31st Symposium on Naval Hydrodynamics

Monterey, California, 11-16 September 2016

Experimental Investigation of the Hydro-Elastic Response of a

Surface-Piercing Hydrofoil in Multi-Phase Flow

C. Harwood1, J. Ward1, Y.L. Young1, S. Ceccio1,2

(1Dept. of Naval Architecture and Marine Engineering, University of Michigan,

USA, 2Dept. of Mechanical Engineering, University of Michigan, USA)

ABSTRACT

Ventilation of lifting surfaces can cause dramatic

changes in hydrodynamic loads and flow topology,

which may have significant ramifications on the fluid-

structure interactions of marine systems constructed

with compliant materials. In this work, towing-tank

experiments on flexible surface-piercing hydrofoils

were performed to explore the hydroelastic response

of wetted and ventilated lifting-surfaces. A robust,

non-optical method for 3D shape-sensing was

developed to monitor deformations of the flexible

hydrofoils. Hydrodynamic loads are larger in wetted

flow than in ventilated flow, and static deflections are

commensurate with the loads in both flow regimes. A

small increase in hydrodynamic loading was induced

by hydroelastic coupling in the case of the flexible

hydrofoil. In general, modal frequencies were found to

decrease with foil immersion and forward speed, and

to increase with ventilation, due to changing fluid

added mass. The changes in modal frequencies and

damping coefficients varied non-monotonically with

increasing mode numbers.

INTRODUCTION

Ventilation (specifically atmospheric ventilation) is a

multi-phase flow that occurs on hydrodynamic

systems operating in proximity to the free surface –

most frequently lift- or thrust-producing systems such

as hydrofoils, propellers, and control surfaces. When

lifting-surfaces operate at or near the fluid free surface,

low-pressure regions of the flow by which lift is

generated can entrain air from above the free surface

to form partial cavities and supercavities. When this

occurs, the total realizable lift may be reduced

dramatically (Breslin & Skalak, 1959; Swales, et al,

1974) or reversed (Rothblum, et al, 1969), and the

ventilated flow may persist to angles of attack and

speeds much lower than those at which the cavity

incepted (Kiceniuk, 1954; Wetzel, 1957). Harwood, et

al (2014) and Harwood, et al (2016b) defined three

characteristic flow regimes of a surface-piercing

hydrofoil: fully wetted (FW) flow, fully ventilated

(FV) flow, and partially ventilation (PV) flow.

Thorough reviews of relevant literature may be found

in Harwood, et al., (2016b) and Young, et al (2016a).

Hydrofoils, propellers, control surfaces, etc,

fabricated from non-metallic materials offer a number

of potential advantages including lighter weight,

reduced acoustic signature, corrosion resistance, and

tailored elastic response to loading. Such hydroelastic

tailoring has been demonstrated to improve efficiency,

widen operating envelopes, and delay the onset of

vaporous cavitation in marine propellers (Motley, et

al, 2009; Motley & Young, 2011; Young, et al,

2016b). However, lifting surfaces that are prone to

ventilation can experience rapidly- and widely-

fluctuating loads, resulting in structural deflections;

this issue may be more pronounced for lighter-weight,

compliant lifting surfaces.

The field of aeroelasticity is well-established,

addressing the coupled aerodynamic and structural

responses of aerospace structures (Bisplinghoff, et al,

1996; Fung, 1955). Hydroelasticity, though it shares

many of the same features as aeroelasticity, presents a

number of unique challenges. Water has a density

three order of magnitude larger and dynamic viscosity

two orders of magnitude larger than those of air. As a

result, hydrodynamic systems experience larger loads

and larger viscous effects. The dimensionless mass

ratio (𝜇) describes the ratio of the mass of a lifting-

section per unit span (𝑚) to the fluid mass contained

within its circumscribed circle (𝜋𝜌𝑓𝑏2), where 𝑏 is the

radius of the circle, equal to the foil’s semichord,

𝜇 = √𝑚

𝜋𝜌𝑓𝑏2. (1)

Mass ratios of 𝜇 > 4 are typical of aerospace systems,

while marine systems typically fall into the range of

𝜇 < 2. Coincidentally, this low range of mass ratios is

also the range in which the classical tenets of

aeroelastic analysis are least valid (e.g. linear

frequency-domain analysis).

Numerical work (Akcabay, et al, 2014;

Akcabay & Young, 2014, 2015; Chae, et al, 2013,

2016a; Fu & Price, 1987; Harwood, et al, 2016a) and

experimental work (Besch & Liu, 1971, 1973, 1974;

Blake & Maga, 1975; Fu & Price, 1987; Harwood, et

al., 2016a) have demonstrated that fluid inertia,

damping, and disturbing forces can dominate the

governing equations of motion for lightweight lifting

bodies in water. These large fluid forces, combined

with the presence of a free surface, wave generation,

and multi-phase flow, contribute to significant

nonlinearity in the fluid-structure interactions. Recent

reviews are provided by Young, et al. (2016a) and

Young, et al. (2016b).

In this work, we will present the results of

experiments, conducted with a flexible surface-

piercing strut in a towing tank. The purpose of this

research is to advance scientific understanding of

fluid-structure interactions in the presence of multiple

phases. Additionally, we seek to demonstrate the

efficacy of new experimental techniques for

investigating such problems. The specific objectives

of this paper are as follows: (1) develop and validate a

non-optical method of measuring the static and

dynamic deformations of a flexible surface-piercing

hydrofoil in wetted and ventilated flows; (2) quantify

the effects of wetted and ventilated flow regimes on

the steady and unsteady hydro-elastic response of the

flexible hydrofoil; and (3) quantify the effects of

spanwise bending and twisting deformations on

steady-state hydrodynamic loads and flow regime

stability boundaries.

EXPERIMENTAL SETUP

Ventilation on a vertically-cantilevered, flexible

hydrofoil was studied experimentally. Hydrodynamic

testing was conducted in the towing tank of the

University of Michigan’s Marine Hydrodynamics

Laboratory. The tank has dimensions of 110 m length

by 3.2 m depth by 6.7 m width (360 ft × 10.5 ft × 22

ft), and the carriage is capable of speeds up to 6.1 m/s

(20 ft/s).

The test specimen was a vertical strut with a

semi-ogival section, a chord length of 27.9 cm (11 in)

and a geometric span of 91.4 cm (36 in). The strut is

geometrically identical to that described by Harwood

et al., (2014) and Harwood, et al., (2016a,b). The strut

was vertically cantilevered, hanging below a steel

frame and piercing the water surface to a prescribed

depth of ℎ. The angle of attack was set by rotating the

strut inside of the frame and clamping it in place.

Forces and moments on the strut were measured using

a 6-DOF load transducer and instantaneous speed was

measured via an optical encoder on the towing

carriage wheels. All data were digitized and recorded

via hardware-synchronized and phase-locked A/D

converters at rates of 1-2 kHz and at 16-bit resolution.

