a study of sloshing absorber geometry for structural control with sph

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Journal of Fluids and Structures

Journal of Fluids and Structures 27 (2011) 1165–1181

0889-97

doi:10.1

n Corr

E-m

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

A study of sloshing absorber geometry for structural controlwith SPH

Adam Marsh a,n, Mahesh Prakash b, Eren Semercigil a, Ozden F. Turan a

a Victoria University, School of Architectural, Civil and Mechanical Engineering, Melbourne, VIC 3011, Australiab CSIRO Mathematical and Information Sciences, Private Bag 33, VIC 3168, Australia

a r t i c l e i n f o

Article history:

Received 16 August 2010

Accepted 28 February 2011Available online 30 April 2011

Keywords:

Liquid sloshing

Absorber shape

Structural control

Smoothed Particle Hydrodynamics

Fluid–structure interaction

46/$ - see front matter & 2011 Elsevier Ltd. A

016/j.jfluidstructs.2011.02.010

espondence address: LMF Ecole Centrale de

ail addresses: [email protected], adam

a b s t r a c t

A liquid sloshing absorber consists of a container, partially filled with liquid. The absorber is

attached to the structure to be controlled, and relies on the structure’s motion to excite the

liquid. Consequently, a sloshing wave is produced at the liquid free-surface within the

absorber, possessing energy dissipative qualities. The primary objective of this work is to

numerically demonstrate the effect of a sloshing absorber’s shape on its control perfor-

mance. Smoothed Particle Hydrodynamics (SPH) is used to model fluid–structure interac-

tion of the structure/sloshing absorber system in two dimensions. The structure to be

controlled is a lightly damped single degree-of-freedom structure. The structure is subjected

to a transient excitation and then allowed to respond dynamically, coming to rest either due

to its own damping alone or with the added control of the sloshing absorber. It is identified

that the control performance of the conventionally used rectangular container geometry can

be improved by having inward-angled walls. This new arrangement is robust, and of

significant advantage in situations when the external disturbance is of uncertain magnitude.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Sloshing is the oscillation of a liquid within a partially full container. In the study of sloshing, efforts are usually made inthe direction of suppression, due to the damaging effects it can impose (Koh et al., 2007; Shekari et al., 2008; Faltinsen,1993; Panigraphy et al., 2009). On the other hand, sloshing has an inherent ability to dissipate large amounts of energy. Forthis reason, it is possible to employ liquid sloshing as an effective energy sink in structural control applications, providingprotection for structures exposed to excessive levels of vibration (Sun et al., 1989; Modi et al., 1996; Sun and Fujino, 1994;Modi and Munshi, 1998; Marsh et al., 2010a).

Generally, a sloshing absorber is tuned so that the frequency of sloshing coincides with the natural frequency of thestructure (Kareem, 1990; Banerji et al., 2000). When designed properly, the sloshing fluid oscillates out of phase from thestructure, creating a counteracting pressure force on the side of the container. Shearing of the fluid is the primary form ofmechanical damping, providing that the liquid level is low (Kareem, 1990).

Liquid sloshing in rectangular tanks has long been an area of study (Sun et al., 1989; Ikeda and Nakagawa, 1997; Rafieeet al., 2010), with significant effort being focused on increasing the energy dissipation performance of this widelyused conventional design. Variations on the conventional design have included the introduction of wedge shaped objectson the container bottom (Modi and Akinturk, 2002), baffles on the container walls (Anderson et al., 2000), wall flexibility

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[email protected] (A. Marsh).

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A. Marsh et al. / Journal of Fluids and Structures 27 (2011) 1165–11811166

(Gradinscak et al., 2002), floating solid particles at the free-surface (Sun and Fujino, 1994) and submerged nets within thefluid volume (Kaneko and Ishikawa, 1998). Studies involving varying liquid depth have been undertaken, focusing onincreasing the amount of energy dissipation produced (Anderson et al., 2000; Guzel et al., 2005; Reed et al. 1998; Marshet al., 2010b). These studies have demonstrated that shallow depths are more effective at dissipating energy than deepliquid levels, on a per unit mass basis.

Alternative container shapes to that of a rectangle have been studied in order to assess their control performance(Tamura et al., 1996; Casciati et al., 2003). Tait and Deng (2008) conducted a comparative study between different absorbershapes using Linear Long Wave Theory. This study outlined the structural control characteristics of the shapes investigated;however, such a numerical model is limited to the analysis of small structural displacements only, due to theapproximations made. Therefore, the effect that an absorber’s shape has on its control performance is still largelyunknown, particularly when the system is energetic enough to produce breaking waves within the absorber.

