Experimental Investigation on Vibration Control of a Suspended Particle-Tuned Liquid Damper

Abstract

The particle-tuned liquid damper (PTLD) can combine the functions of baffles and energy-dissipating materials, such as highly viscous liquids, by integrating the particle dampers into a conventional tuned liquid damper (TLD). However, the particles distributed only at the bottom of the container cannot drive the motion of water in the middle layer to function effectively. Therefore, a suspended particle-tuned liquid damper (SPTLD) is proposed in this study and its effectiveness and reliability are examined compared with the conventional TLD through shaking table tests. Based on the experimental results, a parametric analysis of the SPTLD is further conducted to investigate the damping mechanism of the SPTLD, including the number of particles, the excitations with various amplitudes, and the use of suspended versus floating particles in liquid. The test results revealed that SPTLD successfully controlled the structural acceleration responses under seismic excitations with good reliability; the peak acceleration response was reduced by 67.4% and the RMS value was reduced by 75.9%. In the SPTLD, the particles filled in the container can drive more liquid to effectively participate in the sloshing motion, and the superimposed damping effects involving collisions and the energy-dissipation mechanisms of buoyancy and hydraulic resistance in the SPTLD lead to an improvement in the vibration control performance. Furthermore, the comparison of SPTLD and the floating particle-tuned liquid damper (FPTLD) demonstrates the better availability of SPTLD in practical applications, especially for some slender structures with limited plane space on the top floor.

1. Introduction

With the height of engineering structures and the frequency of disaster occurrence increasing, structures are vulnerable to vibrations excited by external dynamic loadings such as seismic excitations and other hazardous events. To improve the seismic resistance of structures, some methods comprising increasing cross sections, adding plates to structural members [1], and using reinforced composites [2,3,4] have been proposed. Recently, structural vibration control strategies have attracted increasing attention devoted to attenuating the damage and enhancing the comfort of structures [5,6]. The tuned liquid damper (TLD) is one of the most effective anti-vibration technologies, as it is economic, easy to design, and independent of external energy supply. A TLD is usually a container filled with a fixed volume of liquid which will undergo a sloshing motion when the container is excited by lateral excitation. The concept of the liquid damper was pioneered by Froude [7] for reducing the rolling of a ship, and later the TLD was applied to engineering structures and proved to have favorable control effects when properly designed [8,9]. However, only a portion of liquid in the container will participate in the reduction in structural vibration in most TLDs, and the contained water viscosity alone is insufficient to dissipate energy. Therefore, baffles are usually implemented inside the container to drive more liquid mass to participate in the motion [10], or energy-dissipating materials are added to enhance the vibration suppression effect of the damper [11].

To effectively combine the functions of baffles and the energy-dissipating materials, such as highly viscous liquids, a method of integrating the particle dampers into conventional TLDs was proposed. The particle damper possesses the advantages of good robustness, a wide damping frequency band, and low sensitivity to extreme temperatures. Considering that the particle damper can be arranged flexibly, it can also be regarded as a component integrating into the conventional linear dampers such as TMD and TLD. The basic configuration of a particle damper consists of a container at an appropriate location inside the primary structure, with moving particles filled inside the container. Fu et al. [12] and Lu et al. [13] conducted comparative studies to investigate the superiority of the particle dampers, and the results show that the particle dampers have better vibration attenuation capacity than conventional TLD or TMD for elastic and nonlinear performance indexes of the primary structure. Furthermore, the particle dampers have also proved to be effective in suppressing the vibrations of multi-drum ancient columns and tall buildings [14,15]. Through the momentum transfer between the damper and primary structure, and the energy dissipation during particle–particle and particle–container collisions, the damping effects of particle dampers are generated.

