Incompressible smoothed particle hydrodynamics modeling of tuned liquid dampers for structural vibration control

Abstract

This study extends previous experimental investigations of TLD-equipped frame structures by integrating Incompressible Smooth Particle Hydrodynamics (ISPHs) simulations with calibrated analog models, thereby bridging numerical, analytical, and experimental approaches. TLDs dissipate vibrational energy through sloshing, and the ISPH formulation is enhanced with kernel gradient correction and particle shifting techniques to improve stability and free-surface accuracy. Results of the dam-break benchmark test confirm consistency with established experimental and numerical findings. The ISPH model was further validated against a 3-story frame and its modified analog counterpart. When the tank liquid height is tuned near the resonance frequency of the frame, structural vibrations are significantly reduced, demonstrating the strong energy absorption capacity of sloshing. Damping arises not only from lateral wall pressures but also from base friction and free-surface wave dynamics. In particular, accurate modeling of resonance-induced surface waves is critical for reliable numerical representation of TLD performance.

Keywords

structural vibration control incompressible smooth-particle-hydrodynamics tuned liquid dampers sloshing

1. Introduction

Structural control systems are damping systems that absorb or redirect external vibrations to the structure. These damping systems were originally designed as mechanisms that eliminate energy by friction (Habib et al., 2025; Son et al., 2025; Zayas et al., 1989, 1990; Çabuk et al., 2020) or provide damping by the viscous properties of viscoelastic fluids (Cai et al., 2025; Xu et al., 2023). Apart from these types of dampers, tuned mass dampers (TMD) (Belykh et al., 2021; Kourakis and Gy, 2007; Nishitani, 2019; Pawar and Salgar, 2025; Tuan and Shang, 2014; Twardoch et al., 2025) with counter-acting restoring forces have also been very useful. In this type of systems, the opposite force of the object connected to the spring is used by adjusting the resonance frequency of the structure and the natural frequency of the damper system. The long-term performance of seismic isolation systems is well clarified at recently published authoritative reports (ASCE/SEI 7-22, 2022; Constantinou et al., 2007).

TLDs are also highly effective in controlling wind-induced vibrations that have been developed primarily for such vibrations (Fujino and Sun, 1993; Kareem et al., 1999; Sun et al., 1995; Xu et al., 1992). TLDs function in a similar way with TMDs and are used in high-rise buildings to mitigate wind-induced vibrations (Zhang et al., 2016). Unlike TMDs that work with a single frequency, TLDs contain more than one frequency and can absorb effective energy in extreme loading situations. TLDs performance depends on the liquid in the tank, the height of the liquid, and the tank geometry and can be enhanced using multiple tanks. Sloshing restrains structural motion, reducing lateral displacements and resonance effects. Nonlinear damping arises from wave breaking, fluid-wall impacts, and viscous friction. Early studies used semi-analytical models (Fujino et al., 1992; Modi et al., 1990) based on shallow water theory, with experiments confirming significant vibration reduction—up to two-thirds in real structures like the Nakayawa Airport and Yokohama Marine towers (Tamura et al., 1995). Subsequent research incorporated nonlinear damping terms and energy dissipation mechanisms into analytical models (Koh et al., 1994; Sun et al., 1989, 1992; Yu et al., 1999). Experimental and numerical studies, including those using boundary element methods and analog models, validated TLD effectiveness (Li et al., 2025; Sierikova et al., 2025). Fluid-structure interaction is modeled either by integrating tank dynamics into structural models or by coupling separate fluid and structural solvers (Dou et al., 2020; Dou et al., 2020).

