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Journal of Sound and Vibration

Journal of Sound and Vibration 333 (2014) 5004–5017

http://d0022-46

n Corr

journal homepage: www.elsevier.com/locate/jsvi

Added mass matrix estimation of beams partially immersedin water using measured dynamic responses

Fushun Liu a,n, Huajun Li a, Hongde Qin b, Bingchen Liang a

a Shandong Provincial Key Lab of Ocean Engineering, Ocean University of China, Qingdao 266100, Chinab College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China

a r t i c l e i n f o

Article history:Received 4 March 2014Received in revised form29 April 2014Accepted 17 May 2014

Handling Editor: L.G. Tham

which means that the mass and stiffness of the FEM and the added mass of water could be

Available online 13 June 2014

x.doi.org/10.1016/j.jsv.2014.05.0360X/& 2014 Elsevier Ltd. All rights reserved.

esponding author.

a b s t r a c t

An added mass matrix estimation method for beams partially immersed in water is proposedthat employs dynamic responses, which are measured when the structure is inwater and in air.Discrepancies such as mass and stiffness matrices between the finite element model (FEM) andreal structure could be separated from the added mass of water by a series of correction factors,

estimated simultaneously. Compared with traditional methods, the estimated added masscorrection factors of our approach will not be limited to be constant when FEM or theenvironment of the structure changed, meaning that the proposed method could reflect theinfluence of changes such as water depth, current, and so on. The greatest improvement is thatthe proposed method could estimate added mass of water without involving any water-relatedassumptions because all water influences are reflected in measured dynamic responses of thestructure in water. A five degrees-of-freedom (dofs) mass-spring system is used to study theperformance of the proposed scheme. The numerical results indicate that mass, stiffness, andadded mass correction factors could be estimated accurately when noise-free measurementsare used. Even when the first two modes are measured under the 5 percent corruption level,the added mass could be estimated properly. A steel cantilever beamwith a rectangular sectionin a water tank at Ocean University of China was also employed to study the added massinfluence on modal parameter identification and to investigate the performance of theproposed method. The experimental results demonstrated that the first two modal frequenciesand mode shapes of the updated model match well with the measured values by combiningthe estimated added mass in the initial FEM.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

A slender beam is considered to be a simple yet very frequently used engineering structure. The frequency of using thebeam “in air” is usually higher than “in water”, mainly due to the influence of added mass and the “contact with water”. Onecan also conclude that the added mass will influence many other factors, such as area coefficients, beam/draft ratio,boundary conditions of a water tank (or sea), slenderness ratio, environmental conditions, etc. For offshore structures,environmental conditions such as waves and currents change all the time, which implies that the added mass will not beconstant throughout a structure's service life.

F. Liu et al. / Journal of Sound and Vibration 333 (2014) 5004–5017 5005

When submerged in a fluid, the dynamic response of a solid body is altered by the effect of the added mass of the fluid(Ma). Consequently, the ratio between the natural frequency of a given mode of vibration in water (f w) and in air (f v) can beapproximated as follows:

f w;i

f v;i�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

ð1þðMa=miÞÞ

s(1)

where i denotes each particular mode shape and m is the corresponding modal mass.From Eq. (1), an added mass coefficient, Cm, can be defined with:

Cm ¼ f v;if w;i

� �2

�1 (2)