In terms of particle dampers serving as components in conventional linear dampers, many attempts have been conducted where the nonlinear characteristic of the particle damping is utilized to broaden the frequency band of the conventional TMD and TLD. Studies show that the particle-tuned mass damper (PTMD) can achieve better control performance than conventional particle dampers [16] and TMDs [17], accompanied by good robustness in controlling high-rise buildings subjected to earthquakes and wind loads [17,18], seismically excited continuous viaducts [19], structural vibration with small acceleration in vertical direction [20], etc. Additionally, some studies have been implemented on the particle-tuned liquid damper (PTLD) and have preliminarily demonstrated the feasibility of enhancing TLD by employing particle dampers [21,22]. The existing research is very limited and mainly concentrates on the TLDs with particles that sink to the bottom of the container. When the available planer space of the container is small, the damping effect of the PTLD will be disturbed as the number of filled particles is restricted. In addition, for the PTLD with deep water in the container, the particles distributed only at the bottom of the container cannot drive the motion of water in the middle layer to function effectively.

Therefore, a suspended particle-tuned liquid damper (SPTLD) is proposed in this paper, aiming to improve the control performance and reliability of the conventional TLD. The improved vibration attenuation effect is realized by integrating the damping mechanism of collision and driving more liquid to participate in the sloshing motion through the filled particles. Enhanced reliability is achieved with the nonlinear characteristics of the particle damping. To examine the effectiveness and reliability of the proposed SPTLD, shaking table tests are implemented, where the vibration suppression effects of SPTLD are compared with the conventional TLD. Based on the experimental results, a parametric analysis of the SPTLD is further conducted to reveal the damping mechanism of the proposed SPTLD, including the number of particles, the excitations with various amplitudes involving sweep wave and seismic waves, and the use of suspended versus floating particles in liquid. The forms of the particles used in this study can further expand the application scenarios of PTLDs.

2. Configuration of SPTLD and Experimental Setup

2.1. Configuration of SPTLD

The SPTLD consists of a basic configuration of TLD with particles suspended in the liquid of the container. Figure 1a presents the structural schematic diagram of SPTLD, where $G$, $F_{\mathrm{b}}$, and $F_{\mathrm{r}}$ represent the gravity, buoyancy force, and hydraulic resistance of the particle, respectively. $X_{\mathrm{s}}\left(t\right)$ denotes the motion of the SPTLD. When the vibration of the primary structure is generated under external excitations, the container moves with the primary structure and the internal fluid and particles are excited. Through the tuning effect and the energy-dissipation mechanism mainly comprising the impact of particles, friction, and wave breaking, the SPTLD is expected to suppress the vibration of the primary structure.

In the SPTLD, it is critical to keep the particles suspended in the liquid. An achievable configuration of suspended particles is proposed in this paper, as seen in Figure 1b. The suspended particle is designed as a lightweight polyfoam ball containing two smaller steel balls. The outer diameter of the polyfoam ball is 18 mm, and the polyfoam ball is drilled through. One end of the hole is embedded with a steel ball with a diameter of 6.25 mm, then sand and latex are filled into the gap, and another end is embedded with a steel ball with a diameter of 8 mm. The buoyancy force acting on the particle $F_{\mathrm{b}}$ (N) can be calculated as follows.

$$F_{\mathrm{b}}=G_{\mathrm{p}}={\rho }_{\mathrm{p}}g{V}_{\mathrm{p}}$$

(1)

where $G_{\mathrm{p}}$ denotes the gravity of the particle (N); $g$ is the gravitational acceleration (m/s²); and $V_{\mathrm{p}}$ represents the volume of the particle (m³). ${\rho }_{\mathrm{p}}$ is the overall density of the particle (kg/m³). For each particle suspended in water, the sand filled in the gap needs to be adjusted so that ${\rho }_{\mathrm{p}}$ is equal to the density of water ${\rho }_{\mathrm{w}}=1\times {10}^3 \mathrm{kg}/{\mathrm{m}}^3$.

Then, the mass of each particle $m_{\mathrm{p}}$ (kg) is given:

$$m_{\mathrm{p}}=\frac{3\pi r_{\mathrm{p}}^3}{4}\times {\rho }_{\mathrm{p}}$$

(2)

where $r_{\mathrm{p}}=0.5d_{\mathrm{p}}$; $r_{\mathrm{p}}$ and $d_{\mathrm{p}}$ indicate the radius (m) and diameter of the particle (m), respectively.