Smooth Particle Hydrodynamics (SPH) is a particle-based method well-suited for modeling large deformations, free-surface waves, and wave refraction, outperforming traditional Eulerian methods (Monaghan, 1992; Monaghan and Gingold, 1983; Monaghan and Thompson, 1994). It calculates fluid properties like velocity and pressure through interpolation using a smoothing function. However, the weakly compressible SPH approach can produce inaccurate pressure results and density fluctuations due to particle disorder (Cummins and Rudman, 1999). To address these issues—especially in long simulations—diffusive terms were added to the governing equations, forming the delta-SPH method (Antuono et al., 2010; Marrone et al., 2011; Molteni and Colagrossi, 2009), which improves stability and reduces numerical artifacts. Alternatively, incompressible SPH (ISPH) is a reliable method for conserving mass and momentum, especially effective in suppressing spurious pressure fluctuations in free-surface flows. It solves a Pressure-Poisson equation derived from mass and momentum conservation, typically using the Conjugate Gradient method for efficiency with large sparse matrices (Ihmsen et al., 2014). However, due to particle disorder—especially near boundaries—ISPH can suffer from uneven particle distributions. Kernel gradient corrections (Bonet and Lok, 1999; Chen et al., 1999) and particle shifting techniques (PST) (Huang et al., 2018, 2019; Liu et al., 2022), based on Fick’s Law, help maintain uniformity. These enhancements have enabled ISPH to perform well even in high-speed impact scenarios (Shao et al., 2012; Zhang and Liu, 2017), making it a robust and accurate numerical tool.

This work contributes an experimentally calibrated ISPH framework—enhanced with kernel gradient correction and particle-shifting—and a matched analog model to quantify TLD performance, revealing the role of wall pressures and nonlinear free-surface dynamics (including traveling and breaking waves) in structural vibration control. It is observed that breaking waves occur near resonance that coincide with increased energy loss, but a quantitative comparison of damping into wall pressure, friction, and breaking-wave contributions remains for future work.

2. Methods

2.1. SPH formulation

In SPH methods, using the particle approximation at point i and taking volume of any neighbor particle as $V_{j} = m_{j} / ρ_{j}$ where $m_{j}$ and $ρ_{j}$ denote mass and density of the j^th particle, any arbitrary f at $x = (x, y)$ :

f (x_{i}) = \sum f (x_{j}) w (r_{i j}, h) V_{j} = \sum f (x_{j}) w (| r_{i j} |, h) m_{j} / ρ_{j}

(1)

where

w (x - x^{'}, h)

is a compact Kernel function, with a smoothing length, h, the property of unity and delta function.

r_{i j} = x_{i} - x_{j}

is the position vector. The derivative of the function is defined properly as

\nabla f (x_{i}) = \sum (f (x_{j}) - f (x_{i})) \nabla_{i} w (| r_{i j} |, h) m_{j} / ρ_{j}

(2)

The $w$ is corrected by the procedure in the study of Chen et al. (1999).

2.2. Discretization of governing equations

Tank sloshing is modeled through conservation laws, expressed with the material derivative ( $D / D t$ ):

D ρ / D t = - ρ \nabla u

(3)

D u / D t = - \nabla P / ρ + ϑ \nabla^{2} u + g

(4)

ρ

ϑ

, t and P are scalars for density, kinematic viscosity, time, and pressure;

u

and

g

is the velocity and the body force. ISPH velocity-pressure coupling is solved in two steps (Chorin, 1968):

Predictor step : (u^{*} - u^{n}) / Δ t = ϑ \nabla^{2} u^{n} + g^{n}

(5)

Corrector step : (u^{n + 1} - u^{*}) / Δ t = - \nabla P^{n + 1} / ρ_{0}

(6)

Taking the gradient of equation (6) with ${\nabla u}^{n + 1} = 0$ yields Poisson’s equation:

\nabla^{2} P^{n + 1} = (ρ_{0} / Δ t) {\nabla u}^{*}

(7)

A second-order approximation (Cummins and Rudman, 1999; Morris et al., 1997) is used where $ε = 0.01 h$ to avoid division by zero:

\nabla^{2} {P_{i}}^{n + 1} ≅ \frac{2 m_{0}}{ρ_{0}} \sum_{j} \frac{(P_{j}^{n} - P_{i}^{n}) r_{i j}^{n} \cdot \nabla_{i} w_{i j}}{{| r_{i j}^{n} |}^{2} + ε^{2}}