The primary objective of this work is to investigate the effects that an absorber’s shape has on its control performance,particularly during circumstances of large structural displacements. In such conditions, violent fluid behaviour is observedat the free-surface. Such behaviour is known to be responsible for dissipating large amounts of structural energy (Marshet al., 2009), and is therefore a desirable phenomenon. Potential enhancements are explored, with shapes different than arectangle, through numerical investigation. The relationship between container shape and fluid behaviour is assessed.

Due to complex free-surface behaviour, Smoothed Particle Hydrodynamics (SPH) is used as a numerical modelling toolin this study. SPH is a Lagrangian method for solving the equations of fluid flow. It is suitable for modelling liquid sloshingdue to its grid free nature, and inherent ability to capture free-surface behaviour accurately (Monaghan, 1992). SPH hasbeen successfully applied to a wide range of industrial fluid flow applications involving complex geometries in manyinstances (Cleary et al., 2007a, 2007b, 2002, 2006; Marsh et al., 2009).

In order to validate the predictions of fluid–structure interaction with the SPH model, simple experimental observations(Marsh, 2009) are also presented here. SPH is found to provide an accurate prediction of structure’s oscillations, employing awide range of liquid depths. A brief description of the experiments, along with comparisons of two representative cases isgiven in the Appendix.

2. Numerical model

Smoothed Particle Hydrodynamics (SPH) is used in this study to model fluid–structure interaction. CSIRO’s(Commonwealth Scientific and Industrial Research Organisation) Mathematical and Information Sciences Division hasdeveloped the code used here. SPH is a Lagrangian method of solving the equations of fluid flow, suitable for modellingliquid sloshing due to its grid-free nature, and inherent ability to model complex free-surface behaviour. This particularSPH code has been successfully applied to a wide range of industrial fluid flow applications (Cleary et al., 2006).A description of the method and the governing equations used in the code is presented here for the sake of completeness.A more detailed description of the method can be found in Monaghan (1992).

In SPH, the fluid being modelled is discretized into fluid elements or particles, the properties of which are attributed totheir centres. The method works by tracking particles and approximating them as moving interpolation points. These fluidparticles (or moving interpolation points) have a spatial distance over which field variables such as density, velocity andenergy are smoothed. This is achieved via an interpolation kernel function.

The fundamental concept of the integral representation of a function used in the SPH method comes from the identityshown in Eq. (1)

f ðxÞ ¼Z

Vf ðx0Þdðx�x0Þdx0, ð1Þ

where f(x) is a function of the three-dimensional position vector x. V is the volume of the integral that contains x andd(x�x0) is the Dirac delta function defined by

dðx�x0Þ ¼1 x¼ x0,

0 xax0:

(ð2Þ

Identity (1) implies that a function can be represented in integral form. This integral representation is exact since thedelta function is used, providing that f(x) is defined and continuous in V (Liu and Liu, 2003). In SPH the Dirac delta functionis replaced by the smoothing function W(x�x0,h) so that the integral representation of f(x) is specified as

/f ðxÞS¼Z

Vf ðx0ÞWðx�x0,hÞdx0, ð3Þ

where W is the interpolation kernel and h is the smoothing length that defines the region in which the smoothing functionoperates. A cubic smoothing kernel has been used here for W(x�x0,h), approximating the shape of a Gaussian profile buthaving compact support, so that W(x�x0,h)¼0 for x�x04h.

A. Marsh et al. / Journal of Fluids and Structures 27 (2011) 1165–1181 1167

The integral representation of the spatial derivative of a function in SPH is performed in the same way f(x) issubstituted for rUf(x) in Eq. (3) to produce

/rUf ðxÞS¼Z

VrUf ðxÞWðx�x0,hÞdx0: ð4Þ

From the identity in Eq. (5)

½rUf ðx0Þ�Wðx�x0,hÞ ¼rU½f ðx0ÞWðx�x0,hÞ��f ðx0ÞUrWðx�x0,hÞ, ð5Þ

we obtain

/rUf ðxÞS¼Z

V½rUf ðx0ÞWðx�x0,hÞ�dx0�

ZV

f ðx0ÞrUWðx�x0,hÞdx0: ð6Þ

The first integral on the right-hand side of Eq. (6) is converted using the divergence theorem of Gauss to give