According to the linear water wave theory, the frequency of the water sloshing motion $f_{\mathrm{w}}$ (Hz) can be estimated as [21]:

$$f_{\mathrm{w}}=\frac{1}{2\mathsf{\pi }}\sqrt{\frac{g\mathsf{\pi }}{L}\tanh \left(\frac{h\mathsf{\pi }}{L}\right)}$$

(3)

where $L$ denotes the length of the container in the direction of excitation (m) and $h$ represents the depth of water in the container (m).

2.2. Experimental Setup

The test model for the study of vibration control is a single-story steel frame structure manufactured by Quanser (Toronto, ON, Canada), and the structure installed with SPTLD in the shaking table test is presented in Figure 2a. The primary structure consists of two columns made of 2 mm thick steel plates and a floor slab made of 13 mm thick plexiglass. The dimensions of the steel frame are 32 cm long × 11 cm wide × 50 cm high. The weight of the steel frame is 1.6 kg, the structural lateral stiffness is 500 N/m, and the natural frequency of the structure is 1.7 Hz. In addition, an accelerometer is mounted on the top of the structure and the dynamic response is captured through a vibration signal acquisition and analysis system (SVSA).

To obtain effective control performance using SPTLD, $f_{\mathrm{w}}$ is taken as equal to the natural frequency of the steel frame used in the test. Considering the sizes of the test equipment and the particles, a container measuring 160 mm × 60 mm × 60 mm is defined, of which $L$ = 160 mm and $h$ = 35 mm. Figure 2b presents a picture of the SPTLD utilized in the test, where the particles suspended in the water are made according to the calculated properties and $m_{\mathrm{p}}$ = 0.003054 kg.

The excitation inputs to the steel frame in the test are generated by a Shaker II mini-Shaker manufactured by Quanser (Toronto, ON, Canada). The maximum load capacity of the equipment is 15 kg, which can provide a maximum acceleration of 2.5 g. To conduct the vibration control performance of SPTLD, harmonic excitation, sweep wave, and seismic waves comprising the CPM wave, H-E wave, and SYL wave are adopted in this study. Note that the amplitude of the harmonic excitation is 1.7 cm, and the amplitudes of the other waves are initially scaled to 2.0 cm. The acceleration time histories of the scaled seismic waves are illustrated in Figure 3.

3. Test Results and Discussion

To quantitatively evaluate the vibration attenuation performance of SPTLD, the reduction effects are defined and calculated as follows:

$${\eta }_{\mathrm{p}}=\frac{A_{\mathrm{po}}-A_{\mathrm{p}}}{A_{\mathrm{po}}}\times 100\%$$

(4)

$${\eta }_{\mathrm{r}}=\frac{A_{\mathrm{ro}}-A_{\mathrm{r}}}{A_{\mathrm{ro}}}\times 100\%$$

(5)

where ${\eta }_{\mathrm{p}}$ (%) and ${\eta }_{\mathrm{r}}$ (%) are the peak and root mean square (RMS) reduction ratios of the damper, respectively; $A_{\mathrm{po}}$ and $A_{\mathrm{ro}}$ are the peak and RMS accelerations of the primary structure without control (m/s²); and $A_{\mathrm{p}}$ and $A_{\mathrm{r}}$ are for the structure under the control of the damper, correspondingly (m/s²).

3.1. Control Effectiveness of SPTLD

In the preliminary effectiveness verification of the SPTLD, three particles are filled in the liquid, where the mass ratio of the particles to the liquid ${\mu }_{\mathrm{p}}$ = 2.73%. The attenuation effects on structural acceleration responses under harmonic excitation are evaluated compared with TLD. Note that the TLD maintains the same parameters as SPTLD, except that no particles are added.

Table 1. Structural acceleration response and reduction ratio under harmonic excitation.