(8)

The viscosity term is formulated with $u_{i j} = u_{i} - u_{j}$ , and $Π_{i j}$ is an artificial viscosity term (Monaghan, 1992) as

ϑ \nabla^{2} u^{n} = \frac{m_{0}}{ρ_{0}} \sum_{j} (\frac{2 ϑ r_{i j}^{n} \cdot \nabla_{i} w_{i j}}{{| r_{i j}^{n} |}^{2} + ε^{2}} u_{i j}^{n} + α h c Π_{i j}^{n})

(9)

Due to the sparse, asymmetric matrix in Poisson’s equation, the Stabilized BiCG method (Dincer et al., 1996; Dinçer et al., 1994) is utilized. After computing $P^{n + 1}$ , the corrector step yields $u^{n + 1}$ , and particle positions are updated by

r_{i}^{n + 1} = r_{i}^{n} + (\nabla t / 2) (u_{i}^{n + 1} + u_{i}^{n})

(10)

At the end of each time step, the particle positions are shifted slightly by the amount of $δ r$ to obtain a uniform distribution (Lind et al., 2012).

2.3. Particle classification and repositioning

The 5^th order Wendland kernel is employed. The kernel’s influence radius extends to 2h (Figure 1). With an initial particle spacing of d₀, the smoothing length h is chosen as 1.3d₀ within the recommended 1.3d₀ to 1.6d₀ range (Liu and Liu, 2003; Monaghan, 1992) to meet resolution requirements. This ensures that, under a uniform distribution, at least 14 neighboring particles lie within the influence domain of an interior particle (I). This condition is verified at each time step using the specified procedure.

• Interior particles are defined and remaining particles are assigned as surface particles(S).

• In the next step, if any interior particle interacts with any surface particle (as shown in Figure 1 by particle k), this interior particle is reassigned as a subsurface particle (SS).

Figure 1.

Classification of particles.

3. Validation of the code by sloshing tank test

The accuracy of the ISPH code was ensured by an experimental study (Xue et al., 2017). Initial height of the liquid is taken as 0.18 m in the prismatic tank with a length of L = 0.57 m. In these studies, excitation displacement was taken as $x_{G} (t) = 0.01 \cos (0.94 ω_{1} t)$ where the first natural frequency of the liquid in the tank is $ω_{1} =$ 6.403 rad/s. The origin of the coordinate system is located horizontally in the middle of the tank’s width and vertically at the tank’s base. x = 285 mm and y = 65 mm is selected as the probe location (P1) as in the experiment. n_x is the initial number of evenly spaced particles in the x-direction, and n_y is the number of particles in the y-direction. Four individual tests were conducted using 57 x 18, 95 x 30, 133 x 42, and 171 x 54 particles.

Due to its implicit operation, ISPH can work with larger time delays than other SPH methods. Time stepping is selected as

Δ t = 2 \cdot 10^{- 4}

sec. Time stability criterion

Δ t < C_{C F L} h / v_{\max}

(Violeau and Leroy, 2014, 2015) is compared by taking

C_{C F L} = 0.25

and

v_{\max} =

5 m/s as observed in simulations (Table 1).

Table 1.

Time stability for various resolution sets.

n_x x n_y	Initial spacing (d₀)	h = 1.3d₀	$C_{C F L} h / v_{\max}$
57 x 18	0.01000	0.01300	6.50 x 10⁻⁴
95 x 30	0.00600	0.00780	3.90 x 10⁻⁴
131 x 42	0.00435	0.00566	2.83 x 10⁻⁴
171 x 54	0.00333	0.00433	2.17 x 10⁻⁴

At P1, the pressure history (Figure 2) was obtained by the test when 57 x 18 (d₀ = 10 mm) corresponding to the weakest resolution case. Pressure values include two peak points, as in experimental results (Xue et al., 2017), the first occurs when the liquid hits the wall for the first time, and the second occurs when it leaves the wall. The maximum pressure at P1 is 1.67 kPa which is 1.42 kPa in the given experimental work.