One problem in Eq. (2) is that all parts of the structure in fluid will be assumed to have the same value Cm once Eq. (2) issolved, which opposes practical situations. Most of the existing relevant works have been obtained from experiments in whichthe most popular technique is evaluation of the added mass at the resonant frequency corresponding to the peak of a frequencyresponse curve obtained from the 'forced' vibration analysis. The semi-circular cylinder is often adopted to simplify an object'scross-section in the analysis of added mass. The added masses of circular cylinders and cylinders with other cross-sectionalshapes in deep water are available (Newman [1]). De Tarso et al. [2] investigated the added mass and damping of rectangularcylinders mounted at the seabed and presented the shallow water effect on added mass and damping. Clarke [3] used conformalmapping and calculated the added mass of a circular cylinder in shallow water. He demonstrated the effect of water depththrough a comparison of results from different methods based on conformal mapping techniques and concluded that theapproach of using a row of distributed dipoles gave the best accuracy. Clarke [4] also provided the added mass for the complexcase of an elliptical cylinder in shallow water using a general mapping technique based on the Schwartz–Christoffel method.Clarke [5] used a similar method to calculate the added mass of an elliptical cylinder with a vertical fin stabilizer in shallowwater. Lin and Liao [6] applied the fast multiple boundary element method (FMBEM) to calculate the added mass coefficients ofcomplicated three-dimensional (3D) underwater bodies calculated by the FMBEM. The FMBEM is much more computationallyefficient than the traditional boundary element method. Therefore, the FMBEM provides an effective numerical method topredict added mass coefficients of complicated underwater bodies.

Recently, Benaouicha et al. [7] addressed a theoretical study of the added mass effect in cavitating flow. The cavitation isconsidered to induce a strong time–space variation of the fluid density at the interface between an inviscid fluid and a threedegrees-of-freedom rigid section. The added mass coefficients decrease as the cavitation increases, which should induce anincrease of the natural structural frequencies. Torre et al. [8] used a non-intrusive excitation and measurement system basedon piezoelectric patches mounted on the hydrofoil surface to determine the natural frequencies of the fluid–structuresystem. Kramer et al. [9] investigated the effects of material anisotropy and added mass on the free vibration response ofrectangular, cantilevered composite plates/beams via combined analytical and numerical modeling. The results show thatthe natural frequencies of the composite plates are 50–70 percent lower in water than in air due to large added mass effects.Torre et al. [10] studied the influence of the boundary conditions on the added mass of a NACA0009 cantilever hydrofoil,including experiments. A detailed fluid–structure model has been built for both cases, and a modal analysis has been carriedout. The obtained results are in reasonably good agreement with experimental data.

In practice, most added mass determination techniques involve solving Laplace's equation, which governs the inducedwater field; in other words, water-related assumptions are involved. In this paper, we try to estimate the added mass ofbeam structures from the view of structure (i.e., using only dynamic responses of the structure in water and in air). A fivedegrees-of-freedom (dofs) mass-spring systemwill be used to identify the performance of the proposed scheme, and a steelcantilever beam with a rectangular section in a water tank will be employed to demonstrate the approach.

2. Transverse vibration of a cantilever beam

Overlooking shear, damping, and axial-force effects, the solution for free bending transverse vibration of a beam isobtained by solving the differential equation of motion of a Bernoulli–Euler beam, which can be written as follows [11]:

m∂2v∂t2

þ ∂2

∂x2EI

∂2v∂x2

� �� �¼ 0 (3)

where m is the mass distribution of the unit of length, EI is the flexural rigidity, and v is the transverse displacement, whichis a function of the spatial coordinate (x) and time (t).

The solution of Eq. (3) is assumed in the form of a product of two functions:

vðx; tÞ ¼wðxÞYðtÞ (4)

Substituting Eq. (4) into Eq. (3) results in

wðxÞm∂2YðtÞ∂t2

þ ∂2

∂t2EI∂2wðxÞ∂x2

� �YðtÞ ¼ 0 (5)

F. Liu et al. / Journal of Sound and Vibration 333 (2014) 5004–50175006

Then, the nth circular or natural frequency (f ) of the considered uniform beam is obtained in the form of

f ¼ χnπ2

L2

ffiffiffiffiffiEIm

r(6)

where χn are constants that depend on actual boundary conditions and L is the length of the beam.

