deflections represent, based on the assumption that the blade loading varies with the square of the angular velocity, is given by

w' = ./w?

-where is the higher speed and w2 is the lower speed.

In the vibration analysis, the speed and time delay were unchanged for each exposure. The first exposure was made with the blade rotating in an acoustic pressure field, generated by a loudspeaker, oscillating at the resonant frequency of the blade and the second exposure

was made after the pressure field was turned off. The particular resonant frequency of the blade being investigated was previously determined by real-time holography on the stationary propeller. Additional details of the vibration analysis technique have been published.3

DISPLACEMENT ANALYSIS

A preliminary experiment was derived to determine the accuracy of the holographic dis-placement technique. An aluminum cantilevered beam 11.5 x 2 x 0.079 in. was rotated about its center in the vacuum chamber shown in Figure 2 so that only inertial loading was applied to the beam. A 48 gm mass fastened to the beam near its free end provided a bend-ing moment, causbend-ing deflections in the axial direction, in the beam which increased with the square of the angular velocity. The foreground of Figures 3 and 4 show the fringe patterns when the beam rotated at an effective speed of 32. 1 and 45.6 rpm, respectively, and a pro-peller blade showing contour fringes appears in the background. The normal displacements of the beam were calculated as previously reported1 and were found to be in excellent agree-ment with beam theory. This comparison is presented in Figure 5. The propeller models used in the experiments were NSRDC Propellers P4383 and P4498. These twelve inch diameter, five-bladed aluminum propellers are identical except for rake. Figure 6 shows Propeller P4498.

Holograms were made of Propeller P4498 while it was rotating in air in both the for-ward and reverse directions throughout a range of speeds. The fringe patterns for three speeds are shown in Figures 7, 8, and 9. Each fringe which crosses the blade chordwise

corresponds approximately to a line of constant displacement since the blade deflection is predominantly in the axial direction. An additional set of fringes which run out the length of the blade are contour fringes and are not related to displacements. The holographic dis-placement analysis' was used to determine the three-dimensional disdis-placement components. The blade deflection pattern for the reverse direction of rotation at an effective speed of

197 rpm is shown in Figure 7. The blade deflection for the forward direction of rotation

at effective speeds of 237 rpm and 301 rpm, respectively, are shown in Figures 8 and 9. The normal blade deflections obtained at four different speeds for Propeller P4498 as a function of chord length and as a function of blade length were computed from the holo-graphic data and are presented in Figures 10-13.

Although the normal deflections consist of components from the three-dimensional dis-placement field, the dominant component of 5 is in the axial direction for Propeller P449 8. It was found that the axial and normal deflections agreed within 5 percent beyond the 80 percent

3Sikora, J. P. and F. T. Mendenhall, Jr., "Holographic Vibration Study of a Rotating Propeller Blade," in Experimental Mechanics, Vol. 14, No. 6 (Jun 1974).

blade radius when Propeller P4498 was statically loaded. Even though the in-plane displace-ments may not be identical for both static and dynamic loads, the error induced by calcu-lating 5 from only the axial components is probably small.

The change in angle of attack, Ø, due to load is a function of the normal deflection at the leading and trailing edges and of the chord length. It is defined as

TE -

6LE

C

The changes in angle of attack for different speeds, directions of rotation and radii of pro-peller blade were obtained from normal deflection data and appear in Table 1.

TABLE I - CHANGE IN ANGLE OF ATTACK ( RADIANS)

The experimental setup represents crash-ahead and crash-astern operating conditions; i.e., the propeller rotates at full speed while the ship has zero velocity. The change in angle of attack (see Table 1) was found to increase from the hub out to the tip for both forward and reverse speeds. For forward speeds, Ø decreased with increasing speeds. The opposite effect was found to occur when the propeller spun in reverse. The change in the angle of attack of rotating blades was found to be less than half as severe as that based on static uni-form air pressure tests of similar blades.4

The deflections for reverse speeds had nearly the sanie magnitude (in the opposite direc-tion) as for equivalent forward speeds in the vicinity of 200 rpm. At such low speeds, the small changes in angle of attack did not produce a difference in deflection.

STRESS ANALYSIS

Surface strains are defined in terms of displacements in the plane of the surface. However, in plate-like structures subjected to bending such as propeller blades, large

out-of-plane displacements occur compared to in-plane displacements. These displacements can

be measured holographically with more accuracy than in-plane displacements.