Figure 2.

The comparison of ISPH pressure history (red) at P1 with the experimental (black) work (Xue et al., 2017), when $ω = 0.94 ω_{1}$ .

The net force is obtained on the tank by the sum of the pressure distribution on the wall surface. The difference, $F_{T L D} = F_{R I G H T} - F_{L E F T}$ represents the sloshing effect on the tank. Coarse and dense particle resolutions are compared in Figure 3. Around t = 10 sec., a transition begins where the free-surface motion becomes nonlinear. While the dense resolution (171 x 54) exposes finer structures and localized instabilities in transition, the coarser resolution (95 x 30) smooths out them. This shows the importance of the particle density in capturing sloshing-induced energy dissipation.

Figure 3.

F_TLD for n_x x n_y = 57 x 19 (black), 95 x 30 (green), 133 x 42 (red), and 171 x 54 (blue) when $ω = 0.94 ω_{1}$ .

In Table 2 at t = 3.3, relative error of F_TLD drops to 3.3 % in the densest case. The simulations were performed with the Intel Ultra 9 processor. When the number of particles (N) and elapsed time per process (El.T) were examined on a logarithmic scale, it was observed that there was a relationship between the increase in the number of parts and the elapsed time, approximately as El.T = O(N^1.4).

Table 2.

F_TLD (t = 3.2 sec) for various resolution sets and corresponding elapsed time per step.

n_x x n_y	F_TLD at t = 3.2 sec	$ε_{R E L}$ (%)	Elapsed time per step (sec) x10³
57 x 18 = 1026	268.2	-	20.586
95 x 30 = 2850	305.3	12.2	77.875
131 x 42 = 5586	324.3	5.9	217.418
171 x 54 = 9234	335.3	3.3	481.113

ISPH images were turned horizontally to be compatible with the experimental (Xue et al., 2017) images (Figure 4). In the experiment, the refracted waves are more pronounced. Nevertheless, a great deal of agreement was achieved with the sloshing snapshots included in the experimental study.

Figure 4.

ISPH when n_x x n_y = 171 x 54 (d₀ = 3. $\bar{3}$ mm) and experimental (Xue et al., 2017) snapshots at (a) t = 1.97 s, (b) t = 4.96 s, (c) t = 6.03 s, (d) t = 8.21 s, (e) t = 9.33 s, (f) t = 11.54 s when $ω = 0.94 ω_{1}$ .

4. Modeling 3-story frame structure

In sloshing experiments (Li et al., 2025; Vafaei et al., 2021), a tank (L_x = 0.4 m, b = 0.2 m, H = 0.12 m; H/L_x = 0.3) was mounted on a 3-story frame (Figure 5) to measure story displacements. The structure had 1 m spans, with story heights L₁ = L₃ = 0.8 m and L₂ = 1.05 m. Beams in the excitation direction were 2 cm × 2 cm box sections, perpendicular elements 4 cm × 4 cm angles, and columns of 5 cm width with thicknesses 12 mm (1^st), 9 mm (2^nd), and 6 mm (3^rd). Steel properties were ρ = 7850 kg/m³ and E = 200 GPa (ANSI/AISC 360-16, 2016). The total weight was ∼40 kg, plus added loads of 180 kg (1^st floor) and 80 kg (3^rd floor).

Figure 5.

3-Story Frame model (Li et al., 2025; Vafaei et al., 2021).

4.1. TLD analogy

This 3-storey structure is modeled as a lumped mass system (Chopra, 1995). The horizontal displacements and velocities of the stories are defined as $(x_{1}, x_{2}, x_{3})$ and $(v_{1}, v_{2}, v_{3})$ , respectively. When the TLD tank is located to the story, j, kinematic relationships link displacements to velocities:

{\dot{x}}_{1} = v_{1}

(11)

{\dot{x}}_{2} = v_{2}

(12)

{\dot{x}}_{3} = v_{3}

(13)

