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Optimal Parameters for Tuned Mass Dampers and Examination of Equal Modal Frequency and Damping Criteria

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Abstract

Purpose

This work developed an idea and systematically explains ‘equal modal frequency and damping’ (EMFD) technique to derive the complete closed form solutions of tuned mass damper (TMD), to reduce dynamic response of structural system.

Method

The optimum parameters for the damped system till date required numerical search technique or shootout technique, which is the trial-error method, of-course add complexity in selecting parameter obtained at optimality conditions. The study of optimum parameter based on EMFD shows the complete closed-form solution, independent to another optimum parameter.

Result

The parameters solely depend on structural damping and mass ratio. The study reveals the multiple optimal solutions. This work investigates the various optimal parameters under harmonic excitation, also compare same with relevant researchers. To confirm its robustness under random excitation various real earthquake time history loads are applied to investigate behavior of different solutions. A demonstration of the shear building for harmonic and random excitation for various solutions provides valuable practical insights for the research endeavors.

Conclusion

Furthermore, the article presents a novel method for finding optimal parameters efficiently without requiring numerical calculations at any stages. The significance of multiple optimal solutions becomes evident when structural systems face random excitation.

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Data availability

The datasets used and analyzed during the current study are available from the corresponding author on reasonable request.

Abbreviations

TMD:

Tune mass damper

EMF:

Equal modal frequency

EMD:

Equal modal damping

EMFD:

Equal modal frequency and damping

FRF:

Frequency response function

SDOF:

Single degree of freedom

MDOF:

Multi degree of freedom

\({\omega }_{0}\) :

Natural frequency of system

\({\omega }_{t}\) :

Natural frequency of TMD

\(k\) :

Stiffness of system

\({k}_{t}\) :

Stiffness of TMD

\(c\) :

Damping of system

\({c}_{t}\) :

Damping of TMD

\(m\) :

Mass of system

\({m}_{t}\) :

Mass of TMD

\(\beta\) :

Structural damping ratio

\(\mu\) :

Mass ratio

\(\xi\) :

Damping ratio of TMD

\(\gamma\) :

Frequency ratio

\(\Omega\) :

Excitation frequency ratio

\(t\) :

Time

\(\omega\) :

Excitation frequency

\({H}_{x}\left(i\Omega \right)\) :

Frequency response function for displacement of system

\({H}_{{x}_{t}}\left(i\Omega \right)\) :

Frequency response function for displacement of TMD

\({H}_{{x}_{d}}\left(i\Omega \right)\) :

Frequency response function for TMD stroke of system

\({\ddot{x}}_{g}\) :

Input ground acceleration

\(x,\dot{x},\mathrm{and}\,\ddot{x}\) :

Displacement, velocity and acceleration of main mass respectively

\({x}_{t},{\dot{x}}_{t},{\text{and }} {\ddot{x}}_{t}\) :

Displacement, velocity and acceleration of TMD mass respectively

\(\overline{{\upxi }_{1}}\), \(\overline{{\upxi }_{2}}\),\(\overline{{\upxi }_{3}},\mathrm{and}\,\overline{{\upxi }_{4}}\) :

Roots of TMD damping

\(\overline{{\upgamma }_{1}}\), \(\overline{{\upgamma }_{2}}\),\(\overline{{\upgamma }_{3}},\overline{\mathrm{and}\,{\upgamma }_{4}}\) :

Roots of tuning frequency damping

\({R}_{1}\), \({R}_{2}\), \({R}_{3}\), and \({R}_{4}\) :

A close form solution pairs {\(\overline{{\xi }_{1}},\overline{{\gamma }_{1}}\)}, {\(\overline{{\xi }_{2}},\overline{{\gamma }_{1}}\)}, {\(\overline{{\xi }_{1}},\overline{{\gamma }_{2}}\)}, and {\(\overline{{\xi }_{1}},\overline{{\gamma }_{2}}\)} of optimal TMD

\(\uplambda\) :

Characteristic value

\(\phi\) :

Modal displacement vector

\({\omega }_{1 }{\text{and}}\, {\omega }_{2}\) :

Modal frequency of Mode 1 and Mode 2

\({\xi }_{1} {\text{and}}\, {\xi }_{2}\) :

Modal damping of Mode 1 and Mode 2

\(\overline{{\gamma }_{1}}\) :

Optimal tunning frequency \({\gamma }^{{\text{opt}}}\)

\(\overline{{\xi }_{1}}\) :

Optimal TMD damping \({\xi }^{{\text{opt}}}\)

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Author notes

  1. Vishalkumar Bhaskarbhai Patel and Radhey Shyam Jangid have contributed equally to this work.

Authors and Affiliations

  1. Department of Civil Engineering, IIT Bombay, Powai, Mumbai, India

    Vishalkumar Bhaskarbhai Patel & Radhey Shyam Jangid

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  1. Vishalkumar Bhaskarbhai Patel
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Correspondence to Vishalkumar Bhaskarbhai Patel.

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Patel, V.B., Jangid, R.S. Optimal Parameters for Tuned Mass Dampers and Examination of Equal Modal Frequency and Damping Criteria. J. Vib. Eng. Technol. (2024). https://doi.org/10.1007/s42417-024-01351-x

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