The model described herein differs from the

rigid aluminum foil tested in previous work (Harwood

et al., 2014; Harwood, et al., 2016b) in the construction

and instrumentation of the strut itself. In the present

work, two variations of a flexible hydrofoil strut were

considered. The first was briefly described by

Harwood, et al (2016a), and is constructed of PVC,

with two instrumentation-channels located along the

chord-line. Shape-sensing spars, of the authors’ design

were inserted into these channels, with spar “A”

located near the trailing edge and spar “B” located just

forward of mid-chord. The shape-sensing spars,

described later, use strain-gauges to infer the 3D

deflections of the strut. Affixed to the end of each spar

was a single-axis accelerometer, used to measure tip-

accelerations.

The second variation on the flexible strut

used the same model, with an aluminum strip 0.6 cm

thick × 2.79 cm wide (0.25 in × 1.1 in) affixed to the

trailing edge. The strip provided additional rigidity in

bending and moved the elastic axis of the strut aft. The

elastic axis is defined as the locus of sectional shear

centers, such that loads applied along the elastic axis

do not induce any twisting motion of the strut. Figure

1a depicts the sections of both flexible struts (with and

without the aluminum strip). Figure 1b shows the

experimental setup, with the locations of the sensors

indicated by callouts. In the following text, the PVC

strut without the reinforcement will be referred to as

model 1, and the reinforced PVC strut will be referred

to as model 2. The beam-section properties of the two

models are given in Table 1.

A shaker motor (Modal Shop model 2007E)

was hung from the mounting-structure and attached to

the hydrofoil via a nylon stinger, as shown in Figure

1b. The motor has a frequency response of 0-10kHz

and a maximum force amplitude of 31 N (7 lbf). An

impedance-head, mounted between the stinger and the

hydrofoil, was used to measure forces and

accelerations at the drive-point.

Table 1. Beam section properties of hydrofoil models

Model 1 Model 2

Chord, m 0.2794 0.2858

Geometric span, m 0.9144 0.9144

Area, m^2 0.00588 0.00606

Mass/unit length, kg/m 8.74 9.22

𝑿𝑺𝑪/𝒄 (fwd of mid-chord) -0.106 -0.259

𝑿𝑪𝑮/𝒄 (fwd of mid-chord) -0.045 -0.07

GJ (N-m2 / rad) 1795 2446

EI (N-m2) 1217 1947

(a)

(b)

Figure 1. (a) 2D section of flexible hydrofoil with the

locations of the shape-sensing spars. For steady

hydroelastic testing, the unmodified PVC hydrofoil

was used (model 1, pictured on top). For modal

testing, an aluminum plate was attached to the trailing

edge of the hydrofoil (model 2, pictured on bottom).

Locations of the center of gravity and shear center are

shown for each version of the foil section. (b)

Rendering of PVC hydrofoil and instrumentation.

Shape-sensing spars are installed into interior channels

of the strut, and a linear shaker-motor provides

excitation to the structure. The coordinate system, as

pictured, has an origin located at the intersection of the

mid-thickness, mid-chord, and root planes.

As in previous studies, the test matrix

canvassed a three-dimensional parametric space

described by the yaw-angle (identically referred to as

the geometric angle of attack, 𝛼), the depth-based

Froude number,

𝐹𝑛ℎ =𝑈

√𝑔ℎ, (2)

and the immersed aspect ratio,

𝐴𝑅ℎ =ℎ

𝑐,

(3)

where ℎ denotes the immersed depth of the foil’s free

tip, 𝑐 is the chord-length of the foil, 𝑈 is the forward

speed of the foil, and 𝑔 is gravitational acceleration.

These quantities are shown in Figure 2.

A general method of inferring the static and

dynamic motions of the 3D hydrofoil was sought,

which was not reliant upon optical measurement of the

foil’s displacement. To this end, a type of shape-

sensing spar was developed. The method entails

instrumenting one or more slender aluminum beams,

or “spars” with strain gauges, which are then placed

inside of a parent structure. The deflections of the

spars are inferred from the measurements of strain,

which in turn permits the motions of the parent

structure to be inferred. A preliminary description of

the device and analysis method was presented by

Harwood, et al (2016), and is summarized below.

Figure 2. Sketch of the experiment installed on the

carriage of the University of Michigan towing tank.

The variables of interest are indicated in the sketch.

The shape-sensing methodology is based

upon the beam-bending equation,

𝜖𝑧 = −𝑡

2

𝜕2𝑌

𝜕𝑍2, (4)

where 𝜖𝑧 is the bending strain measured on the surface

of a beam of thickness 𝑡. A slim aluminum beam is

instrumented with single-axis strain-gauges in half-

cosine spacing along the Z-axis. In the present study,

eight gauges were used on each beam, arranged in four

half-bridges, but in general, 𝑁 locations may be

instrumented. Measuring the strain directly at 𝑁

locations and assuming a strain-free boundary

condition at the free tip (𝜖|𝑍=𝐿 = 0), an 𝑁𝑡ℎ order

polynomial is fitted to the measured strains, yielding,

𝜖�̂�(𝑍) = ∑ 𝑎𝑖𝑍𝑖

𝑁

𝑖=0

. (5)

The deflection of the beam in the 𝑌 direction is, from

equation 4, the second-integral of the strain-

distribution. The deflection may therefore be

estimated by the polynomial,

�̂�𝑍(𝑍) = ∑ 𝐴𝑖𝑍𝑖

𝑁+2

𝑖=0

, (6)

where

𝐴𝑖 = {𝑎𝑖−2(𝑖 − 2)!

𝑖!, 𝑖 = 2,3, … . 𝑁 + 2. (7)

Coefficients 𝐴0 and 𝐴1 are determined from the

boundary conditions at the root of the spar. If the spar

is perfectly clamped at the root, then 𝐴0 = 𝐴1 = 0.

However, early testing demonstrated that the root

boundary condition of the spars in-situ was better-

described by a torsional spring, which permits a

nonzero slope at the root, i.e. �̂�′(𝑍)|𝑍=0

≠ 0, while no

displacement occurs at the root. 𝐴0 = 0 satisfies the

zero-displacement conditions, and 𝐴1 is defined as,

𝐴1 = −𝜅𝜖�̂�|𝑍=0. (8)

Here, 𝜅 is a dimensionless compliance factor, which

permits the root of the spar to take on a nonzero slope,

proportional to the applied moment at the root.