The TLD acts as an auxiliary spring attached to k^th mass with stiffness $k_{0}$ referenced to $x_{0}$ . Kronecker delta, $δ_{j k}$ ensures the force only applies to the specific coupled degree of freedom such that

F_{T L D}^{k} = δ_{j k} (- k_{0} (x_{k} - x_{0}))

(14)

And the dynamic relationships would be

{\dot{v}}_{1} = (- (k_{1} + k_{2}) x_{1} + k_{2} x_{2} - (c_{1} + c_{2}) {\dot{x}}_{1} + c_{2} {\dot{x}}_{2} - F_{T L D}^{(1)}) / (m_{1} + δ_{j 1} m_{r}) + a_{G}

(15)

{\dot{v}}_{2} = (k_{2} x_{1} - (k_{2} + k_{3}) x_{2} + k_{3} x_{3} + c_{2} {\dot{x}}_{1} - (c_{2} + c_{3}) {\dot{x}}_{2} + c_{3} {\dot{x}}_{3} - F_{T L D}^{(2)}) / (m_{2} + δ_{j 2} m_{r}) + a_{G}

(16)

{\dot{v}}_{3} = (k_{3} x_{2} - k_{3} x_{3} + c_{3} {\dot{x}}_{2} - c_{3} {\dot{x}}_{3} - F_{T L D}^{(3)}) / (m_{3} + δ_{j 3} m_{r}) + a_{G}

(17)

In the analog model (Figure 6), the ground acceleration is denoted as a_G. Story stiffness, $k_{j} = \sum_{c o l u m n s} 12 E I_{c} / {L_{j}}^{3}$ are calculated by assuming fixed column ends where $E$ is the modulus of elasticity, $I_{c}$ the column moment of inertia, and $L_{j}$ the story height. Damping coefficients at each story are given by

c_{j} = 2 ξ_{j} \sqrt{m_{j} k_{j}}

(18)

with

ξ_{j}

, the damping ratio and

m_{j}

, the story mass. The stiffness corresponding to the first dynamic mode of the sloshing liquid in the tank,

k_{0}

, is defined following Abramson (1966):

k_{0} = {ω_{0}}^{2} m_{0}

(19)

Figure 6.

Story frame analogy with TLD system, TLD system is at 1^st, 2^nd and 3^rd stories, from top to bottom, respectively.

The damping coefficient of the sloshing liquid $c_{0} = 2 ξ_{0, L} \sqrt{m_{0} k_{0}}$ where $ξ_{0, L}$ is the unitless damping parameter of 1^st mode of the sloshing. Here, the mass value corresponding to this first mode is $m_{0}$ and referring natural frequency $ω_{0}$ :

m_{0} = 8 m_{L} (\frac{\tanh (π H / L_{x})}{π^{3} (H / L_{x})})

(20)

ω_{0} = \frac{g}{L_{x}} π \tanh (\frac{π H}{L_{x}})

(21)

where m_L is the mass of the liquid,

m_{L} = ρ L_{x} H b

. Accordingly, the mass of the liquid is 9.6 kg and the frequency values of the first 3 modes of the liquid in the tank are 1.20 Hz., 2.41 Hz. and 3.12 Hz.

4.2. Determination of sloshing damping coefficient, ξ_0,L

This section determines the damping coefficient of sloshing liquid in ISPH simulations by analyzing free decay after applying 0.05 Hz harmonic displacements 0.10,0.12 and 0.14 m for 5 seconds. The free-surface elevation decays exponentially as $e^{- ξ_{0, L} ω_{0} t}$ (Faltinsen, 1978), with the dominant frequency 1.1982 Hz. The resulting dimensionless damping ratios are 0.047, 0.044, and 0.043, yielding an average first-mode damping value of 0.045.

4.3. Calibration for the numerical models

The primary natural frequency of the structure given in the experimental study is 1.18 Hz. Both the mass values of the floors and the flexural stiffness of the structural elements were calculated directly using the experimental model. However, due to some reasons such as fixed modeling of joints in the lumped system, the first natural frequency of the lumped system is 1.61 Hz. Here, the analogy is made more realistic by modifying the stiffness values equally. The stiffness values shown in Table 3 were scaled by the number 0.56 by trial-and-error method, and the primary frequency value of the numerical model was obtained as 1.20 Hz. In this way, the numerical model is harmonized with what is in the experiment.