3. Added mass estimation using dynamic responses

The mass matrix of a real structure in water is expressed as

~M ¼MþΔMfasþΔMv

asþΔMfa (7)

where M is the mass matrix from the finite element model (FEM). ΔMfas and ΔMv

as represent the mass discrepancy of theportion of the structure in water and in air, respectively. ΔMf

a represents the added mass due to the presence of water.Likewise, the stiffness matrix of the real structure in water could be written by

~K ¼KþΔKfasþΔKv

as (8)

where K is the stiffness matrix from FEM. ΔKfas and ΔKv

as denote the stiffness discrepancy of the segment of the structure inwater and in air, respectively.

The main purpose of this paper is to the estimate inevitable modeling errors from FEM ΔMfas, ΔM

vas, ΔK

fas and ΔKv

as, andthe added mass due to the presence of water ΔMf

a simultaneously.We assume that the added mass is a modification of the related mass matrix Mf

asf via:

ΔMfas ¼ ∑

Nfasf

asf ¼ 1εfasfM

fasf (9)

where Nfasf is the number of elements in water and εfasf is the corresponding correction factors.

Similarly:

ΔMvas ¼ ∑

Nvas

asv ¼ 1εvasvM

vasv (10)

ΔKfas ¼ ∑

Nfksf

ksf ¼ 1εfksfK

fksf (11)

ΔKvas ¼ ∑

Nvksf

ksv ¼ 1εvksvK

vksv (12)

ΔMfa ¼ ∑

Nfaf

af ¼ 1εfafM

faf (13)

where Mfasf , M

vasv are the asf th and asvth mass matrices from FEM in global coordinate; and Kf

ksf , Kvksv are the ksf th and ksvth

stiffness matrices from FEM in global coordinate; and Mfaf is the af th mass matrix from FEM in global coordinate.

Substituting Eqs. (9)–(13) into Eqs. (7) and (8) yields:

~M ¼Mþ ∑Nf

asf

asf ¼ 1εfasfM

fasf þ ∑

Nvas

asv ¼ 1εvasvM

vasvþ ∑

Nfaf

af ¼ 1εfafM

faf (14)

~K ¼Kþ ∑Nf

ksf

ksf ¼ 1εfksfK

fksf þ ∑

Nvksf

ksv ¼ 1εvksvK

vksv (15)

If modal parameters of the structure without consideration for the added mass of water could be obtained as well as thejth eigenvalues and eigenvectors denoted as λ

!j and Φ

!j, then their relationship can be expressed as

Kþ ∑Nf

ksf

ksf ¼ 1εfksfK

fksf þ ∑

Nvksf

ksv ¼ 1εvksvK

vksv

0@

1AΦ

!j� λ

!j Mþ ∑

Nfasf

asf ¼ 1εfasfM

fasf þ ∑

Nvas

asv ¼ 1εvasvM

vasv

0@

1AΦ

!j ¼ 0 (16)

Obtaining the pth eigenvalues and eigenvectors of the structure in water, i.e., ~λp and ~Φp, then:

Kþ ∑Nf

ksf

ksf ¼ 1εfksfK

fksf þ ∑

Nvksf

ksv ¼ 1εvksvK

vksv

0@

1A ~Φp� ~λp Mþ ∑

Nfasf

asf ¼ 1εfasfM

fasf þ ∑

Nvas

asv ¼ 1εvasvM

vasvþ ∑

Nfaf

af ¼ 1εfafM

faf

0@

1A ~Φp ¼ 0 (17)

F. Liu et al. / Journal of Sound and Vibration 333 (2014) 5004–5017 5007

Pre-multiplying Eqs. (16) and (17) by ΦTi , respectively, yields:

ΦTi Kþ ∑

Nfksf

ksf ¼ 1εfksfK

fksf þ ∑

Nvksf

ksv ¼ 1εvksvK

vksv

0@

1AΦ

!j� λ

!jΦT

i Mþ ∑Nf

asf

asf ¼ 1εfasfM

fasf þ ∑

Nvas

asv ¼ 1εvasvM

vasv

0@

1AΦ

!j ¼ 0 (18)

and

ΦTi Kþ ∑

Nfksf

ksf ¼ 1εfksfK

fksf þ ∑

Nvksf

ksv ¼ 1εvksvK

vksv

0@

1A ~Φp

� ~λpΦTi Mþ ∑

Nfasf

asf ¼ 1εfasfM

fasf þ ∑

Nvas

asv ¼ 1εvasvM

vasvþ ∑

Nfaf

af ¼ 1εfafM

faf

0@

1A ~Φp ¼ 0 (19)

Combining Eqs. (18) and (19), one obtains:

∑Nf

asf

asf ¼ 1εfasf Aijpþ ∑

Nvas

asv ¼ 1εvasvBijpþ ∑

Nfksf

ksf ¼ 1εfksf Cijpþ ∑

Nvksv

ksv ¼ 1εvksvDijpþ ∑

Nfaf

af ¼ 1εfaf Eip ¼ �ðFijþFipÞ (20)

where

Aijp ¼ �ΦTi M

fasf ð λ

!jΦ!

jþ ~λp ~ΦpÞ (21)

Bijp ¼ �ΦTi M

vasvð λ!jΦ

!jþ ~λp ~ΦpÞ (22)

Cijp ¼ΦTi K

fksf ðΦ

!jþ ~ΦpÞ (23)

Dijp ¼ΦTi K

vksvðΦ!jþ ~ΦpÞ (24)

Eip ¼ � ~λpΦTi M

faf

~Φp (25)

Fij ¼ΦTi KΦ

!j� λ

!jΦT

i MΦ!

j (26)

Fip ¼ΦTi K ~Φp� ~λpΦT

i M ~Φp (27)

Eq. (20) can also be rewritten as

Aijp Bijp Cijp Dijp Eiph i

εfasfεvasv

εfksfεvksv

εfaf

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

¼ �ðFijþFipÞ (28)

For clarity, Eq. (28) can be rewritten in a matrix form:

ZΔ¼ F (29)

where

Z¼ Aijp Bijp Cijp Dijp Eiph i

(30)

and

Δ¼

εfasfεvasv

εfksfεvksv

εfaf

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

(31)

A standard inverse operation could be used to solve Eq. (29). Detailed information has been discussed in reference (Li[12]).

F. Liu et al. / Journal of Sound and Vibration 333 (2014) 5004–50175008

From Eqs. (13) and (29), one can obtain the added mass distribution for each element in water. For example, for the nthelement, i.e., af ¼ n in Eq. (13), the added mass could be expressed as

ΔMna ¼ εfnM

fn (32)

The global added mass matrix can then be obtained:

ΔMa ¼ ∑Nf

af

n ¼ 1εfnM

fn (33)

In general, the proposed method includes four sequential steps:

(1)

Step 1: Measuring and obtaining the jth eigenvalues and eigenvectors λ!

j and Φ!

j, of the structure without considerationfor the added mass of water, and the pth eigenvalues and eigenvectors of the structure in water, ~λp and ~Φp, respectively.

(2)

Step 2: One determines the mode shapes Φi of the FEM of the measured structure and computing Eqs. (21)–(27), thensolve Eq. (31) by substituting Eqs. (21)–(27) into Eq. (29).

(3)

Step 3: Computing Eq. (13) in terms of Eq. (31), and then substituting it into Eq. (33), one can obtain the added massΔMa.

(4)

Step 4: Computing Eqs. (9)–(12) in terms of Eq. (31), and then substituting Eqs. (9)–(12) into Eqs. (14) and (15), one canget the updated FEM and make a comparison with measured modal parameters.

4. Numerical study: a 5-dof mass-spring system

To illustrate the procedure and demonstrate the performance of the proposed scheme, a 5-dof mass-spring system wasused to represent a lumped mass beam fixed at the ground and some portion of it in water, as shown in Fig. 1. The uniformmass and stiffness coefficients were taken to be Mn ¼ 50 kg and Kn ¼ 2:9� 107 N/m, n¼ 1;2;3;4;5, respectively. Thecoordinates of the 5-dof model are denoted by xn, with x1 at the fixed end and x5 at the free end. Mass 1, mass 2, and mass 3are assumed to be in water; mass 4 and mass 5 are assumed to be in air.