Once 𝜅 is determined, as described in the

following text, then the deflection along the Y-axis

along the beam is given by EQ 6. The process is

illustrated in Figure 3. With two beams installed into

the strut at different chord-wise locations, the two

deflections can be re-cast as bending and twisting

deformations at each section. The two spars are

respectively denoted by A and B, and the twisting

deformations of the strut about the elastic axis (the

locus of sectional shear centers) are expressed,

𝜃(𝑍) = arcsin (�̂�𝐴(𝑍) − �̂�𝐵(𝑍)

𝑋𝐴 − 𝑋𝐵

), (9)

Where 𝑋𝐴 and 𝑋𝐵 are, respectively, the chordwise

locations of spars A and B, measured forward from

mid-chord. The bending deformations at the mid-

chord position and the elastic axis may then be

expressed

𝛿𝑀𝐶(𝑍) = �̂�𝐴(𝑍) − 𝑋𝐴 sin(𝜃(𝑍)); (10)

𝛿𝐸𝐴(𝑍) = �̂�𝐴(𝑍) − ⋯

(𝑋𝐸𝐴 − 𝑋𝐴) sin(𝜃(𝑍)). (11)

The root compliance factors 𝜅𝐴 and 𝜅𝐵 were

determined using an in-situ calibration. The strut was

clamped at the root and concentrated tension loads

were applied at a number of known chord-wise and

span-wise positions. The lateral deflections at the foil

leading and trailing edges were measured at another

known span-wise position, straight linear rules, as

shown in Figure 4. A variety of loading conditions

were applied, including pure bending (load applied

along the elastic axis), pure torsion (a couple applied

such that the lateral motion at the elastic axis was

zero), and eccentric loading (loading at chord-

positions not coincident with the elastic axis). The

measurements of the leading and trailing edge as

bending and twisting deformations at the elastic axis.

A nonlinear minimization algorithm was used to find

the vector {𝜅𝐴, 𝜅𝐵} that minimized the 𝐿2 norm of the

error in the bending and twisting deformations

predicted at the same location by the shape-sensing

spars across all tests. This process yielded a final root

compliance vector of {𝜅𝐴, 𝜅𝐵} = {7.56, 4.77}.

The results are shown in Figure 5 for the

point-load testing of the hydrofoil, with the optimized

root-compliance vector. The measured and inferred

deflections of the hydrofoil in bending and twisting,

when subjected to bending, twisting, and combined

loads, are shown. The percentage error was calculated

as the ratio of the root-mean-square error to the root-

mean-square of the deflections. In bending, for

example,

% 𝐸𝑟𝑟𝑜𝑟 ≡𝑅𝑀𝑆(𝛿 − �̂�)

𝑅𝑀𝑆(𝛿). (12)

Figure 3. Functional diagram of the shape-sensing

method used in this work. A polynomial estimator of

the strain is fitted to point-measurements and

integrated to yield the deformation. A compliance

factor permits a non-zero slope at the clamped root of

the beam.

Figure 4. In-situ calibration of the 3D shape-sensing

beams. The bending and twisting deflections of the foil

were measured using linear rules at the leading and

trailing edges, measured at a known spanwise station.

The root-compliance factors were optimized to

minimize the error between the deflections inferred

from the shape-sensing beams relative to those

measured using the rulers (reference).

Figure 5. Characterization of error in shape-sensing

spars, after optimization of the root-compliance

values. Reference data were measured using the setup

depicted in Figure 4. Loads were a combination of

pure-bending, pure-torsion, and combined bending

and torsion. Measurements were taken at two

spanwise locations: 78.4 cm and 87 cm from the root

of the foil (hence the two distinct slopes evident in the

plot of 𝛿𝐸𝐴 vs 𝐹𝑌). The grey shading indicates the range

of loads measured during towing-tank tests.

The performance of the shape-sensing spars

is very good in bending, with an RMS error of less than

2.5% of the RMS of deflections at the elastic axis. In

twisting, the performance is less robust, with a ratio of

approximately 20%. Note that the largest proportional

error in twist measurements occurs near zero twist,

corresponding to cases where the load was applied

along the elastic axis to induce pure bending. In these

cases, a finite twisting deformation was inferred by the

shape-sensing beams. The reason for the spurious

twist measurement is not precisely known, but it is

Tension

Linear Rule

Linear Rule

believed to follow from the imperfectly-clamped

boundary condition at the foil’s root and, possibly,

from imperfect mounting of the strain gauges. It is

unsurprising that the performance of the shape-sensing

spars in torsion is less reliable than in bending because

the twist angle is computed from the small difference

between two relatively large deformations along the

chord. Thus, even negligible errors in the lateral

deflection along either spar are amplified in the twist-

measurement error.

The performance of the shape-sensing spars

is actually quite promising. They permit deflections to

be reliably inferred without a line-of-sight to the

hydrofoil and without interfering with the flow.

MODAL ANALYSIS OF FLEXIBLE FOIL

To identify the resonant frequencies, mode shapes, and

modal damping factors of the reinforced PVC

hydrofoil (model 2), modal analysis was conducted.

The shaker motor was used produce harmonic

excitation of the hydrofoil over a range of frequencies

of interest. The ratio of output to input signals may be

represented in the frequency domain by a complex-

valued frequency response function (FRF). When the

output has units of acceleration, the FRF is known as

an inertance function. When the output is a structural

deflection, the FRF is a compliance function. The

excitation force was measured as the system input.

Output measurements consisted of tip-accelerations

and the inferred deflections from the shape-sensing

spars. The two types of outputs were used to construct

estimates of the inertance and compliance FRF of the

hydrofoil, respectively. Tests were conducted with the

hydrofoil in air and at an immersed aspect ratio of

𝐴𝑅ℎ = 1, in still water and with forward speed in

wetted and ventilated flow.

The excitation signal was a linearly-swept

sine wave. The data-acquisition PC was used to

generate a pulse train at the desired frequency, to

which a pair of lock-in amplifiers (Stanford Research

Systems SR830) were phase-locked and used to

generate a sinusoid at the commanded frequency. The

sine wave was amplified through a conventional

analogue power-amplifier to drive the shaker motor.

The accelerometers at the tip of each shape-sensing

spar were connected to inputs of the lock-in amplifiers,

permitting the magnitude and phase of the structural

accelerations to be monitored on-line.

The excitation force at the drive-point of the

shaker motor was recorded as the system input. The

foil-tip accelerations were recorded as system

acceleration outputs. The inferred deflections, from

the shape sensing spars, were interpolated at 20 points

and recorded as the deflection outputs. The over-

interpolation of the deflections provided robustness

against numerical issues in the following analyses.

Simultaneous measurements of the

respective outputs at different points on the structure

are combined to form FRF matrices. With the shaker-

motor’s location fixed, the inferred motions of the strut

and the tip accelerations were used to estimate

columns of the compliance and inertance FRF

matrices. Estimation of each FRF was accomplished

using smoothed auto- and cross-spectral densities of

the input and respective output. The resulting FRF

estimates are denoted 𝐇𝟏𝐢𝐧𝐞𝐫𝐭 and 𝐇𝟏

𝐜𝐨𝐦𝐩 for the

inertance and compliance FRF’s, respectively. The

EASYMOD library for MATLAB was used to

generate the FRF estimates (Kouroussis, et al., 2012).