Case	Acceleration Response (μg)		Reduction Ratio (%)
	Peak	RMS	ηp	ηr
Uncontrolled	795.3	422.5	/	/
TLD	606.7	306.2	23.7	27.5
SPTLD	583.0	262.4	26.7	37.9

lists the structural acceleration responses and reduction ratios. It is seen that the two dampers both present good reduction effects on the peak and RMS acceleration responses of the primary structure. To further conduct a comparison of the two dampers, the acceleration responses of the primary structure under harmonic excitation using different dampers are illustrated in Figure 4. It can be observed from Figure 4a that the SPTLD performs slightly better than TLD in terms of reducing the peak acceleration of the primary structure, while for the attenuation of RMS acceleration, a more obvious improvement is reflected in SPTLD. Exactly, SPTLD achieves a reduction ratio of 38.8% on RMS acceleration response, while the value for TLD is only 29.0%. Moreover, it can be observed from Figure 4b that SPTLD exhibits a better reduction effect on the vibration dominated by the first structural frequency than TLD, where the first-order vibration of the structure can be reduced by about 64%. This is because the incorporation of particles increases the energy dissipation capacity of the damper through particle–particle and particle–container collisions. The violent motion of particles further drives the water in the lower part of the container to participate in the vibration damping process more effectively. Accompanied by the energy dissipation mechanism of buoyancy and hydraulic resistances, the superimposed damping effects finally lead to an improvement in vibration control using SPTLD compared with TLD.

3.2. Parametric Analysis

3.2.1. Particle Numbers

Figure 5 illustrates the vibration control effects of SPTLD with various particle numbers under harmonic excitation, and the corresponding structural responses are tabulated in

Table 2. Structural response and reduction ratio with various particle numbers under harmonic excitation.

Number of Particles	μp (%)	Acceleration Response without Control (μg)		Acceleration Response under SPTLD (μg)		Reduction Ratio (%)
		Peak	RMS	Peak	RMS	ηp	ηr
1	0.91	795.3	422.5	664.3	351.4	16.5	16.8
2	1.82			586.1	250.7	26.3	40.7
3	2.73			583.0	262.4	26.7	37.9
4	3.64			620.2	264.2	22.0	37.5
5	4.55			646.4	287.4	18.7	32.0
6	5.46			653.8	300.6	17.8	28.9
7	6.37			668.0	321.8	16.0	23.8
8	7.28			678.6	323.1	14.7	23.5

. It can be seen that as the number of particles increases, the damping effect of SPTLD first increases and then shows a trend of a slow decrease. When there is only one particle in the liquid, no collision is available between the particles, and the motion of the particle is greatly limited by hydraulic resistance; hence, the vibration reduction capacity of SPTLD is not exploited. After adding two particles to the liquid, the damping effect on the structural acceleration response has been greatly improved. The damping rates of SPTLD reach the optimum values when 2–3 particles are added, where the peak and RMS control effects can achieve around 26% and 40% under harmonic excitation, respectively. Nevertheless, when the number of particles is too large, the particles in the container tend to be packed together and the collisions of the particles are restricted. Furthermore, the addition of too many particles causes a significant increase in the depth of liquid and produces a large disturbance in the tuning effect of the damper. Consequently, the number of suspended particles should not be too large in practical applications.

3.2.2. Excitations with Various Amplitudes

In this section, the vibration attenuation effects of SPTLD filled with three particles under various excitations are evaluated to investigate the reliability of the control performance, where the displacement amplitude of each input wave varies from 1.0 cm to 3.0 cm. Figure 6 illustrates the acceleration time histories of the primary structure under different excitations with the amplitude of 2.0 cm as an example. From the plot, it can be seen that SPTLD can achieve good vibration suppression effects on structural dynamic responses under various excitations comprising the sweep wave and seismic waves, of which the attenuation effects under H–E and SYL waves reach a higher level. Figure 7 shows the reduction effects of SPTLD under four excitations with various amplitudes, and quantitative results are included in

Table 3. Vibration control effects of SPTLD under various excitations.