Considering the inevitable modeling errors from FEM, a series of quantities as defined from Eq. (9) to Eq. (13) areemployed to calculate these errors. Here, we randomly generate a series of numbers using the Matlab function, assumingεfasf ¼ �0:0087 �0:0333 0:0025

� �and εvasv ¼ 0:0058 �0:0230

� �. Thus, the second mass modeling error is

�0:0333�M2 ¼

0 0 0 0 00 50 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

26666664

37777775

(34)

Likewise, εfksf ¼ 0:0119 0:0119 �0:0004� �

and εvksv ¼ 0:0033 0:0017� �

are used to represent stiffness modelingerrors for elements in water and in air, respectively. For simulating added mass of water, we assumeεfaf ¼ 0:15 0:25 0:20

� �. The purpose of the following work is to study whether the added mass caused by water could

be estimated properly, including modeling errors and using measured modal parameters. By implementing Eigen analysis,one can obtain the frequencies of the FEM: 34.5, 100.7, 158.75, 203.93, and 232.60 Hz. The frequencies of the model, withouttaking into account the added mass of water, are 35.006, 102.98, 160.12, 206.5, and 235.66 Hz. When the added mass isconsidered, the frequencies changed to 32.687, 92.635, 145.96, 190.00, and 212.77, respectively.

From Eq. (28) and Fig. 1, one can see that there are five mass, five stiffness, and three added mass coefficients that shouldbe estimated, which means that at least 13 equations are needed to solve Eq. (28). Therefore, we assume two modes fromthe model that do not consider the added mass of water and two that do, and all analytical modes from FEM will be used.As such, 20 equations could be constructed, which are sufficient to solve the above 13 coefficients. However, though 20equations were constructed, the rank of coefficient matrix Z in Eq. (29) was 12, which means that the 13 unknowns couldnot be solved uniquely. Therefore, we can assume that mass 4 could be modeled precisely and take it as a constraint. Fig. 2shows the comparison of the estimated and preset coefficients plotted against the number of unknowns. Fig. 2 clearlyindicates that the added mass of water could be estimated accurately, including modeling errors, when the measured modesfrom dynamic testing are noise free.

Water depth may have an influence on the added mass of a structure in water, and that added mass on the real structurewill be variable with the change in environmental conditions such as water depth, marine growth, and so on. Thus, we willstudy whether our approach could properly estimate the added mass when environmental conditions change. Here, we takethe change in water depth as an example, assuming that masses 1 and 2 are in water and that masses 3 through 5 are in air,which means that 12 unknowns should be calculated in this scenario. Based on the same considerations that yielded Fig. 2,one can obtain a comparison of the estimated and preset coefficients plotted against the number of unknowns as shown inFig. 3. The numerical results demonstrated that our approach could identify and estimate added masses when operationalconditions of the structure changed.

Fig. 1. A 5-dof mass-spring system: three in water and the other two in air.

F. Liu et al. / Journal of Sound and Vibration 333 (2014) 5004–5017 5009

Another problem to note is that measured modes usually include noise; thus, the remaining numerical study focused onimplementing the proposed method with corrupted modes. Values of corrupted modes were generated by multiplying thetrue value by a factor ð1þNlÞ, where Nl is called a corruption level. Fig. 4 shows a comparison of the estimated and presetcoefficients when the first two modes were measured under a 5 percent corruption level. One can conclude from Fig. 4 thatthe added mass could be properly estimated with the exception of the estimation at mass 1, which had relatively largererrors. One can also conclude that many errors will be introduced as the corruption level increases; thus, the following is toinvestigate the performance of added mass estimation with increasing corruption levels.