Mode shapes are estimated from the

operating deflection shapes (ODS) of the hydrofoil.

Resonances may be identified as peaks on a plot of the

FRF matrix. At a given frequency, the imaginary

component of the FRF matrix yields a vector of the

structure’s deflections or accelerations at that

respective frequency. Thus, an ODS taken at a

frequency near one of the resonant modes of the

structure will be dominated by the underlying linear

normal mode shape of the structure.

Figure 6 shows the ODS of the reinforced

PVC hydrofoil at each of the identified modes in dry

conditions. The shaker motor was used to excite the

hydrofoil with a linearly-swept sine wave between 2

Hz and 150 Hz. The complex-valued compliance FRF

matrix is shown; at each identified mode, the

imaginary components of the FRF matrix are used to

form the ODS, which approximates the underlying

mode shape. Undamped modal frequencies and mode

shapes, computed with the commercial finite element

code ANSYS Mechanical, are shown below the ODS

reconstructions. In general, they compare well with the

ODS estimates from the shape-sensing spars.

The rational fractional polynomial (RFP)

method is a curve-fitting approach for identifying the

modes of multi-degree-of-freedom (MDOF) systems

with closely-coupled modes (Richardson & Formenti,

1982). The approach fits an analytical transfer

function model to the complex-valued FRF, using

orthogonal polynomial basis functions. The resulting

poles and zeros are used to yield the undamped natural

frequency, damping ratios, and modal residues for

each of a specified number of modes. With this

approach, the mode shapes, resonant frequencies, and

damping ratios for each mode may be identified

independently, even while those modes are not well-

separated in the frequency domain. In the present

work, the RFP method was used to identify the

resonant frequency and damping factor of each mode

from each of the elements in the compliance and

inertance FRF matrices. The means and standard

deviations are then computed from the fitted values.

Figure 6. Operational deflection shapes (ODS) of the reinforced PVC hydrofoil (mode 2) in dry conditions. The real

and imaginary components of the compliance FRF matrix, 𝐇𝟏𝐜𝐨𝐦𝐩

, are shown in the top two axes; twenty lines are

overlaid, corresponding to twenty interpolated points on the foil’s surface at which the lateral deflections were

estimated. The 3D surfaces represent the ODS of the hydrofoil with bold black lines indicating nodal lines of the

respective modes. The undamped linear normal modes and modal frequencies, computed by 3D finite-element method

(FEM) are shown at the bottom for comparison. The mode shapes and resonant frequencies obtained using the shape-

sensing spars both agree well with the FEM analysis.

RESULTS

The results this work are presented in two subsections.

First, the effects of ventilation and hydrodynamic

loading on the static structural response will be shown,

along with the coupled effects of the structural

deflections on the hydrodynamic response. The static

response corresponds to cases without excitation from

the shaker motor. The dynamic hydroelastic response,

including external excitation via the shaker motor, will

follow. These results will be compared with the

following theoretical model.

Consider the two-dimensional, two-degree-

of-freedom (2D, 2-DOF) model in Figure 7, which

depicts a section taken through the flexible hydrofoil

at some spanwise station. Bending and twisting

deformations are respectively denoted as 𝛿 and 𝜃, both

measured at the elastic axis of the section, formed by

the locus of sectional shear-centers. The structural

stiffness values of the hydrofoil in bending and

twisting are respectively denoted 𝐾𝑠,𝛿 and 𝐾𝑠,𝜃, while

damping values are denoted by 𝐶𝑠,𝛿 and 𝐶𝑠,𝜃.

Figure 7. 2D, 2-DOF model of a flexible hydrofoil

section.

The general equations of motion for the 2D

model may be written in matrix form as,

[𝑚 𝑆𝜃

𝑆𝜃 𝐼𝜃] {�̈�

�̈�} + [

𝐶𝑠,𝛿 0

0 𝐶𝑠,𝜃] {�̇�

�̇�} …

+ [𝐾𝑠,𝛿 0

0 𝐾𝑠,𝜃] {

𝛿𝜃

}

= 𝑭𝑬𝑿 + 𝑭𝐻 .

(13)

𝑚 is the section’s mass per unit span, 𝐼𝜃 is the polar

mass moment of inertia about the elastic axis, and 𝑆𝜃

is the static unbalance.

𝑭𝑬𝑿 is a vector of the disturbing force and

moment exerted by an external source. 𝑭𝑯 is a general

form of the fluid disturbing force vector,

𝑭𝑯 = {𝑙

𝑚𝑍}. (14)

Where 𝑙 and 𝑚𝑍 are respectively the lift and moment

per unit span. Steady, quasi steady, and unsteady

hydrodynamic theory may be used to model 𝐹𝐻. The

resulting hydrodynamic forces can augment or detract

from the effective system mass, damping, and stiffness

matrices, and can induce coupling between otherwise-

decoupled degrees of freedom (Chae, et al, 2016a,

2016b).

Static Hydroelastic Response (Model 1)

In steady conditions, the hydrodynamic force

components are given by steady-state hydrodynamic

theory. Using linear theory, the lift per unit span is

given by,

𝑙 = 𝐶𝑙𝑞𝑐 = 𝑎0𝛼𝑒𝑞𝑐, (15)

where 𝐶𝑙 is the 2D lift coefficient, 𝑞 =𝜌

2𝑈2 is the fluid

dynamic pressure, and 𝑎0 is the linearized slope of the

lift-coefficient as a function of the angle of attack, 𝛼. The twisting moment is given by,

𝑚𝑍 = 𝐶𝑚𝑞𝑐2 = 𝑎0𝛼𝑒

2𝑞𝑐2, (16)

where 𝑒 is the distance from the elastic axis to the

center of pressure, nondimensionalized by the semi-

chord. With EQs 15 and 16 substituted into 𝑭𝑯, EQ 13

may be solved for the steady-state deflections,

𝛿 =𝑎0𝛼𝑞𝑐

𝐾𝑠,𝛿

(1 +𝑎0

𝑒2

𝑞𝑐2

𝐾𝑠,𝜃 − 𝑎0𝑒2

𝑞𝑐2) ; (17)

𝜃 =𝑎0𝛼𝑒

𝑒2

𝑞𝑐2

𝐾𝑠,𝜃 − 𝑎0𝑒2

𝑞𝑐2. (18)

Contours of the 3D lift and moment

coefficients, 𝐶𝐿 and 𝐶𝑀 are shown in Figure 8, along

with contours of the dimensionless static bending and

twisting deformations measured the tip of the

hydrofoil. The 3D lift and moment coefficients are

defined as,

𝐶𝐿 =𝐿

𝑞𝑐ℎ; (19)