	Reduction Ratio (%)	Sweep Wave		CPM Wave		H-E Wave		SYL Wave
Amplitude of the Excitation (cm)		ηp	ηr	ηp	ηr	ηp	ηr	ηp	ηr
1.0		48.4	54.2	42.0	27.2	6.7	43.8	19.1	48.4
1.5		57.7	60.9	34.7	28.8	29.0	61.5	35.3	57.7
2.0		56.5	65.0	41.7	29.9	55.5	73.9	67.4	56.4
2.5		60.2	68.4	51.9	75.9	34.2	66.8	30.8	60.2
3.0		63.5	69.7	39.3	66.5	29.4	61.0	27.7	63.5

. SPLD successfully controlled the structural acceleration responses; ${\eta }_{\mathrm{p}}$ can reach 67.4% (under SYL wave with an amplitude of 2.0 cm) and ${\eta }_{\mathrm{r}}$ can reach 75.9% (under CPM wave an amplitude of 2.5 cm).

For a brief comparison of the proposed SPTLD and the existing enhanced TLDs, some of the existing typical techniques are stated here. Zahrai et al. [10] experimentally investigated a TLD with rotatable baffles on a five-story benchmark building, and the acceleration response under scaled-down seismic waves can be reduced up to 27.24%; Shad et al. [23] found that when a TLD with baffles of an appropriate vertical blockage ratio is subjected to harmonic vibrations, the structural acceleration is decreased by up to 51%. In contrast, it can be concluded that the proposed SPTLD exhibits performance advantages to a certain extent, even though the studied structures and excitations are not exactly the same as the literature above.

In addition, it can be observed from Figure 7 that the damping effects under the sweep wave are relatively more stable. Due to the complexity of the seismic excitation components, the control performance of SPTLD varies more obviously with the excitation amplitude changing. In general, as excitation amplitude increases, the vibration suppression effects on structural peak and RMS acceleration responses both increase and reach the maximum at the excitation amplitude of 2.0 cm–2.5 cm. This is because the increase in excitation amplitude makes the liquid and particles in the container vibrate more intensely, and the energy dissipation mechanism involving collision and friction can be fully exploited. When the excitation magnitude is too large, the capacity of SPTLD is insufficient to absorb more input energy with a limited motion of particles and sloshing of water in the container. However, even though the control effects of SPTLD may be diminished under seismic waves of too-large excitation magnitudes, SPTLD still exhibits effectiveness on structural responses. For example, the peak damping effect of SPTLD is most affected by the SYL wave with increasing amplitude, while a reduction ratio of 27.7% is still achieved under the largest excitation amplitude studied in this paper. In terms of RMS reduction ratios, the values are maintained above 61.0% under the excitation amplitude of 3.0 cm. To conclude, it is demonstrated that by utilizing the nonlinear characteristics of the particle damping, the SPTLD is able to provide reliable control performance under various seismic excitations.

3.2.3. Suspended vs. Floating Particles

In this section, the vibration control performance of SPTLD is compared with that of the floating particle-tuned liquid damper (FPTLD) to investigate the influence of the particle status on the attenuation effects and reliability. In the FPTLD, particles are lightweight with ${\rho }_{\mathrm{p}}<{\rho }_{\mathrm{w}}$ and floating on the surface of the liquid in the container. Maintaining the same ${\mu }_{\mathrm{p}}$ as the SPTLD, the number of floating particles in FPTLD is inevitably increased compared to the number of suspended particles in SPTLD. When there are 2 suspended particles in SPTLD, 12 floating particles are added to the container of FPTLD under the same ${\mu }_{\mathrm{p}}$ = 1.82%. In this case, there are already enough floating particles in the container, and collision space will be greatly limited with additional particles. Therefore, ${\mu }_{\mathrm{p}}$ = 1.82% is determined for conducting the comparative study of SPTLD and FPTLD.

Table 4. Vibration control effects of SPTLD and FPTLD under various excitations.