Table 3.

Lumped system numerical values after the stiffness approximation.

Stories	Mass (kg)-m	Stiffness (N/m)-k	Damping Coef. (Ns/m) - c
1^st story	196	75,600	462
2^nd story	16	14,106	57
3^rd story	88	9450	109

The ξ was set uniformly at 1.7% (Li et al., 2025; Vafaei et al., 2021), though slightly higher values improved agreement in the analog model. Under 1.2 Hz excitation (near resonance) without a TLD, story displacements were 9.5 mm, 51.5 mm, and 84 mm, with ground motion limited to 10 cm. The optimal damping was estimated by minimizing differences (Equation (22)) from experimental data. S reaches its smallest value when $ξ =$ 6.09%. The displacement values given by this dumping value are 8.4 mm, 46.2 mm and 98 mm, respectively.

S = {(1 - 46.874 / 9.5 x^{- 0.951})}^{2} + {(1 - 262.83 / 51.5 x^{- 0.962})}^{2} + {(1 - 561 / 84 x^{- 0.966})}^{2}

(22)

5. Tests on 3-story frame

In the tests performed with ISPH, the liquid in the tank was represented by 100 particles in horizontal direction and was taken as d₀ = 0.004 m. In the vertical direction, 30 particles were used, so tests were carried out using a total of 3000 particles. The $Δ t$ value is taken as $2 \cdot 10^{- 4}$ as in Chapter 2.

5.1. Free harmonic motion

Free harmonic motion tests were run for both analog and ISPH models with initial displacements of 4 mm, 35 mm, and 55 mm for the 1^st–3^rd stories, released for 10 sec oscillations. In the bare frame, displacements decayed below 6 mm at t = 10 sec. With the TLD on the 1^st story, damping was negligible and ISPH/analog curves overlapped (Figure 7). At the 2^nd story, beating occurred: amplitudes decreased, regrew near t = 4 sec, then subsided (Figure 8), consistent with low-damping energy exchange (Yalla and Kareem, 2001). ISPH differences stem from wall-pressure modeling, while the analog model uses only the first sloshing mode. On the 3^rd story, beating was stronger (Figure 9); at t = 4 sec displacements rose across all floors, with the 3^rd reaching ∼27 mm (≈½ of its initial 55 mm). Temporal mismatches appeared: story peaks just before t = 1 sec vs. free-surface peaks at t = 1.05 s; between 3 and 4 sec story displacements rose while the free surface was calm; at t = 4.95 sec displacements dropped while free-surface motion increased (Figure 10). These results show sloshing-structure interaction evolves over time rather than instantaneously.

Figure 7.

Displacements of stories when TLD is at 1^st story by Analog model and ISPH methods compared with the bare system (dashed).

Figure 8.

Displacements of stories when TLD is at 2^nd story by Analog model and ISPH method compared with the bare system (dashed).

Figure 9.

Displacements of stories when TLD is at 3^rd story by Analog model and ISPH method compared with the bare system (dashed).

Figure 10.

Free-surface level (η) on the left wall and the wave profile at instants t = 1.10 sec, t = 4.95 sec and t = 9.50 sec, in the case of free harmonic motion and the TLD is at 3^rd story.