/rUf ðxÞS¼Z

Sf ðx0ÞWðx�x0,hÞUn

*dS�

ZV

f ðx0ÞrUWðx�x0,hÞdx0; ð7Þ

here S is the surface of the integration domain V and n*

is the unit normal to the domain surface S. When the supportdomain of W is located within the problem domain, the first integral on the right-hand side of Eq. (7) is zero. However,when the support domain overlaps with the problem domain, W is truncated by the problem domain boundary; hence, thesurface integral is no longer zero. For all points in space whose support domain lies within the problem domain, Eq. (7)simplifies to

/rUf ðxÞS¼�Z

Vf ðx0ÞrUWðx�x0,hÞdx0: ð8Þ

For all other points in space, modifications need to be made to treat the boundary effects if the surface integral is to beequated to zero (Liu and Liu, 2003). Eq. (8) states that the spatial gradient of a function is determined from the values ofthe function and the derivative of the smoothing function, rather than the derivative of the function itself.

Discretization is performed by converting the integral representations in Eqs. (3) and (8) into summations over all theparticles that lie within the support domain of W. This is achieved by replacing the infinitesimal volume dx0 by the finitevolume of particle j, DVj. The mass of particle j (mj) is then related to this volume by

mj ¼ rjDVj, ð9Þ

where rj is the density of particle j. The discretized particle approximation can then be written as

/f ðxiÞS¼XN

j ¼ 1

mj

rj

f ðxjÞWðxi�xj,hÞ: ð10Þ

Eq. (10) states that the value of a function at particle i is approximated using the average of the same function at all j

particles within the support domain of particle i, weighted according to the smoothing function. The same approach isused to produce the particle approximation of the spatial derivative of a function

/rUf ðxiÞS¼�XN

j ¼ 1

mj

rj

f ðxjÞUrWðxi�xj,hÞ; ð11Þ

rW is taken with respect to particle j in Eq. (11), when taken with respect to particle i the negative sign is removed,producing

/rUf ðxiÞS¼XN

j ¼ 1

mj

rj

f ðxjÞUriWðxi�xj,hÞ: ð12Þ

The SPH approximations in Eqs. (10) and (12) are applied to the field variables and their derivatives within theLagrangian equations of fluid flow. This yields the continuity equation

dri

dt¼XN

j ¼ 1

mj

rj

vijUriWij, ð13Þ

where Wij¼W(rij,h) and is evaluated for the distance 9rij9. rij is the position vector from particle ‘j’ to particle ‘i’ and is equalto ri�rj, and the momentum equation

dvi

dt¼�

XN

j ¼ 1

mj

Pj

r2j

þPi

r2i

!�

xrirj

4mimj

ðmiþmjÞ

vijrij

r2ijþZ2

" #riWijþg: ð14Þ

The first term in Eq. (14) within the square brackets is the pressure. The term on the right, without x, is the artificialviscosity; this term is used to increase the stability of the numerical algorithm (Colagrossi, 2005). x is a proportionalityfactor that relates the artificial viscosity to the real SPH viscosity, and has a theoretical value of 4, but has been modified


empirically to 4.96333 (Cleary, 1998). x has values of between 4 and 5 for most applications. Pi and mi are the pressure andviscosity of particle ‘i’, the same applies for particle ‘j’. vij¼vi�vj. Z is a parameter used to smooth out the singularity atrij¼0, and g is the body force acceleration due to gravity.

The code uses a compressible flow method for determining the fluid pressure. It is operated near the incompressiblelimit by selecting a speed of sound that is much larger (around 10 times) that of the velocity scale expected in the fluidflow. The relationship between particle density and fluid pressure is expressed by

P¼ P0rr0

� �g�1

� �; ð15Þ

a modified version of that used in Batchelor (1967), to accurately describe sound wave propagation. In Eq. (15), P is themagnitude of the pressure and r0 is the reference density. g¼7 is used to represent water (Batchelor, 1967). P0 is thereference pressure. The pressure the equation of state solves for P, is then used in the SPH momentum equation governingthe particle motion.

The time stepping in this code is explicit and is limited by the Courant condition modified for the presence of viscosity

Dt¼mina 0:5h= csþ2xma

hra

� �� , ð16Þ

where cs is the local speed of sound.Solid boundaries are constructed with boundary particles. The force the boundary particles exert on the fluid is

approximated with a Lennard–Jones form, acting in the boundary particle’s normal direction (Monaghan, 1992).Interpolation of this force at each particle produces a smoothly defined repulsive boundary force, aimed at restrictingthe fluid’s ability to penetrate the boundary.

