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Optimized parameters of toroidal tuned liquid column dampers for multidirectional pitching vibration mitigation of structures

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Abstract

Toroidal tuned liquid column dampers (TLCDs) are recently designed devices that extend the application of TLCDs to multidirectional vibration control. Toroidal TLCDs are promising in suppressing the horizontal vibration response of structures. This study further explores the potential and optimization scheme of toroidal TLCDs for multidirectional pitching vibration mitigation. Firstly, equations of motion for a toroidal TLCD-structure system in pitching motion are presented. A non-iterative analytical closed-form solution for calculating the dynamic response of the system under harmonic loading is developed. Subsequently, optimized frequency tuning ratio and flow resistance coefficient can be obtained. The optimization results theoretically confirm the direction-independent control performance of toroidal TLCDs. A design example of toroidal TLCDs is presented in detail as a reference. Finally, a pendulum rotational structure for TLCDs testing in pitching motion is constructed. The multidirectionality, effectiveness and robustness of toroidal TLCDs in mitigating pitching vibration are verified by free vibration tests.

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Abbreviations

A H :

Equivalent cross-sectional area of the horizontal section

A V :

Equivalent cross-sectional area of the vertical section

A y :

Amplitude of the liquid displacement in the first equivalent TLCD

B 1 :

Length of the cylinder outer shell along axis

B 2 :

Length of the cylinder inner shell along axis

c f :

Liquid damping

c 𝜃 :

Structural damping

D Y :

Frequency response function of y1

D 𝜃 :

Frequency response function of 𝜃

DMF :

Dynamical magnification factor of the damper-structure system

e :

Distance between the rotational center and the horizontal column

g :

Acceleration of gravity

H i :

Horizontal liquid length of the i th equivalent TLCD

IND :

Performance index

j :

Imaginary unit

J f :

Mass moment of inertia of the liquid

J 𝜃 :

Mass moment of inertia of the structure

k f :

Liquid torsional stiffness

k 𝜃 :

Structural torsional stiffness

L e :

Effective length of the liquid column for TTLCD

M 0 :

Amplitude of the harmonic external moment

n :

Notation for the numeration of equivalent TLCDs

N :

Half the number of baffles in TTLCD

p :

Ratio of the horizontal length to total length of the liquid column for the first equivalent TLCD

q :

Ratio of the distance between the rotational center and horizontal column versus the horizontal column length of the first equivalent TLCD

R 1 :

Radius of the cylinder outer shell

R 2 :

Radius of the cylinder inner shell

V :

Vertical liquid length of the TTLCD

\(y_{i}, \dot {y}_{i}, \ddot {y}_{i}\) :

Displacement, velocity and acceleration of liquid in the i th equivalent TLCD

Y 0 :

Complex amplitude of the liquid displacement

α i :

Ratio of the horizontal section length of the i th equivalent TLCD to that of the first TLCD

α ∗ :

Correction factor related to αi

β :

Frequency ratio of the harmonic loading versus structure

δ :

Equivalent flow resistance coefficient for the TTLCD

ε :

Non-dimensional parameter related to μ

ζ f :

Equivalent damping ratio of the TTLCD under harmonic loading

ζ s e q :

Equivalent damping ratio of the damper-structure system

ζ 𝜃 :

Structural damping ratio

η :

Cross-sectional area ratio of the vertical section to the horizontal section

\(\theta , \dot {\theta }, \ddot {\theta }\) :

Rotational angle, rotational angular velocity, rotational angular acceleration of structure

𝜃 0 :

Complex amplitude of the structural rotation angle

𝜃 s t :

Static displacement value

μ :

Mass ratio of the moving liquid versus structure

μ r :

Mass ratio of the whole liquid versus structure

μ 1 :

Correction factor for the liquid mass

ρ w :

Liquid density

φ :

Frequency tuning ratio of the TTLCD versus structure

ω :

Circular frequency of the harmonic load

ω f :

Natural circular frequency of the liquid in TTLCD

ω 𝜃 :

Natural circular frequency of the structure

m f :

Liquid mass

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Acknowledgements

The authors acknowledge Mr. Jun-Feng Wang and Mr. He-Ping Jiang for their assistance during the experiments.

Funding

This work was supported by the National Natural Science Foundation of China (Nos. 51725901 and 51639006).

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Authors and Affiliations

  1. State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing, China

    Hao Ding, Jin-Ting Wang, Jing-Wen Wang & Ruo-Fan Jia

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  1. Hao Ding
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Correspondence to Jin-Ting Wang.

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Responsible Editor: Makoto Ohsaki

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All data generated or analysed during this study are included in this article and its supplementary information files.

Replication of results

The MATLAB codes for the calculation of the optimized frequency tuning ratio and flow resistance coefficient of the design example are available at the Supplemental Material 6.

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Ding, H., Wang, JT., Wang, JW. et al. Optimized parameters of toroidal tuned liquid column dampers for multidirectional pitching vibration mitigation of structures. Struct Multidisc Optim 64, 3401–3421 (2021). https://doi.org/10.1007/s00158-021-03015-w

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