Wave	Excitation Amplitude (cm)	Damper	Acceleration Response under SPTLD (μg)		Reduction Ratio (%)
			Peak	RMS	ηp	ηr
Harmonic excitation	1.7	FPTLD-12	589.5	266.4	25.9	36.9
		SPTLD-2	586.1	250.7	26.3	40.7
Sweep wave	2.0	FPTLD-12	2058.5	487.6	41.2	49.7
		SPTLD-2	1597.1	349.7	54.3	63.9
CPM wave	2.0	FPTLD-12	2009.3	915.6	43.3	32.5
		SPTLD-2	1941.9	891.3	45.2	34.3
H-E wave	2.0	FPTLD-12	673.1	156.4	47.8	72.1
		SPTLD-2	661.9	137.9	48.7	75.4
SYL wave	2.0	FPTLD-12	591.4	162.8	65.8	59.1
		SPTLD-2	571.2	141.9	66.9	64.3

lists the vibration control effects of SPTLD and FPTLD under various excitations, where SPTLD-2 denotes the SPTLD with 2 particles filled in the container, and FPTLD-12 represents the FPTLD with 12 floating particles. In terms of the SPTLD and FPTLD with the same ${\mu }_{\mathrm{p}}$, slightly better vibration attenuation effects on structural acceleration responses are found in the SPTLD under the harmonic excitation and the selected seismic waves. However, when the structure is excited by the sweep wave, SPTLD performs obvious improvement in the control of both structural peak and RMS acceleration responses. Concretely, ${\eta }_{\mathrm{p}}$ and ${\eta }_r$ are 54.3% and 63.9% under the control of SPTLD-2, while the values for FPTLD-12 are only 41.2% and 49.7%, respectively.

In general, the SPTLD filled with only two suspended particles can achieve better damping effects compared with the FPTLD with 12 floating particles, greatly increasing the availability of SPTLD in practical applications. The reason is that 12 floating particles are packed together on the surface of the liquid, and the collisions between the particles are greatly restricted, which has been demonstrated in Section 3.2.1. Furthermore, the floating particles fail to drive deep water in the container to effectively participate in the sloshing motion; thus, the energy dissipation capacity of FPTLD is reduced. For SPTLD, both shortcomings mentioned above are overcome by integrating suspended particles to the liquid; hence, the SPTLD has more potential in damping performance when facing random external excitation.

4. Conclusions

In this paper, a suspended particle-tuned liquid damper is proposed by integrating suspended particles into the conventional tuned liquid damper. The effectiveness of SPTLD under harmonic excitation is verified compared with the conventional TLD. Based on the experimental results, parametric analysis of the SPTLD is also conducted, including the number of particles, the excitations with various amplitudes, and the use of suspended versus floating particles.

The results show that the SPTLD performs superiorly in reducing structural acceleration responses of the steel frame structure compared with the conventional TLD under harmonic and seismic excitations with good reliability. The peak acceleration response can be reduced by 67.4% and the RMS value can be reduced by 75.9%. In the SPTLD, the particles filled in the container can drive more liquid to effectively participate in the sloshing motion, and the superimposed damping effects involving collisions and the energy dissipation mechanism of buoyancy and hydraulic resistances in the SPTLD lead to an improvement in the vibration control performance. Moreover, the SPTLD filled with only two suspended particles can achieve better damping effects compared with the FPTLD with 12 floating particles, greatly increasing the availability of SPTLD in practice. The reason is that when too many particles fill in the liquid, they will be packed together, and the collisions between the particles will be greatly restricted. Moreover, the floating particles fail to drive deep water in the container to effectively participate in the sloshing motion, and thus the energy dissipation capacity of FPTLD is reduced. Meanwhile, the proposed SPTLD can overcome these shortcomings by utilizing suspended particles, and has more potential in damping performance when facing random external excitation.

SPTLD is expected to reduce the seismic and wind-excited vibrations of structures in practical applications, especially for some slender structures with limited plane space on the top floor, owing to its distributed arrangement of particles in liquid and water-driving ability. The forms of the particles used in this study can further expand the application scenarios of PTLDs. The damping mechanism of the proposed SPTLD should be further investigated theoretically due to the high nonlinearity of the TLDs and the particle dampers, and more structural performance indicators should be examined. Further studies regarding integrating particle damping into conventional TLDs should be focused on the enhancement of dampers with a perturbing effect on deep water and applicability in planar space-constrained cases. Moreover, design guidelines involving multiple design parameters should be investigated to optimize the dampers for practical applications.