𝐶𝑀 =𝑀𝑍

𝑞𝑐2ℎ. (20)

The FW and FV flow regimes are respectively

represented by the blue and red contour lines. As also

described in Breslin & Skalak, (1959); Harwood et al.,

(2014); Harwood, et al, (2016b); Rothblum et al.,

(1969); Rothblum, (1977); Swales, et al, (1973);

Swales et al, (1974), the transition from FW to FV

flow is accompanied by dramatic reductions in the lift

and moment coefficients. In general, the lift is reduced

because the ingestion of air along the suction-surface

of the strut limits the development of low-pressures

which are instrumental in generating large forces. The

moment coefficient is attenuated both because the lift

is reduced and because the center of pressure translates

from its location of approximately 0.25𝑐 forward of

mid-chord in FW flow towards a position at mid-chord

in FV flow, causing both the lift and its moment arm

to be reduced. The result of the attenuated lift is a

proportional attenuation of the tip deflections, and a

result of the reduction in twisting moment is a

reduction in the static twist angle.

Manipulating EQ 18 yields the dimensionless

scaling expression,

𝜃

𝜃 + 𝛼=

𝑐

2𝐾𝑠,𝜃

𝑒 𝑎0 𝑞 ∝ 𝑒𝑎0𝐹𝑛ℎ2 . (21)

For a given lift-slope, 𝑎0, and a fixed center of pressure

(constant 𝑒), EQ 21 predicts a linear relationship

between the squared Froude number and the twist

ratio. The dimensionless ratio on the left-hand side of

EQ 21 is plotted in Figure 9 as a function of 𝐹𝑛ℎ2 . The

data in all three flow regimes (FW, PV, and FV) are

bounded above by the FW data, which lie along a line,

and bounded below by the FV data at 𝛼 = 25∘. It was

shown by (Harwood, et al, 2016b) that, as 𝛼 and 𝐹𝑛ℎ

increase, the mean cavity length increases, causing

both 𝑎0 and 𝑒 to decrease. The FV data behave in a

sublinear fashion with respect to 𝐹𝑛ℎ2 , and the

dimensionless twist decreases with increasing angle of

attack. Both characteristics are consistent with the

notion that both 𝑒 and 𝑎0 decrease as the ventilated

cavity grows with increasing 𝛼 and 𝐹𝑛ℎ.

Figure 8. Contour plots of (a) 3D lift coefficient, 𝐶𝐿, (b) 3D yawing moment coefficient (about mid-chord), 𝐶𝑀, (c)

Bending deformation at free tip (measured at mid-chord), and (d) Twisting deformation at free tip. All forces and

deformations are steady-state. Blue contours indicate values in the fully-wetted (FW) flow regime, and red contours

indicate the fully-ventilated (FV) flow regime.

Figure 9. Dimensionless twist ratio plotted against the

squared Froude number. Data are bounded above by

FW results, which collapse along a line, and below by

the FV data at 𝛼 = 25∘. The FV data asymptotically

approach the lower bound as 𝛼, 𝐹𝑛ℎ increase,

indicating reductions in 𝑎0 and 𝑒 caused by an

increasing cavity length.

Hydroelastic Coupling Effects

The results of the preceding section

demonstrate that the hydrodynamic loading has a

measurable effect on the structural response of the

flexible hydrofoil (model 1). When the hydrofoil

assumes a static twisting deformation, the elastic twist

angle, 𝜃, contributes toward the total effective angle of

attack, 𝛼𝑒 = 𝛼 + 𝜃. It is apparent from EQ 15 and EQ

16 that the induced twist will result in larger lift and

moment coefficients compared to those of a rigid foil.

The 3D lift, drag, and yawing coefficients are shown

in Figure 10. The static twist angle causes a small

increase in the lift and yawing moment coefficients,

relative to the rigid hydrofoil. The change is most

pronounced in the FW flow regime, where the moment

arm of the lift is greatest.

The drag coefficient is insensitive to the

effect of foil flexibility. As discussed by Breslin &

Skalak, (1959) and Harwood, et al, (2016b), the drag

coefficient is a sum of profile, spray, and lift-induced

components. Only the lift-induced drag has a first-

order dependence upon the effective angle of attack.

Thus, small changes in 𝛼𝑒 are expected to have only a

small effect on the aggregate drag coefficient.

Figure 10. (a) 3D lift coefficient, (b) 3D drag

coefficient, and (c) 3D moment coefficient (about

mid-chord) as functions of 𝛼 for 𝐹𝑛ℎ = 3, 𝐴𝑅ℎ = 1. Results shown were measured on the rigid aluminum

hydrofoil from Harwood, et al. (2014) and Harwood,

et al. (2016b) and the flexible PVC hydrofoil (model

1). The lift and moment coefficients are larger for the

flexible hydrofoil – a consequence of the elastic twist

angle 𝜃. The drag coefficient is less sensitive because

the viscous and form drag are insensitive to small

changes in 𝛼𝑒.

Harwood, et al, (2016) and Young, et al,

(2016) defined the unsteady transition processes of

ventilated flow: ventilation formation (comprised of

inception and stabilization) and ventilation elimination

(comprised of washout and re-wetting). These serve to

delineate the FW, PV, and FV flow regimes from one

another in a parametric space defined by 𝛼 and 𝐹𝑛ℎ by

forming transition boundaries. The formation and

elimination boundaries for the rigid and flexible

(model 1) hydrofoils are shown in Figure 11a and

Figure 11b, respectively. As in the steady

hydrodynamic loads, the effect of foil flexibility is

subtle, but consistent with the trends observed thus far.

All four boundaries (inception, stabilization, washout,

and re-wetting) for the flexible hydrofoil fall slightly

below and/or to the left of the boundaries for the rigid

hydrofoil. This indicates that for a given initial angle

of attack, 𝛼, the deformation of the hydrofoil causes

ventilation transition mechanisms to occur at

somewhat smaller Froude numbers than in the case of

the rigid hydrofoil, with the potential for slightly

greater hysteresis in the transition between regimes.

Dynamic Hydroelastic Response (Model 2)

In unsteady flows, the hydrodynamic force

vector in EQ 14 may be computed from a variety of

unsteady aerodynamic or hydrodynamic theories,

including Theodorsen’s function (Theodorsen, 1949)

or a convolution of indicial response functions based

upon Sears and Wagner functions (Bisplinghoff et al.,

1996; Brunton & Rowley, 2010; Fung, 1955; Shen &

Fuhs, 1997). The hydrodynamic disturbing force can

manifest as apparent changes to the effective mass,

damping, and stiffness matrices of the dynamical

system. As a result, the modal frequencies and

parameters of the system can change as functions of

the flow conditions and structural motions.