A schematic representation of the structure/sloshing absorber system to be modelled is given in Fig. 1. In the model, therigid container of a sloshing absorber is represented by a partially constrained moving boundary, allowed dynamictranslation in one-direction only. Tethers are attached to the boundary, representing the force relationship between thestructure and the absorber. This relationship exists due to the structure’s stiffness (k) and viscous damping (c) properties.The structure’s stiffness (k) and mass (m) are 4260 N/m and 60.5 kg, respectively, to give a natural frequency of 1.33 Hz.The viscous damping coefficient is chosen to give a critical damping ratio of 1%, to represent a lightly damped resonantstructure. The sloshing absorber attached on the structure consists of a container partially full of liquid. The sloshing fluidis water with a density of 1000 kg m�3 and dynamic viscosity of 0.001 Pa s.

The interaction forces between the fluid and the structure are calculated via boundary particle acceleration

FbpðiÞ ¼mbpðiÞ

dvbpðiÞ

dt, ð17Þ

where the subscript bp(i) represents boundary particle i, m is particle mass.The total force the fluid exerts on the structure is calculated by

Ff ¼XbpðiÞ

FbpðiÞmstP

bpðiÞmbpðiÞ, ð18Þ

where mst is the structure mass.The motion of the structure is then updated via temporal integration of the equation of motion

mstd2x

dt2¼X

Fst ¼ Ff þFsxþFddx

dt: ð19Þ

In Eq. (19), Fs and Fd, respectively, the spring force and the damping force. x is the displacement from the structure fromits static equilibrium position.

m

k

c

Fig. 1. Tuned rectangular liquid damper coupled to a single degree of freedom structure.


Time stepping is explicit and is limited by the Courant condition modified for the presence of viscosity (Monaghan,1992). The time step of integration is 1 ms, corresponding to approximately 75,000 integration steps for each naturalperiod of oscillation. A resolution study revealed that a particle size of 0.8 mm by 0.8 mm is fine enough to model thewater within the container in a two-dimensional space (Marsh, 2009).

Initially fluid within the container is allowed to settle under gravity for one second, so that it reaches an initial state ofrest. The structure is then given an initial velocity of 0.5 m/s over one time step. The structure is allowed to respond freely,and its motion excites the water within. Liquid depth is kept relatively shallow in order to promote travelling free-surfacewaves. In all cases, mass of the sloshing water is 6.05 kg, 10% of the structure’s mass. Such a high mass ratio is used toproduce significant control by the sloshing absorber, minimising the time taken for the structure to come to rest. Theauthors do not consider this mass ratio to be representative of a typical dynamic vibration absorber, however is deemedinconsequential due to its invariability.

3. Rectangular geometry

The dashed line in Fig. 2(a) is the displacement history of the uncontrolled structure. Maximum displacement reachedby the structure, is around 60 mm. The light damping characteristics of the structure result in almost unrestricted freevibration.

The response of the structure is significantly improved with a tuned rectangular sloshing absorber, indicated by thesolid line in Fig. 2(a). An approximately 95% reduction in the time taken for the structure to come to rest, is achieved hererelative to the uncontrolled case. For the rectangular container, the container width and liquid depth have been chosen(tuned) to produce a 1.33 Hz fundamental sloshing frequency equal to the natural frequency of the structure using thestandard approach (Milne-Thomson, 1968). This tuning corresponds to a container width of 200 mm and a liquid depth of30 mm.

Fig. 2. (a) Uncontrolled structure history (- - -) vs. the same structure controlled with a tuned rectangular liquid damper (—) and (b) first 10 s of

displacement (- - -) and fluid power (—) histories of the tuned rectangular absorber.


Strong fluid–structure interaction occurs within this type of absorber. Such interaction transfers energy from thestructure to the fluid through the sloshing force. This force is the resulting net effect from the pressure and shear stress thefluid exerts on the boundary. The control characteristics produced by the sloshing force can be demonstrated by examiningthe fluid power exerted on the container. Fluid power is the dot product of the sloshing force and the structure’s velocity.Control is achieved when the product is negative. When the product is positive, on the other hand, fluid excites thestructure, sometimes resulting in an undesirable beat in the structure displacement history.

The histories of fluid power and structure’s displacement of the first 10 s of simulation for the tuned rectangularsloshing absorber are shown in Fig. 2(b). All peaks in power correspond to instances when a wave impacts on a containerwall, near maximum structure velocity (or zero displacement). Significant negative power and almost no positive powerare seen through the first two cycles of structural oscillation. Control is highly effective as a result. Fluid begins to excitethe structure in the 2nd half of the 2nd cycle continuing through to the 5th. After the 5th cycle, power magnitude becomesalmost insignificant, with minimal control effect.