5.2. Forced harmonic motion

The 3-story frame was tested with/without TLDs on each story under 15 sec harmonic excitation, 10 cm lateral displacement, and 0.9–1.7 Hz frequencies (0.1 Hz steps). ISPH damping was tuned to match bare-frame Peak Response Displacements (PRDs) (Vafaei et al., 2021) of 8.4 mm, 46.2 mm, and 98 mm (Equation (22)). A 3^rd-story TLD gave strongest damping (Figures 11 –13): at 1.2 Hz PRDs decreased by ∼5%, 25%, and 60% for 1^st–3^rd stories, halving displacements even at 1.1 Hz. Eigenvectors (8.8, 48.7, 103.7 mm) confirmed largest PRD at the 3^rd story, enabling stronger coupling. This was partly due to the traveling waves accompanying the standing waves and increasing the pressure on the walls (Figure 14(a)). Resonance peaks appeared at 1–1.3 Hz, max near 1.2 Hz. At 1 Hz, ISPH vs. analog differed by ∼10% (27 mm vs 31 mm), while experiments showed 36 mm (∼30% higher). At ≥1.4 Hz, both models agreed: PRDs ∼2.5 mm (1st), 10–17 mm (2nd), 21–35 mm (3rd) due to the standing waves corresponding to the first mode of sloshing, which corresponds to the ISPH model (Figure 14(b)). Largest ISPH–analog difference was ∼12% at 1.4 Hz with a 2^nd-story TLD. Experimental PRDs deviated at low/high frequencies (∼1 mm, <5 mm, 10–20 mm). At 1.1 Hz, ISPH damping was stronger (22 mm vs 35 mm analog). At 1.2 Hz, PRD reductions matched experiments (Table 4), with analog showing max effectiveness (60–70%) for a 2^nd-story TLD.

Figure 11.

PRDs at 1^st story.

Figure 12.

PRDs at 2^nd story.

Figure 13.

PRDs at 3^rd story.

Figure 14.

(a) Standing and traveling waves together at w = 1.0 Hz and (b) standing waves at w = 1.4 Hz when TLD is at 3^rd story.

Table 4.

PRDs of 2^nd and 3^rd stories for TLD is at 1^st, 2^nd and 3^rd stories and corresponding differences in percentage according to corresponding model’s bare PRDs at resonance (w = 1.2 Hz).

		Bare	TLD@1	$\| \frac{u_{T L D} - u_{B a r e}}{u_{B a r e}} \|$ (%)	TLD@2	$\| \frac{u_{T L D} - u_{B a r e}}{u_{B a r e}} \|$ (%)	TLD@3	$\| \frac{u_{T L D} - u_{B a r e}}{u_{B a r e}} \|$ (%)
u₂ (mm)	ISPH	41	39	5	28	32	15	63
	Exp.	52	45	13	39	25	22	58
	Analog	46	43	7	17	63	10	78
u₃ (mm)	ISPH	88	84	5	62	30	30	66
	Exp.	84	84	0	68	19	39	54
	Analog	98	90	8	34	65	23	77

At 1.3 Hz, analog PRDs align with experiments, while ISPH shows less damping since it considers only lateral wall pressures. Sloshing at this frequency also involves breaking waves, visible in ISPH but absent in the simplified model (Figure 15). Table 5 shows pressure at the left bottom corner relative to the initial hydrostatic value (1.18 kPa); even under violent waves at 1.3 Hz, this ratio remains stable and lower than at 1.4 Hz. Thus, wall pressures alone cannot capture TLD damping—accurate modeling must include energy absorption from breaking waves.

Figure 15.

Breaking waves at w = 1.3 Hz and when TLD is at 3^rd story.

Table 5.

Peak values of P₀/P_static at the lowest point at the left wall.

w (Hz)	0.9	1.0	1.1	1.2	1.3	1.4	1.5	1.6	1.7
P₀/P_static	1.15	1.36	1.82	1.98	1.98	2.30	1.51	1.40	1.36

6. Comparative analysis and structural control

The calibrated analog model bridges experiments and ISPH simulations, confirming consistency and reliability for vibration control. When the exact same experimental model is taken with fixed nodes, the first natural frequency value of 1.18 Hz, which is specified in the experimental study, cannot be reached, and a higher natural frequency value of 1.61 Hz is obtained. In this study, when the stiffness and the masses of the stories are tested in various ranges, the value of 1.18 Hz and the next value 2.0 Hz cannot be reached with certainty. For this purpose, it was tried to match the structure used in the experiment with the ISPH model by reducing practically all stiffness values by 0.56, considering that all nodes of the multistorey structure used in the experiment were of similar semi-rigidity and considering that the first mode was dominant.