Figure 12 shows the averaged compliance

FRF magnitude of the reinforced PVC hydrofoil

(model 2) as a function of Froude number in the FW

and FV flow regimes. The frequency spectra of the

hydrofoil in dry conditions is also shown. The

locations of the frequency peaks, indicative of

resonance, are projected on the frequency − Fnh

plane, and are reproduced in Figure 13. Relative to the

dry resonant frequencies, the wetted modal

frequencies are significantly reduced, and the

percentage reduction varies from mode to mode. The

same observation was made by Harwood, et al,

(2016a); Kramer, et al, (2013); Motley, et al, (2013).

The reason for the reduction in frequency is an

increase in the fluid added-mass as the foil moves from

a light fluid (air) to a dense one (water). In general, the

wetted modal frequencies slightly decrease with

increasing forward speed or Froude number. This

trend is not consistent across all modes or Froude

(a)

(b)

(c)

numbers, however. Certain modes are probably

affected also by a fluid stiffness term in the unsteady

hydrodynamic load that increases with speed

(Theodorsen, 1949; Chae, et al,. 2016a). When

ventilation occurs, the resonant frequencies

significantly increase, relative to their wetted values.

This is caused by the replacement of water by air along

the suction side of the hydrofoil, and thus a decrease

in the fluid added-mass.

Figure 14 shows estimates of the modal

damping factor, 𝜁, for each mode as a function of

Froude number in FW and FV flow regimes. There is

considerable uncertainty in the data, but the results

suggest a slight increase in damping with both

increasing forward speed and with ventilation. It is

possible that the presence of the cavity interface

provides additional mechanisms of mechanical energy

dissipation (i.e. wave generation on the cavity wall).

We intend to explore damping measurements further

in future experiments.

Figure 11. (a) Ventilation formation boundaries for

the rigid aluminum and flexible PVC hydrofoil (model

1, without reinforcing plate). (b) Ventilation

elimination boundaries for rigid aluminum and

flexible PVC (model 1) hydrofoils.

Figure 12. Compliance frequency response function

estimator, 𝐇𝟏𝐜𝐨𝐦𝐩

, of reinforced PVC hydrofoil (model

2) as a function of Froude number. The black spectrum

corresponds to the foil in dry conditions, the modal

frequencies of which are indicated by dashed black

lines. FW and FV data are for runs at 𝐴𝑅ℎ = 1, shown

as blue and red lines, respectively. Empty regions

indicate a lack of experimental data.

Figure 13. Variation in the modal frequencies of the

first three modes of model 2 with Froude number at

𝐴𝑅ℎ = 1. The modal frequencies of the foil are

significantly smaller in wetted conditions (blue lines)

than in dry conditions (black lines). Ventilation causes

an increase in modal frequencies (red lines). Changes

in modal frequencies with forward speed are non-

monotonic, and are probably attributable to fluid-

stiffness terms in the unsteady hydrodynamic force

(Theodorsen, 1949; Chae, et al., 2016a).

(a)

(b)

Figure 14. Estimates of the modal damping factor, 𝜁, for the first three modes as a function of Froude

number. Data are for the reinforced PVC hydrofoil

(model 2) at 𝐴𝑅ℎ = 1. Solid lines correspond to FW

flow; dashed lines correspond to FV flow.

DISCUSSION AND CONCLUSIONS

A series of towing-tank experiments were conducted

on two flexible hydrofoil models to examine the static

and dynamic fluid-structure interactions in wetted and

ventilated flows. To do this, a novel and robust 3D

shape-sensing methodology was developed, using

polynomial-fitting of measured strain distributions on

modular aluminum spars. This allows real-time

reconstruction of bending motions of up to ±10 cm

(0.11 S) with a measurement uncertainty of 2.5%.

Twisting deformations up to ±4∘ were tested with a

measurement uncertainty of 20% (though the error

was smaller under all loading conditions except pure

bending). The maximum deflections during towing-

tank tests were within |𝛿𝐸𝐴| ≤ 10 cm and |𝜃| ≤ 1.5∘

at the free tip. The shape sensing spars were also used

to successfully recover the expected mode shapes from

the compliance FRF matrix. The ODS extraction can

operate on-line, and has been found to be a valuable

tool for identifying excited modes during

hydrodynamic experiments.

The static hydroelastic response of the

flexible PVC foil (model 1) was consistent with

measured loads; bending deflections were

approximately proportional to the measured lift and

twisting deflections were proportional to the yawing

moment. Formation of ventilated cavities reduced

hydrodynamic loads, and hence the bending

deflections, by up to 50%. Twisting motions remained

relatively small for the flexible hydrofoil. The ratio of

twisting deflection was a linear function of 𝐹𝑛ℎ2 in the

FW regime, and collapsed to a different linear trend as

the cavity length grew in the FV regime.

A small increase in steady-state lift was observed in

the flexible hydrofoil, relative to the rigid hydrofoil,

consistent with the small twist angle induced in the

flexible hydrofoil. Ventilation transition mechanisms

also occurred at slightly lower 𝐹𝑛ℎ and 𝛼 on the

flexible hydrofoil, relative to the rigid aluminum foil.

Experimental results revealed changes in the

1st, 2nd, and 3rd resonant frequencies of the reinforced

flexible hydrofoil (model 2), both with forward speed

and with ventilation. Frequencies decreased with

increasing immersion because of increased fluid

added-mass. Frequencies increased with ventilation as

a result of a reduction in the added-mass. Added-mass

is mode-dependent, so changes in resonant frequencies

with immersion and ventilation varied from mode to

mode. Frequencies could increase or decrease with

forward speed, depending on the degree of coupling

with other nearby modes and the presence of fluid-

stiffness terms in the unsteady hydrodynamic force.

Additionally, the evolution of the unsteady

hydrodynamic loading with increasing speed,

ventilation, and/or changing immersion depth may

also alter the modal frequencies.

Modal damping factors increased with

increasing Froude number for all modes. Ventilation

resulted in a small increase in modal damping factors,

though significant uncertainty in the damping

measurements leaves the question open to future

investigation.

In future studies, we will strive for improved

modal parameter estimation by using longer-duration

frequency sweeps and improved signal processing.

Shaker testing will be performed under a variety of

operating conditions, including wetted flow, ventilated

flow, and vaporous cavitating flow. Additionally, we

will attempt to replace ODS estimates with higher-

quality estimates of the normal modes shapes. Future

work will also seek to identify the effects of unsteady

vaporous cavities on the hydrodynamic loading, as

well as the phenomenon of lock-in with periodic

cavity and vortex shedding. The shape-sensing spars

also offer a promising avenue for further development.

A second generation should make use of one member

instrumented for bending strain and the other for shear

strain – such that bending and torsion can be more-

directly resolved.