Stresses in the propeller blade may be derived by using certain assumptions from the pure bending of plates. The pure bending theory does neglect torsional loading which may

4Boswell, R. J., S. K. Dhir and J. P. Sikora, "The Effect of Skew on Elastic Deflections in a Propeller Blade

Influence on Divergence under Astern Operation," NSRDC Test and Evaluation Report 437-H-02, Ship Performance (Feb 1972).

s

Direction Speed (rpm) 60 Percent Radius 80 Percent Radius 90 Percent Radius

Forward 237 7.42 7.73 10.42

Forward 301 1.24 6.49 7.74

Reverse 176 1.48 2.74 4.17

become significant in skewed propeller blades. Since the propeller blade is thin compared to its radius of curvature and deflections are small compared to the blade thickness, the middle surface of the propeller blade is also its neutral surface. Then, as in the case for a

beam where z is the distance from the neutral axis, the strains in a plate are given by

z

ey =

Using Hooke's Law, the stresses in the coordinate directions are then

Ez

'1

_1"

oX I +

l-2

rx Ez

Il

_I'

0=

1

+P-\r,

r

The external moments M and M are

/1

1

M =D(+

-r)

y

'i

M =D(±v---1

_(r

r

_xJ

It can be shown that for any new orientation of the axis system, say x' and y', the external

bending moment M acting on the cut section is given by

1

l\

M = D 1-;- + L)

y /

This demonstrates that for any element orientation, the external moment, and therefore coordinate stresses, are related to the orthogonal curvatures for the element given above.

The radius of curvature of the neutral surface is given by

Ii+

d2w

r

"dw" 21 -3/2 r

_{dx2 L}

6 z = X r

Neglecting the second order term

I

dw

r_{X uX}

The holographically obtained surface deflections _{are assumed to be equal to the neutral}

sur-face deflections. Since propeller blade sursur-faces are curved, it is further _{assumed that the}

normal component of the surface deflections produce _{blade stresses.}

Along some direction (a), select three consecutive normal deflections _{2' 3) that}

are equal distances apart. The radius of a circle passing through the three points can be shown to be

s2

Fa

- 4(6i

- 22 +

ô3)

by using either analytic geometry _{or by taking the second derivative of a quadratic}

polyno-mial through the three points. _{Since the principal radii of curvature are not known, it is} necessary to obtain radii of curvature in three different directions. The normal strain in any direction is proportional to the curvature in that _{direction, thus}

z z z

E=

_Cr

where a, b, c denote three different directions. From _{three-element strain rosette analysis,} the principal strains and their directions can be calculated. At free boundaries, the direc-tions tangent and normal to the boundary _{are principal directions; hence, the principal strain} tangent to the boundary may be calculated directly. _{Principal stresses may be obtained from} Hooke's law.

To determine the validity of the procedure_{a single-blade version of Propeller P4383 was}

loaded by uniform air pressure in a specially designed pressure box. The three-dimensional dis-placements for a load of 1.0 psi were determined holographically and were found to be in good agreement with those determined from _{a finite element analysis.1'5 The stresses in the}

5Ma, J. H., "Stress Analysis of Complex Ship Components by a Numerical Procedure Using Curved Finite Elements," NSRDC Report 4057 (Jul 1973).

radial and tangential directions across the chord of the blade face at the 30, 50, and 70 per-cent blade radius were determined and are presented with those of the finite element analysis and strain gage results in Figure 14. The tangential stresses were found to be nearly identical for all three methods at the 50 percent radius and in fairly good agreement elsewhere. The

holographic radial stresses more closely agreed with the strain gage results than with the finite

element stresses.

The stresses of the five-bladed Propeller P4498 were determined from the deflections

obtained while the propeller rotated at 301 rpm in air. Figure 15 presents these stresses at the 60 and 80 percent blade radius. The stresses are approximately one-tenth of those of P4383, which is a propeller identical to Propeller P4498 except for rake.

VIBRATION ANALYSIS

The primary bending mode resonant frequency of the stationary propeller blade P4383

was determined to be 5 11 Hz by using standard real time-time average techniques. The blade

was then set up in the axially-symmetric system described in this report and excited

acousti-cally at 511 Hz. A series of holograms was made to show the primary bending mode pattern while the blade rotated at speeds up to 400 rpm. Figure 16 is a photograph of a hologram showing the primary bending mode pattern while the blade was rotating at 168 rpm. For the primary bending mode, the node (region of zero displacement) occurs at the hub and the antinode (region of maximum displacement) occurs at the blade tip. It was found that the

frequency of the primary bending mode did not change measureably as the rotational velocity was varied from zero to 400 rpm. No significant changes in the mode patterns due to centrif-ugal effects or twisting were detected throughout the range of speeds.

UNDERWATER FEASIBILITY STUDY

To demonstrate the suitability of the technique of holographic interferometry for

under-water applications, Propeller P4498 was mounted in a 60 gallon tank of water. The light

from the ruby laser was expanded by a lens, passed through a window in the wall of the tank and directed onto the propeller. The remaining optical arrangement and experimental pro-cedure was similar to that previously described. Figure 1 7 shows the deflection fringes due to an equivalent rotational speed of 11 rpm. An analysis of the displacement components is not presented because the validity of the results are affected by boundary conditions due to the small body of water as well as by the low Reynold's Number at such low speeds. Future tests are planned in larger bodies of water and at higher speeds so that structurally and hydro-dynamically useful information may be obtained.