In ISPH, the opposing TLD force derives solely from wall pressures, while simulations reveal additional dynamics such as refracted and reflected waves. The analog model is likewise simplified, using only the first sloshing mode and excluding higher modes or breaking waves. ISPH captures rising/falling wall waves and energy-absorbing free-surface effects, evident near t = 1 sec with a 3^rd-story TLD (Figures 8 and 9), though standing waves dominate overall. Temporal offsets between story displacements and free-surface motion reflect gradual energy transfer (Figure 10). Under harmonic excitation, PRDs differ from experiments and analog results since ISPH neglects bottom friction and surface effects (Figures 11 –13). In fact, the pressure distribution on the base can be effective, especially in cases where there are strong nonlinear oscillations when it is close to the resonance state. Formulating this effect as a dissipation force, taking into account the agitation height of the liquid, the velocity distribution near the bottom, etc., will allow much closer results to be achieved. Apart from this, turbulence models (e.g., k-e model) can also be used. Semi-analytic studies (Koh et al., 1994; Lepelletier and Raichlen, 1988; Sun et al., 1989) attribute damping to shear friction, wall impacts, and contamination. Breaking and refracting waves must therefore be included, especially near resonance (Figure 15). Earlier work modeled damping coefficients non-linearly with free-surface elevation (Yu et al., 1999).Thus, accurate ISPH formulations require damping parameters explicitly tied to surface wave behavior.

7. Conclusions

This study demonstrates that ISPH, validated against experiments and analog models, provides a reproducible framework for analyzing TLDs in vibration control. By integrating ISPH simulations with calibrated analog models and experimental data, the work bridges numerical, analytical, and physical approaches, strengthening confidence in ISPH for practical applications.

• Mechanical properties must be calibrated; with tuning, ISPH and analog results align with experiments.

• Sloshing-structure interaction unfolds gradually. Energy transfer between them occurs with delay, not instantaneously.

• TLD effectiveness is greatest on the 3^rd story, consistent with modal analysis and is efficient around resonance.

In the ISPH model, both traveling waves and breaking waves are included. In the analog model, the sloshing is limited to standing waves only. At w = 1.3 Hz, in parallel with the increasing energy loss, there are breaking waves on the free surface. Further studies should be found out how effective these waves are in TLD damping and how large they are in energy loss compared to wall and bottom frictions. Breaking waves are seen with ISPH, but these are not available in the analog model. These observed waves participate as a plus effect in the damping mechanism, especially when it is close to resonance. However, a quantitative analysis of these waves—accounting for damping, friction, and wall pressure—is beyond the scope of this study and remains an area for future research. While refinements in damping formulations are still required, ISPH provides a robust, realistic, and experimentally consistent framework for analyzing TLD behavior in structural vibration control.

Footnotes

Acknowledgment

The author acknowledges the contributions of prior studies in sloshing dynamics and vibration control, which provided the foundation for advancing the ISPH-based modeling approach presented here. In addition, author declares that the graph of the experimental data used in this study is used to make comparisons by giving proper references which can be found in the references list.

ORCID iD

Ersan Güray

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The author declares that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Incompressible smoothed particle hydrodynamics modeling of tuned liquid dampers for structural vibration control

Abstract

Keywords

1. Introduction

2. Methods

2.1. SPH formulation

2.2. Discretization of governing equations

2.3. Particle classification and repositioning

3. Validation of the code by sloshing tank test

4. Modeling 3-story frame structure

4.1. TLD analogy

4.2. Determination of sloshing damping coefficient, ξ 0,L

4.3. Calibration for the numerical models

5. Tests on 3-story frame

5.1. Free harmonic motion

5.2. Forced harmonic motion

6. Comparative analysis and structural control

7. Conclusions

Footnotes

Acknowledgment

ORCID iD

Funding

Declaration of conflicting interests

References

4.2. Determination of sloshing damping coefficient, ξ_0,L