ACKNOWLEDGEMENTS

This material is based upon work supported by the

National Science Foundation Graduate Student

Research Fellowship under Grant No. DGE 1256260.

The authors would also like to acknowledge the

support of Dr. Ki-Han Kim under ONR grant number

N00014-13-1-0383 and N00014-16-1-2433.

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DISCUSSION #1

Thomas C. Fu, Ship System and Engineering Division

(331)

Office of Naval Research

The authors should be commended for a very complete

and detailed experimental study of a very complicated

flow phenomenon.

The authors state that the three “objectives of

this paper are as follows: (1) Develop and validate a

non-optical method of measuring the static and

dynamic deformations of a flexible surface-piercing

hydrofoil in wetted and ventilated flows; (2) Quantify

the effects of wetted and ventilated flow regimes on

the steady and unsteady hydro-elastic response of the

flexible hydrofoil; and (3) Quantify the effects of

spanwise bending and twisting deformations on

steady-state hydrodynamic loads and flow regime

stability boundaries.

1. It appears that the “validation” of the

development and calibration of the shape-sensing

beam can the authors comment on any steps they

took or concerns they have concerning the

responsiveness of the system to dynamic loading,

and how they fastened the foil to the spars?

2. Additionally, can the authors comment further on

the large uncertainty in the damping

measurements and things that could be done to

try to reduce this uncertainty?

AUTHOR’S REPLY

We wish to thank Dr. Fu for taking the time to review

our submission, and we appreciate his feedback. We

have provided responses to each discussion point in

the following text.

Question 1: The validation & calibration process

reported in this work concerned only static loads, with

the assumption that the shape-sensing spars would

behave in substantially the same manner when

dynamically loaded. Unlike accelerometers, rate

gyros, or other similar transducers, the strain gauges

should not suffer from phasing issues, resonance, or

finite response times because they are not themselves

characterized as dynamical systems.

The fastening of the spars to the foil was

accomplished exclusively through an interference fit.

The passages inside of the foil were machined to very

tight tolerance, so that the fitment of the spars into the

passages is as tight as possible while still allowing the

spars to be pulled out. Figure A1 shows the hydrofoil’s

interior, prior to the two halves being chemically

welded. The spars are inserted through the root of the

foil, as shown in Figure A2(a) and pushed into the

channels until flush with the root, as in Figure A2(b).

We reason that with a spar running nearly the full span

of the foil, reasonably tight tolerances would permit

only negligible differences between the deformation

of the foil and that of the spar in question, whether the

deformation is static or dynamic.

Figure A15. The PVC hydrofoil during construction. The passages for the strain-gauge-equipped spars are visible.

The two halves of the hydrofoil were later chemically welded together to form a waterproof and materially-

homogenous structure.

To further support these assertions, figure A3

shows the time-history of the tip deflections (bending

and torsion) measured with the spars and measured

optically by tracking markers with a high-speed

camera. The deflections inferred from the strain

gauges appear to respond very well to the dynamic

loading, and actually exhibited lower noise and better

spatial resolution than the optical measurements.

Question 2: The uncertainty in damping estimates has

been one of our principal concerns. There are several

reasons that the damping estimates shown in this work

have such large uncertainty. First, our ability to

resolve the phase of the deflections and accelerations

relative to the shaker input assumes that the vibration

is stationary. Additionally, a relatively fine frequency

resolution is required to resolve resonant conditions.

When testing in the towing tank, the duration of a run

during which to execute a frequency sweep and

measure the response is short. Thus, it is difficult to

ensure that transients have died out sufficiently at each

frequency, and the frequency spacing is relatively

coarse as a result of the limited steady-state time. The

problems are exacerbated further at higher speeds,

which explains the growth of the uncertainty bars with

the hydrofoil’s speed.

Recently, we repeated many of the same tests

with the foil mounted in a recirculating flow channel,

so that long-duration sweeps could be achieved.

Additionally, the excitation signal was carefully

stepped through the frequency range, with a sufficient

number of periods at each frequency to ensure a

stationary response. These steps have, so far, helped

significantly reduce the uncertainty in damping

estimates. We are currently working to publish these

recent results in a journal article.

(a) (b)

Figure A16. The shape-sensing spars are held inside the passages by a tight interference fit. The spars are inserted

into the foil’s root, shown in (a), and pushed along the span of the hydrofoil until flush with the mounting tang, as

shown in (b).

DISCUSSION #2

Luca Savio, Research Scientist,

MARINTEK

The paper contains experimental material that is of

general interest for the community of researchers

active in the field of hydroelasticity and more notably

obtained by designing an instrument specifically for

the tests; the former aspect is always welcome while

the latter makes the work very interesting especially in

these days of little interest for new experimental

techniques. Furthermore, the calibration of the

experimental device is thoroughly detailed and many

critical aspects are discussed and addressed.

The work is, therefore, valued as very

positive and consequently my comments aim at giving

the authors some input on how to make the

experimental setup even more robust, at least

according to me. My comments are grouped into two

categories: experimental setup and future work.

Comments on the experimental setup:

The task of designing a strain gauges based sensor is

far from trivial as many aspects come into play and a

good sensor is one that manages to find a good balance

among them. During the calibration the authors

showed that a good balance has been indeed reached.

However, I would recommend considering the

following aspects, which I have further classified as

static and dynamic:

Static:

1. The A/D converter that has been used is a 16-bit

system; however, the research standard is

nowadays 21-bit for strain measurement. It is

worth checking if the bit depth of the converter

has anything to do with the accuracy of the

twisting deformation.

2. The 6 components balance at the root of the foil is

the source of elasticity that results in the root

boundary condition to be a torsional spring. One

might think eliminating the balance to obtain a

perfect clamped condition at the root to check

whether the accuracy of the twisting deformation

measurement is improved.

Dynamic:

3. The A/D converts have been hardware

synchronized. Hardware synchronization provides

a good mean of making the A/D converters

configuration flexible; however, in my experience

the phase consistency is compromised when

channels are logged on different logging cards. In

the paper it is not specified whether any thoughts

was given to this topic. In case channels that have

been logged on different physical cards are then

used to compute derived variables the consistency

of the phase information across channels should be

checked.

4. The A/D converters were operated at 1000-2000

Hz; this would place the cutoff frequency of the

antialiasing filter at 100-200 Hz. If this

corresponds with what has been done, then the

filter would introduce again a phase distortion that

increases with the frequency according to the filter

characteristics. Increasing the sampling frequency

and consequently the antialiasing filter cutoff

frequency should be considered. Typical values for

the frequencies of interest in this work would be

sampling at 10 000 Hz and antialiasing at 1 000 Hz.

Future Work

5. When it comes to future development of the

presented technique no mention is made of the

plans of the authors for improvement. What are the

plans for the future?