Snapshots of some critical physical events within the rectangular container are shown in Fig. 3. Fluid and boundarycolouring relate to the scales shown below each frame. In the left-hand column, this scale indicates fluid velocity in m/s. Inthe right-hand column, it is an indication of the rate of shear energy dissipation in W. Shearing of the fluid is the primaryform of mechanical damping. Initially, fluid and structure motion are highly energetic and out of phase from one another.The opposition of fluid and structure momentum exerts a strong controlling effect on the structure.

Two distinct system states are observed. The first is of high fluid kinetic energy and high structure potential energy; thesecond is of high fluid gravitational potential energy and high structure kinetic energy. The instant shown in Fig. 3(a) is anexample of the second state described. The wave-to-wall interaction shown is responsible for the second peak in fluidpower seen in Fig. 2(b), when the fluid elevation is maximum. At this instant, the rate of shear energy dissipation is alsohigh against the container bottom and right wall, as shown in Fig. 3(b).

The wave-to-wall interaction that follows, shown in Fig. 3(c), causes fluid to rise to the container ceiling against the leftwall. The extensive free-surface deformation seen here prolongs the duration of the interaction, and therefore, the timeover which it has a controlling effect. This type of wave-to-wall interaction is typical within the rectangular container,when subject to such large amplitude excitation. This event produces the third power peak in Fig. 2(b). The rate of shearenergy dissipation continues to increase at the boundary. Significant dissipation is also observed at the free-surface inFig. 3(d).

The large free-surface deformations produced during wave-to-wall interactions cause the phase difference between thefluid and structure to close. The first event responsible for producing positive fluid power is shown in Fig. 3(e). A travellingwave has moved through the container from right to left, impacting on the left container wall and producing a hydraulicjump. At the instant of collision, fluid and structural motion are in phase, albeit for a short period of time. The speed atwhich this wave travels causes very steep velocity gradients to be produced along the bottom of the container. A high rateof shear energy dissipation is seen in this area as a result, now indicated in red in Fig. 3(f).

Fluid and structural motion are mostly in phase from around t¼3.5 s. Positive peaks are now larger than the negativepeaks in fluid power. Energy is passed back to the structure as a result, causing an increase in structural displacementamplitude. Following events display smaller fluid elevations during the wave-to-wall interactions than previously seen. InFig. 3(g), a small, breaking travelling wave moves from wall-to-wall. The rate of shear energy dissipation is generally small;however, it remains somewhat significant at the container bottom. From t¼7.50 s onwards, standing waves are observed,with further restricted control effect. Finally, structure’s displacement decays to 1% of its maximum value at t¼20.0 s.

Peaks in sloshing force are caused by wave-to-wall interactions. The duration of these events along with sloshing forcemagnitude, are responsible for the amount of energy transferred. Duration is long when fluid reaches high gravitationalpotential energy. Therefore, high fluid elevation leads to high energy transfer. Although this is the case, excessive fluidelevation can cause in phase motion between the fluid and the structure. This in phase motion restricts the control effectdue to momentum opposition, and can even cause the fluid to excite the structure.

Energy transferred to the fluid must be dissipated effectively. Viscous dissipation is the primary form of dissipationwithin the working fluid. Wave-to-wall interactions and travelling wave motion produce high shear stress at the containerwalls and at the bottom. Dissipation performance is poor when travelling waves do not exist.

4. Effect of container shape

The effects of implementing angled and curved surfaces as absorber geometry are investigated next. The objective is toimprove on the control characteristics of the tuned rectangular absorber design. The shapes in Fig. 4 have been chosen tocontrol the structure, due to their significant differences from one another, and simplicity in design.

A constant free-surface length of 200 mm (f1) is kept in all cases except for the cylinder in Fig. 4(b). Keeping both liquiddepth and free-surface length constant is not possible, for different shapes. Hence, the liquid depth is allowed to vary asindicated in the figure.

The displacement histories of the structure, in the same order as that in Fig. 4, are shown in Fig. 5. In Fig. 5(a), the tunedrectangular absorber, is the most effective controller. This effective performance is due to the fact that the free-surfacelength of 200 mm and the depth of 30 mm are tuned to the natural frequency of the structure to be controlled. All othershapes in Fig. 4, however, only maintain the same free-surface length. Hence, they cannot be considered ‘tuned’.

t = 1.42 s(a) (b)

t = 1.72 s(c) (d)

t = 2.34 s(e) (f)

t = 3.80 s(g) (h)

Fig. 3. Still frames at instants of interest within the tuned rectangular absorber. Colour scale in the left column represents fluid velocity (m/s), whereas

the scale in the right column represents rate of shear energy dissipation (W). (For interpretation of the references to colour in this figure, the reader is

referred to the web version of this article.)


h

h200 X 200 mm rectangular container.