Here, we define the relative difference between the estimated (�εaf ) and corresponding preset (εaf ) values as an indicator,i.e.:

Rm ¼ j�εaf �εaf jεaf

� 100% (35)

Fig. 5 shows the indicator Rm with a corruption level increase from zero to 10 percent. Numerical results indicate that theapproach could estimate added mass properly when the corruption level is below 5 percent, as shown in Fig. 4 and theaverage line in Fig. 5. As predicted, estimation errors of added masses will increase when the noise becomes more

Fig. 2. Comparison of estimated coefficients and preset values when mass 1 to mass 3 is in water.

Fig. 3. Comparison of estimated coefficients and preset values when mass 1 to mass 2 is in water.

F. Liu et al. / Journal of Sound and Vibration 333 (2014) 5004–50175010

significant, and the estimation of added mass at mass 1 is the most sensitive to noise compared with masses 2 and 3. All ofthese factors imply that accurate modal parameters are expected for the approach.

5. Experimental study of a steel cantilever beam in still water

A steel cantilever beam with a rectangular section in a water tank at Ocean University of China was used to study theadded mass influence on modal parameter identification and to investigate the performance of the proposed methodemploying measured dynamic signals. The experimental setup in which the cantilever beam was fixed on the ground byfour bolts is shown in Fig. 6. The section area of the steel beam is 8:8� 10�4 m2, with a height of 1.9 m. Ten accelerometers(Model 4803A-0002), termed Sensors 1 to 10, were mounted for response measurement and were equally distributed fromthe free end to the fixed end of the beam. A measurement system (PL64-DCB8, Integrated Measurement & ControlCooperation, Germany) was used for data acquisition.

In this experiment, an impulsive load was used to excite the beam, whether it was in water or not. The water depth was set tobe 1.5 m, and a sampling rate of 500 Hz was not changed throughout the whole experiment. Fig. 7 is a comparison of the measuredaccelerations of Sensor 1 in the time domain. By employing the Eigensystem Realization Algorithm (ERA) [13] for modal parameter

Fig. 4. Comparison of estimated coefficients and preset ones when the first two modes were measured under a 5% corruption level.

Fig. 5. The indicator Rm changes with corruption level increases from zero to 10%.

F. Liu et al. / Journal of Sound and Vibration 333 (2014) 5004–5017 5011

identification, the first two frequencies and damping ratios extracted from measured signals in water and in air, respectively,including the corresponding values from FEM, are listed in Table 1. Following the standard formulation for a uniform beam elementin a plane considering axial coordinates, one computes the element stiffness matrix kn and element (consistent) mass matrixmn ofa size 6�6. There are 10 nodal points and each nodal point has 3 DOFs, thus matrices K andM both are with a size 30 � 30. Nodeand element number could be found in Fig. 8 (Model A). Table 1 clearly demonstrates that the frequencies decreased significantly,while the damping ratios improved because of the influence of surrounded water.

In this experiment, two challenges should be properly handled: one is the spatial incompleteness of measured modeshapes, and the other is the supplementation of added constraints to assure the unique solutions of the approach. To addressthe first challenge, the direct mode shape expansion method (Liu[14,15]) was used to improve the accuracy of mode shapeexpansion. Fig. 5 shows that noises that come from dealing with spatial incompleteness, environment, equipment, etc., willinfluence the performance of our approach, so we firstly study the case that only added mass correction factors εfaf wereestimated. Using the first two modes measured when the beam was in water and in air and all modes from the FEM of thebeam, one can obtain correction factors εfaf as shown in Fig. 9, which indicates that the added mass correction factors werenot constants for each element of the beam in water. One should also note that these correction factors were estimated bytaking the FEM as a baseline model, which implies that these factors will change if the baseline model is changed.