6. The shape sensing spars provided a good accuracy

for bending, while the 20% uncertainty of twisting

is somewhat too large for the twist angle

measurement to be fully trusted, at the same time

hydroelastic tailoring relies more on the capability

of the hydrofoil to twist rather than to bend;

therefore as my last comment, I would recommend

the authors to focus on the twisting deformation so

that the proposed experimental technique could

provide useful validation data also for hydroelastic

tailoring.

AUTHOR’S REPLY

We would like to thank Dr. Savio for taking the time

for such a detailed critique. Responses to each of his

points of inquiry are provided below.

Question 1: The use of 16-bit A/D boards certainly

reduces the dynamic range of the sensing setup, since

the quantization error is quite large compared to 21-bit

or 24-bit converters. That said, we strove to optimize

the measurement range such that quantization error

was as small as possible. Vishay 2310 analog signal

conditioning amplifiers were used to excite the bridges

and scale the output voltages, which were then

digitized by the A/D cards. If poor digital resolution

was a dominant source of error, we would expect to

see likewise-poor spatial resolution in the twisting

measurements. That was not the case; in fact, torsional

deflections were inferred with very good precision and

low noise. Figure A3, below, is produced from recent

tests. Deformations were measured with the strain-

gauge arrays as described in this paper, as well as by

optically tracking several markers on the foil’s tip with

a high-speed camera. The foil was excited at the first

bending frequency in still-water conditions. Note that

the optical measurements are quite noisy – a result of

dropouts in visual access to markers during certain

frames. However, the strain-gauge reconstructions are

quite smooth, with very good resolution in both

bending and torsion. This, of course, does not address

the issue of the sensors’ accuracy, but it suggests that

insufficient digital resolution is probably not a source

of twisting error.

Question 2: This is a very good point, and it would

certainly be interesting to perform the calibration

again with a rigid member in place of the load cell. The

load cell used in this test is an overbuilt unit with an

extremely stiff construction. Torsional spring

constants are 1.6×106 − 3.3×106 𝑁𝑚/𝑟𝑎𝑑 for each

axis, so angular deflections are expected to be on the

order of 0.01 degree for the loads reported herein. That

said, small discrepancies in measured deflections are

– as we mention in the report – amplified when

converted into torsional deflections, so the load cell is

quite feasibly a source of error.

Question 3: This is a pertinent question, and one upon

which we would like to expand our discussion

somewhat. There are two obvious reasons that data

channels may be logged asynchronously or out of

phase with one another. The first source is, as Dr.

Savio suggests, potential for phase mismatch between

the clocks on the individual boards in a component

DAQ system with many boards. In an ideal setup, all

sampling would be performed on a single board with

synchronized A/D converters. In reality, the boards

each used a single A/D converter muxed between 8-16

channels. The “switching” delay is another possible

source of phase mismatch between channels.

(a) (b)

Figure A17: (a) Markers on foil tip were tracked in high-speed camera footage during recent experiments. (b) Bending

and twisting deformations at the tip of the reinforced hydrofoil (Model 2) excited in the first bending mode (X-Bend-

1) in still-water conditions (𝐴𝑅ℎ = 1; 𝐹𝑛ℎ = 0; 𝛼 = 0∘), measured using optical marker-tracking and using the

shape-sensing spars. The two measurements agree well overall, but the values derived from strain measurements

demonstrate significantly better spacial resolution and lower noise.

We dedicated significant effort to timing-

considerations. With respect to board-to-board

synchronicity between the boards used, a master-slave

setup was established. A 10 MHz reference clock was

tapped from one board (master) and each of the other

two boards was set to phase-lock their onboard clocks

and A/D conversions with respect to the imported 10

MHz pulse train. “Start” and “stop” TTL triggers were

also used to synchronize the beginning and ending

samples. The muxing delay is impossible to eliminate,

but if one considers the relative timescales of the

sampling and muxing clocks, then the topic is of little

concern. A DAQ board typical of our setup has a

cumulative A/D conversion rate exceeding 106

samples / second. During one nominally synchronized

sample across, for example, 16 channels, the delay

between channel 1 and channel 16 would be at most

1.6×10−5 seconds. At 1000 Hz sampling-rate, this is

on the order of 1% of the sampling period.

We have, in the time since the reported

experiments, verified that synchronicity is maintained

between multiple channels on multiple boards (and

even across multiple DAQ systems) by acquiring

duplicate measurements of a common sine wave at

varying frequencies. The results of these tests showed

no appreciable phase discrepancies between channels

or cards.

Question 4: This is a very interesting point. While we

will surely pay close attention to our anti-aliasing

approaches (and will probably use much higher

sampling rates) in future work, the data in the reported

tests were collected without any anti-aliasing filtering.

While this is a shortfall in our experimental setup, we

do not believe that aliasing errors are too great a

concern. We reason that mean values are unaffected

by aliasing, and that aliased frequencies would simply

elevate our noise floor somewhat, thereby increasing

the uncertainty of steady-state measurements. During

vibration-testing with the electrodynamic shaker, the

shaker itself deposits most energy into a limited range

of frequencies below the Nyquist limit, effectively

band-limiting the response, so that aliased frequencies

are proportionally small, compared to the excited

response of the hydrofoil.

Question 5: We have amended the paper to mention

future plans for the work; the additional text is

reproduced below:

“In future studies, we will strive for improved modal

parameter estimation through the use of longer-

duration frequency sweeps and improved signal

processing. Shaker testing will be performed under a

variety of operating conditions, including wetted flow,

ventilated flow, and vaporous cavitating flow.

Additionally, future work will seek to identify the

effects of unsteady vaporous cavities on the

hydrodynamic loading, as well as the phenomenon of

lock-in when regular cavity and vortex shedding are

present. The shape-sensing spars also offer a

promising avenue for further development. In

particular, a second generation might make use of

separate bending and torsion members – one

instrumented for bending strain and the other for

shear strain – such that bending and torsion can be

more-directly resolved.”

Question 6: We agree with Dr. Savio on this point.

While the uncertainty of 20% is quite high, we suspect

that the number is at least partially inflated by

uncertainty in the reference data itself. Additionally, it

was found that, when subjected to eccentric loading-

cases only (i.e. when the pure-bending and pure-

torsion cases are removed from the validation data),

the RMSE is closer to 10% -- a far-more useable

number. Of course, it is hard to justify an arbitrary

removal of some training data, so we have opted to

report the conservative 20% until a more-thorough

verification study can be performed.

Aside from the fidelity of the benchmarking

data, there are obvious improvements to be made with

the design of the sensor itself. The sensors reported

upon in this work were the first iteration, and their

design highlighted the need to include shear-strain

measurements. Future designs may very well make

use of a bending spar and a torsion link – one

instrumented for bending strain and another for shear

strain – to more-directly resolve the torsional

deflections of the parent structure.