SUMMARY

The deflections of a propeller blade rotating in air were measured for the first time. A new holographic technique, consisting of a spinning hologram and axially-symmetric wave-fronts, was used to obtain the deflections.

In an investigation at 200 rpm, the deflections of a propeller blade in air for crash ahead and crash astern load conditions were found to be equal in magnitude. The change in the angle of attack was found to decrease with increased forward rotational speeds and to in-crease with inin-creased reverse rotational speeds.

Surface stresses of rotating propellers were determined from deflections normal to the blade surface using pure bending-plate theory. The magnitudes and distribution appear to be reasonable when compared with static cases.

The primary bending mode pattern was determined for a rotating propeller. There was

no observable change in the primary bending mode for speeds up to 400 rpm.

CONCLUSIONS AND RECOMMENDATIONS

The suitability of the holographic technique has been demonstrated for structural analysis of marine propeller blades in air, vacuum, and water environments.

Work is in progress to measure blade response of small model propellers operating under water in mean and unsteady flow conditions.

Surface pressure measurements appear to be possible and the holographic technique described in this report lends itself to the measurement of surface pressures on marine pro-peller blades under actual operating conditions.

PROPELLER HOLOGRAM LENS MIRROR MIRROR LENS

-MIRROR REFERENCE BEAM OBJECT BEAM

Figure 1 - Schematic of Optical Arrangements

R OR BEAMSPLITTER

Figure 2a - Front View

Figure 2b - Back View

Figure 2 - Vacuum Chamber

Figure 3 - Fringe Pattern of a Beam Rotating at 32 Revolutions per Minute

200 150

I

C-) z z o I-C-) w 100 -J IJ w o 50

O

HOLOGRAPHY THEORY 13

1jh

Figure 5 Deflection of a Beam Subjected to Inertial Loads

O 2 3 4

4

Figure 6 - Propeller P4498

Figure 7 - Fringe Pattern of Propeller P449 8 Rotating at 197 Revolutions

per Minute

Figure 8 - Fringe Pattern of Propeller P4498 Rotating at 237 Revolutions

per Minute

Figure 9 - Fringe Pattern of Propeller P4498 Rotating at 301 Revolutions

100 (J) u-j

I

_o

z

z

2 50 F-o w -J u-w o O 150 1100 50 16 301 RPM (FORWARD) 237 RPM (FORWARD) 197 RPM (REVERSE) 176 RPM (REVERSE) Fr O 025 0.50 0.75 1.0 LE

FRACTION OF CHORD LENGTH TE Figure 10 - Deflections for Propeller P4498 at 60 Percent Radius Rotating in Air

O 025 0.50 0.75 1.0

LE

FRACTION OF CHORD LENGTH TE

(I) w

I

o z 100 F- O LU -J U- LU o 200 200 150 50 301 RPM (FORWARD) 237 RPM

(FORWARD) 197 RPM (REVERSE) 176 RPM _(REVERSE)

150 100 50 O I I O 1.0 0 0.25 0.50 0.75 1 0 0 2 LE TE HUB 0.4 0.6 0.8 TIP

FRACTION OF BLADE LENGTH

Figure 13

- Midchord Deflections for Propeller P4498

Rotating in Air

FRACTION OF CHORD LENGTH

Figure 1 2 - Deflections for Propeller P4498

(j, (j) w 200 I-(j) 600 400

200

600 400 (i) r 200 w o -200 400 -200 0 RADIAL STRESSES

00

00

E

RADIAL STRESSES

00000

D

O

o

70% BLADE RADIUS 18

-

F.E. D STRAIN GAGE O HOLOGRAPHY TANGENTIAL STRESSES

-o

D

O

D

E

-o

O

70% BLADE RADIUS

Figure 14 - Stress Distribution for Propeller P43 83 Subjected to Uniform Pressure

30% BLADE RADIUS 30% BLADE RADIUS

50% BLADE RADIUS 50% BLADE RADIUS

o

E

o

RADIAL STRESSES

o

o

Do

00000

-o

o

I I i I I I TANGENTIAL STRESSES

-o

o

(j) Q-(J) (J) w H (J,

40 30 20 lo o

lo

40 30 20 10 O

lo

Ü = RADIAL STRESS Go TANGENTIAL STRESS

Figure 15a - 60 Percent Blade Radius

19

LE TE

Figure 1 5b - 80 Percent Blade Radius

Figure 15 - Stress Distribution for Propeller P4498 Rotating at 301 Revolutions per Minute

Figure 1 7 - Fringe Pattern of Propeller P4498 Rotating Underwater at li Revolutions per Minute

INITIAL DISTRIBUTION

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