= 30 mm, fl = 200 mm.

h

h = 50.8 mm, fl

112.8 mm radius cylinder. Equivalent volumeto rectangular container in (a).

= 174 mm.

h

Converging trapezoid 15-degree inward angled walls.h = 30.2mm, fl = 200 mm. Container height = 200 mm

h

Diverging trapezoid 15-degree outward angled walls.h = 32.9 mm, fl = 200 mm. Container height = 200 mm

h

h

Converging trapezoid 30-degree inward angled walls.h = 28.9 mm, fl = 200 mm. Container height = 200 mm

hDiverging trapezoid 30-degree outward angled walls.= 34.9 mm, fl = 200 mm. Container height = 200 mm

fl

h

hHalf cylinder. Radius of 104.8 mm.

= 29 mm, fl = 200 mm.

h

Teacup shape. Wall radius of 120 mm.Flat bottom length = 62.2 mmh = 31.8 mm, fl = 200 mm.

fl

flfl

fl fl

flfl

Fig. 4. Series of shapes investigated. h is the liquid depth used and fl is the free-surface length. The same fluid volume was used in all shapes to keep

added fluid mass constant.


In Fig. 5, rows 2 and 3 show behaviour that may be useful. Row 2, with converging walls, is quite effective in dissipatingenergy rapidly. Row 3, with diverging walls, show clearly beating envelopes indicating the presence of strong interactionbetween the fluid and the structure but lack of effective dissipation. Beating manifests as an increase in structuredisplacement amplitude, caused by the fluid exciting the structure. The implementation of more complex ellipticalsidewalls (in row 4) shows no advantage over the simpler linear arrangements, whereas the circular cylinder in Fig. 5(b) isthe poorest performer.

The difference in control characteristics for four different containers is demonstrated through the fluid power historiesin Fig. 6. The case of the tuned rectangular absorber in Fig. 2(a) is repeated in Fig. 6(a) for the purpose of comparison. Thecircular cylinder in Fig. 6(b) produces significant control during the first cycle of structural oscillation. The first cycle ofstructural vibration produces significant control. After this, the circular surface produces a standing waveform. The fluid

Fig. 5. Structural displacement histories of corresponding shapes in Fig. 4.


now acts mainly as an added mass. The power is positive periodically, exciting the structure around half of the time. Poorcontrol is observed as a result.

Performance is shown in Fig. 6(c), of the converging trapezoid (with 30-degree wall angle). Initially, fluid motion is outof phase from that of the structure. Introduction of positive power is seen from the second cycle of vibration. More positivepower peaks are seen here than in the tuned rectangular absorber, yet no beating is observed in the structure displacementhistory. The lack of the beat is due to the fluid staying in phase with velocity for short periods of time only.

Fig. 6(d) corresponds to the diverging trapezoid (with 30-degree wall angle). Beating is clearly seen in the power historyfor this case. Energy transfer is effective in this shape, however, the fluid’s inability to dissipate all of this energy results init being passed back to the structure periodically, causing a beat envelope.

Dis

plac

emen

t (m

)

Time (s)

Pow

er (W

)

Dis

plac

emen

t (m

)

Time (s)

Pow

er (W

)

Dis

plac

emen

t (m

)

Time (s)

Pow

er (W

)

Time (s)

Dis

plac

emen

t (m

)

Pow

er (W

)

(a) (b)

(c) (d)

Fig. 6. First 10 s of displacement (- - -) and fluid power (—) histories of the (a) tuned rectangular absorber, (b) cylindrical absorber, (c) trapezoidal

damper with converging walls at an angle of 30 degrees and 0.2 m free-surface and (d) trapezoidal damper with diverging walls at an angle of 30 degrees

and 0.2 m free-surface.


5. Tuning trapezoid containers

Tuning of the free-surface length in the diverging and converging trapezoids is analysed next. The objective is toestablish a free-surface length that produces a sloshing frequency equal to the natural frequency of the structure. Optimaleffectiveness occurs when the absorber is tuned in this way. The performance of these shapes can then be compared fairlyto the tuned rectangular geometry. In order to maintain a constant fluid mass when varying free-surface length, liquiddepth is allowed to vary.

Histories of structural displacement, when coupled to a diverging trapezoid (with 30-degree wall angle) are shown inFig. 7. Fig. 7(a)–(e) corresponds to free-surface lengths of 160, 180, 200, 220 and 240 mm, in descending order. The beatgets stronger with shorter beat periods, with increasing free-surface length until 220 mm. Further increase in free-surfacelength results in a reduction in sloshing frequency, to lower than the natural frequency of the structure. Beat strength andfrequency reduce as a result. The diverging trapezoid is considered tuned at the free-surface length of 220 mm.