Though Fig. 9 provides a series of added mass correction factors on each element of the cantilever beam in water, onemay suspect these factors' correctness. One way to demonstrate their accuracy is by studying modal parameters of theupdated FEM, which takes into account the added mass of the water. Table 2 lists the frequencies of the FEM, the beam in

Fig. 6. A steel cantilever beam with rectangle section in a water tank of OUC.

Fig. 7. Measured accelerations of Sensor 1 in time domain: (a) in water, and (b) in air.

Table 1The first two frequencies and damping ratios from FEM when the cantilever beam is in water and in air, respectively.

Frequency (Hz) Damping ratio

FEM In air In water FEM In air In water

1 7.82 7.26 7.07 0 0.0018 0.00752 49.03 49.55 45.51 0 0.0053 0.0066

F. Liu et al. / Journal of Sound and Vibration 333 (2014) 5004–50175012

water, and the updated FEM (Model A). The corresponding first two mode shapes are shown in Figs. 10 and 11, respectively.Results demonstrated that the first two frequencies of the updated FEM considering added mass matched well to measuredvalues, and the first two mode shapes of the updated FEM were improved.

As discussed above, each estimated added mass correction factor is corresponding to a predetermined element in water;one may conclude that the estimation of added mass maybe different when the immersed part of the beam was modeledusing different number of elements. Here we will further study how the estimated added mass correction factors changewith element numbers increasing. In this experiment, four scenarios were studied, i.e., the immersed part of the beam wasdiscrete into 1, 2, 4, and 6 elements, and the corresponding FEM were named Model E, D, C, and B, respectively, as shown inFig. 12. To compare results of above different scenarios, a sensitivity indicator (Sa) was defined,

Sa¼∑

Nfasf

n ¼ 1εfasf ;n

Nfasf

(36)

Fig. 8. Finite element model when the immersed part modeled using 8 elements: Model A.

F. Liu et al. / Journal of Sound and Vibration 333 (2014) 5004–5017 5013

When the immersed part was modeled just using one element, as shown in Fig. 12(d), this FEM has 9 dofs, i.e., 3 masterdofs and 6 slave dofs. The first two modes were employed in implementing the proposed method, then one can get theadded mass correction factor to be 0.72, which means the added mass for each element in water in Fig. 9 is a constant, asshown in Fig. 13(a). By substituting this correction factor back into the FEM, the updated frequencies were 6.90 and 47.83 Hz.Results indicated there were much errors for the second mode if only the immerse part had been modeled using oneelement and regarded as a baseline for added mass estimation. Likewise, added mass correction factors from other scenarioswere plotted in Fig. 13(a), and the sensitivity indicator was plotted in Fig. 13(b). Table 2 lists comparisons of frequencies fromthe FEM, the measured values when the beam in water, and the updated FEM (Models A–E), respectively. From Fig. 13, onecan find that the sensitive indicator becomes stable when the immersed part is modeled more than 6 elements.If substituting the obtained added mass correction factors into the FEM (Model C) and recalculate the first two modalparameters, one can get the two frequencies 7.0473 and 44.0753 Hz, which also demonstrated the correctness of Fig. 9 fromanother view point.

In this experiment, we also tried to simultaneously estimate discrepancies of mass and stiffness from FEM along withadded mass. Based on the same considerations yielding Fig. 9, we used the first two modes measured when the beamwas inwater and in air and all modes from the FEM of the beam in implementing the proposed method for Model A, in which28 unknowns should be estimated and results are shown in Fig. 14(a). One can conclude that the discrepancy between the

1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Number of Unknowns

ε

Fig. 9. Estimation of added mass correction factors using Model A.

Table 2Comparison of frequencies from FEM, beam in water, and updated FEM (Models A-E).

Modal order Frequency (Hz)

FEM In waterUpdated

Model A Model B Model C Model D Model E

1 7.82 7.07 7.09 7.05 7.02 7.09 6.902 49.03 45.51 45.03 44.08 43.28 42.72 47.83

Fig. 10. Comparison of the first mode shape between FEM, measured, and updated FEM.