Fig. 8(a)–(e) corresponds to free-surface lengths of 160, 180, 200, 220 and 240 mm, in descending order, in theconverging trapezoid (with 30-degree wall angle). Narrow free-surface lengths produce an exponential decay in structuraldisplacement. Damping effectiveness increases with free-surface length. In Fig. 8(d), decay envelope appears to be linear,where the converging trapezoid is considered to be tuned. This case is an improvement on the tuned rectangular absorber.From 220 mm onwards, performance deteriorates with increasing free-surface length.

A summary of settling times for varying free-surface length, in a converging trapezoid (with 30-degree wall angle) isshown in Fig. 9. Settling time is defined as the time taken for structural displacement to reach a certain percentage of itsmaximum. The level of residual displacement is indicated in the figure caption. In instances where a data point is notgiven, the structure did not reach this level of residual displacement within the total simulation time. Settling timemagnitudes decrease with increasing free-surface length, until a value of 220 mm. Settling time increases from here withincreasing free-surface length. The absorber is considered tuned at the free-surface length of 220 mm.

Structure displacement and fluid power histories for this tuned case are shown in Fig. 10. At almost all instancesnegative power is observed. Only two significant positive power peaks are seen. The constant opposition of sloshing forceto structure motion is responsible for the linear decay of its displacement.

Fig. 11 shows a free-surface comparison of the tuned rectangular (left column) and tuned converging trapezoid (rightcolumn) absorbers during typical wave-to-wall interactions. Large free-surface deformation is seen in the tunedrectangular absorber, reducing the sloshing frequency during these events. The walls of the converging trapezoid restrictfree-surface deformation by directing the fluid downward, keeping free-surface length within an allowable range. Thisresults in retaining the out-of-phase motion between the fluid and the structure.

Fig. 7. Displacement histories with a trapezoid of 301-diverging walls and free-surface length of (a) 160 mm, (b) 180 mm, (c) 200 mm, (d) 220 mm and

(e) 240 mm.


6. Effect of varying excitation magnitude

The tuned rectangular and tuned converging trapezoid absorbers’ ability to handle varying excitation magnitudes isanalysed next. The left and right columns of Fig. 12 correspond to the structure’s displacement with the rectangular andthe converging trapezoid absorber’s, respectively. Fig. 12(a) and (b) corresponds to an initial velocity of 0.25 m/s; 12(c) and12(d) correspond to 0.5 m/s; and 12(e) and 12(f) to 1.0 m/s.

A relatively strong beat pattern is noticeable in Fig. 12(a) for the tuned rectangular absorber when excited by thesmallest, 0.25 m/s. Significant standing wave development is observed (not shown) at this low level of excitation. Thestructure takes longer to come to rest here than the case when the excitation level is four times as large as shown inFig. 12(e). As excitation magnitude is increased, the beat phenomenon becomes lighter, with quickest settling time for0.5 m/s and only very slightly longer time for the 1 m/s.

In contrast to the rectangular container, the converging trapezoid is able to avoid passing enough energy back to thestructure to cause beating. This is due to its ability to maintain out of phase motion and effective dissipation patterns.Trapezoid container produces a comparable performance to that of the rectangle for the largest disturbance, but becomesclearly more effective for smaller disturbances. Hence, the robustness of the trapezoid may be of some significance whenthe external disturbance is of uncertain nature.

0

5

10

15

20

25

30

35

0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25Free surface length (m)

Setti

ng ti

me

(S)

Fig. 9. Variation of the settling time with free-surface length for 30 degree converging walls. Residual levels of structural displacement as a percentage of

the maximum are indicated as 1% (n), 2% (J), 5% (n), 10% (&) and 25% (þ).

Fig. 8. Displacement histories with a trapezoid of 301-converging walls and free-surface length of (a) 160 mm, (b) 180 mm, (c) 200 mm, (d) 220 mm and

(e) 240 mm.


t = 1.42 st = 1.42 s

t = 1.72 s t = 1.72 s

Fig. 11. Still frames at instances of interest within the tuned rectangular absorber in the left hand column and the tuned converging trapezoid in the right

hand column. Fixed colour scale represents fluid velocity (m/s).

Fig. 10. First 10 s of displacement and fluid power histories of the trapezoidal damper with converging walls at an angle of 30 degrees and 0.22 m

free-surface length.


7. Conclusions

The tuned rectangular sloshing absorber can successfully mitigate vibration of structures subject to impulse type loading.Control is achieved by fluid–structure momentum opposition, and energy dissipation produced via shearing of the fluid.Dissipation via this means is highest at the container surface due to the no-slip condition. Significant energy dissipation occurs atthe free-surface when travelling waves are present. This waveform is discontinuous in nature, having steep velocity gradients andcausing inherent high shear stress. For this reason the travelling waveform should be encouraged.

A series of container shapes have been used to control the structure. The objective is to identify geometry that hassuperior control characteristics to that of the tuned rectangular absorber. Controlling a structure with a circular cylindershaped absorber results in poor control. A dominant standing waveform is seen in this shape, resulting in the fluid acting

Fig. 12. Effect of varying excitation magnitude on the tuned rectangular absorber (left column) and the tuned converging trapezoid (right column). Initial

velocity magnitudes of (a) and (b) 0.25 m/s, (c) and (d) 0.5 m/s and (e) and (f) 1.0 m/s. Note varying y-axis scale down the page.


as added structure mass. The implementation of diverging walls at a 30-degree angle on rectangular geometry maximisesenergy transfer between the structure and the fluid. However, the fluid cannot dissipate this energy, resulting in it beingpassed back to the structure. Undesirable beating is seen in the structure’s vibration history as a result.

Improvements on the tuned rectangular absorber can be made through a simple shape modification. Implementation ofconverging walls at a 30-degree angle eliminates beating, providing the free-surface length is tuned. The inherentrestriction on free-surface deformation ensures out of phase motion between the structure and the fluid is maintained.This results in damping characteristics analogous to Coulomb friction, where the damping force always opposes structurevelocity, producing a linear decay envelope.

The inward angled walls cause the free-surface breakage during wave to wall interaction, producing travelling waves morereadily than in a rectangular container. For this reason, inward angled walls may improve control performance over broaderexcitation levels, due to less effort being required to generate wave breaking and inherent effective energy dissipation.

Acknowledgements

During the course of this work, the first author was a recipient of Victoria University and CSIRO Research Scholarships.


Appendix A

Experiments

The mechanical oscillator used to test the validity of the predictions with SPH was designed to be similar to that usedby Modi and Munshi. The structure is pivoted to ensure angular oscillations. The sloshing absorber is attached on aplatform forming an upside-down pendulum, as shown in Fig. A1. Such a configuration has the advantage to mobilise thesloshing liquid easily and to enhance the suppression performance (Banerji et al., 2000).

The natural frequency and the equivalent viscous damping ratio of the uncontrolled structure are 0.5 Hz70.02 and0.9%70.1, respectively. The container is 340 mm long (in the direction that waves travel), 230 mm wide and 180 mm high.It is mounted 670 mm above the pivot point. The mass moment of inertia of the uncontrolled structure is measured to beapproximately 3.4 kg m2 about the centre of rotation. The ratio of mass moment of inertia of fluid to that of the structure is1/17 for the 5.5 mm depth shown below.

The disturbance is provided from an initial angular displacement of 16 degrees. A simple stop-block allows consistentinitial conditions for all cases. The structure is released from this initial position and allowed to oscillate freely.Experimental observations are video recorded with a standard digital camera, at a speed of 20 frames per second.

In Fig. A2(a), the histories of the angular displacements are shown for the numerical predictions (—) and theexperimental observations (K). There is virtually exact correspondence between the two cases. In Fig. A2(b), thedisplacement histories are shown for the same two cases, this time with 5.5 mm liquid depth in the sloshing absorber. Forthe liquid depth of 5.5 mm, the overall trends between the two sets are very similar. A sharp decay is obtained, with alinear envelope very similar to that of a Coulomb damping case. After the first two cycles, predicted displacements aresmaller than the experimental ones. In an equivalent sense, the predicted viscous damping ratios range between 7% and9.5%, whereas the observed ones are between 6.9% and 7.3%. In addition, the numerically predicted period of oscillations isapproximately 10% longer than the experimental one, after the first two cycles.

Although differences do exist between the predicted displacements and those observed experimentally, thesedifferences are small. Overall, structural behaviour is predicted well, particularly during periods of high fluid kineticenergy. The authors believe that these trends demonstrate the ability of SPH to meaningfully model the severe fluid–structure interactions in a sloshing absorber.

absorber

structure

Fig. A1. Schematic showing the structure and the tuned absorber.

Fig. A2. History of angular displacement of the structure for experimental observations (K) and numerical predictions (—) for (a) the uncontrolled

structure and (b) the structure controlled with 5.5 mm of liquid depth.


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