F. Liu et al. / Journal of Sound and Vibration 333 (2014) 5004–50175014

FEM and the experimental beam mainly comes from the mass modeling errors and the influence of added mass. Fig. 14(b)and (c) shows the mass correction factors εas ¼ ½εfasf ; εvasv� and added mass correction factors εfaf , respectively. ComparingFig. 14 and Fig. 9, we can find that the sixth correction factor of added mass is �0.1179, and the average correction factor isestimated to be 0.1319 for this experiment. If we employ Eq. (2) to compute the added mass factor using the measuredfrequencies, one obtains 0.0545 and 0.1854, respectively; and the average value is 0.1199, which is close but smaller than our

Fig. 11. Comparison of the second mode shape between FEM, measured, and updated FEM.

Fig. 12. Finite element model when the immersed part modeled using: (a) 6 elements (Model B), (b) 4 elements (Model C), (c) 2 elements (Model D), (d) 1element (Model E).

Fig. 13. Estimation of added mass correction factors: (a) Model A–E, and (b) sensitivity indicator Sa.

F. Liu et al. / Journal of Sound and Vibration 333 (2014) 5004–5017 5015

0 5 10 15 20 25−2

0

2x 107

Number of Unknowns

ε

1 2 3 4 5 6 7 8 9 10−1

−0.5

0

0.5

Number of Unknowns

ε as

1 2 3 4 5 6 7 8−0.5

0

0.5

Number of Unknowns

ε aff

Fig. 14. Estimation of correction factors of Model A: (a) mass, stiffness and added mass correction factors when they are estimated simultaneously, (b) masscorrection factors when mass and added mass are estimated simultaneously, and (c) added mass correction factors when mass and added mass areestimated simultaneously.

0 1 2 3 4 5 6 7 8 9 10 110

0.1

0.2

0.3

0.4

0.5

0.6

Node Number

Mod

e S

hape

Val

ue

FEMMeasuredUpdated

0 1 2 3 4 5 6 7 8 9 10 11−0.6

−0.4

−0.2

0

0.2

0.4

Node Number

Mod

e S

hape

Val

ue

FEMMeasuredUpdated

Fig. 15. Comparison of the first and the second mode shape between FEM, measured, and updated FEM when mass and added mass are estimatedsimultaneously: (a) the first order, and (b) the second order.

F. Liu et al. / Journal of Sound and Vibration 333 (2014) 5004–50175016

estimated value 0.1319. To further approve the rightness of these estimated correction factors, the first two frequencies andcorresponding mode shapes (see Fig. 15) of the updated model are studied. So, one can get the first two frequencies 7.0499and 45.9486 Hz of the updated model, respectively. We can find the first two frequencies of the updated FEM are very closeto the measured ones, and the corresponding mode shapes are improved significantly as well.

6. Conclusions

This article tried to estimate the added mass of water from the view of structure using dynamic responses. A series ofcorrection factors was employed to correct and calculate mass and stiffness discrepancies in FEM and the added mass of the

F. Liu et al. / Journal of Sound and Vibration 333 (2014) 5004–5017 5017

water simultaneously. One theoretical development is that the approach could reflect the influence of changes such as waterdepth, current, etc., and provide refined added mass estimation. The other development is that the proposed method doesnot involve much water assumptions because all influences of water can be reflected in measured dynamic responses of thestructure in water. The numerical results show that the mass, stiffness, and added mass correction factors could beaccurately estimated when noise-free measurements are used, even the measured modes with a certain noise. One shouldnote that some constraints may be required for obtaining unique solutions, and these constraints can be easily imposed interms of some inspections. Experimental results also demonstrated that spatial incompleteness is a key issue and that amore accurate and stable mode shape expansion method is preferred when applying the approach to real problems.

Acknowledgments

The authors wish to acknowledge financial support from the 973 project (Grant no. 2011CB013704), the 111 Project(B14028) and the National Natural Science Foundation of China (Grant nos. 51279188 and 51379197).

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