Cross-Floor Vibration Wave Propagation in High-Rise Industrial Buildings Under TMD Control

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper investigates the propagation of cross-floor vibration waves in high-rise industrial buildings under tuned mass damper (TMD) control. The study addresses a compelling and practically significant research topic, offering valuable insights for engineering applications in related fields. However, there are several issues to be addressed before possible acceptance. Below are detailed comments and questions:

The literature review is insufficient and should be expanded to include more relevant studies, particularly those focusing on advanced TMD variants (e.g., Tuned Mass Damper Inerter [TMDI], Negative Stiffness TMD [NS-TMD]) to strengthen the research background and contextualize the study within recent advancements.

This study employs TMDs to mitigate equipment-induced vibrations. However, the underlying working mechanism remains unclear. In practice, vibration isolators are more commonly used to prevent equipment-generated vibrations from transmitting to the structure. The latter one is simpler and more straightforward.

The equipment-induced vibration generally involves many varying frequency contents, while TMD is always designed with a fixed natural frequency. Therefore, how to design TMD to satisfy the requirement.

The quality of Fig .21 should be improve with larger fonts.

Author Response

Comment 1:

The literature review is insufficient and should be expanded to include more relevant studies, particularly those focusing on advanced TMD variants (e.g., Tuned Mass Damper Inerter [TMDI], Negative Stiffness TMD [NS-TMD]) to strengthen the research background and contextualize the study within recent advancements.

Response 1:

Thank you for pointing out this important omission.

In the revised manuscript we have substantially enriched the literature review (page 2 Lines 75-85) by adding a concise summary of recent work on advanced TMD variants and incorporating eight new references ([9]–[16], page 27 lines 608-626). The inserted text is reproduced below for your convenience:

“Two cutting‑edge TMD variants have also attracted considerable attention. The Tuned Mass Damper Inerter (TMDI) augments the classical TMD with an inerter element, markedly boosting control efficiency at low mass ratios; recent studies have optimized its parameters and validated its performance in high‑rise buildings and base‑isolated systems [9-11]. Meanwhile, the Negative‑Stiffness TMD (NS‑TMD) and its hybrids (e.g., TNIMD, TMNSDI) introduce equivalent negative stiffness—via buckled struts, magneto‑elastic devices, or frictional sliders—to widen the effective frequency band and suppress peak responses [12-16]. Comparative analysis indicates that TMDIs excel when the mass budget is limited, whereas NS‑TMDs better handle multi‑modal or broadband excitations; both can be seamlessly integrated into the TMD‑WPA framework proposed herein to furnish more adaptable vibration‑control strategies for industrial high‑rise structures.”

We believe this addition now provides a comprehensive and up‑to‑date context for our study. Thank you again for your constructive suggestion.

Comment 2:

This study employs TMDs to mitigate equipment-induced vibrations. However, the underlying working mechanism remains unclear. In practice, vibration isolators are more commonly used to prevent equipment-generated vibrations from transmitting to the structure. The latter one is simpler and more straightforward.

Response 2:

Thank you for this insightful comment.

A new explanatory paragraph has been inserted at page 11 lines 279–282 to clarify the working principle of the TMD. The inserted text is reproduced below for your convenience:

“When the excitation frequency of heavy machinery approaches the natural frequency of the supporting floor or girder, the TMD develops a 180° phase‑lag response, channeling vibratory energy into the mass–spring–damper sub‑system where it is dissipated through viscous damping.”

The decision to employ floor‑level TMDs rather than vibration isolators from several practical considerations: (i) In a workshop where multiple machines operate simultaneously, each unit must be rigidly anchored for positional accuracy, so the crowded layout leaves neither sufficient space nor adequate stiffness contrast for installing thick isolation pads.; (ii) because vibration sources are spread across different  stories, vibration isolators would only limit transmission at each source while still exciting higher structural modes, whereas a few well‑tuned TMDs can effectively suppress the dominant global modes; (iii) TMDs offer post‑installation retunability—through re‑weighting or hydraulic damping adjustment—to accommodate equipment replacement or structural ageing, a flexibility that fixed‑stiffness isolators lack; and (iv) current design guidance (JGJ/T 478‑2020 and ISO 10137:2007) and successful applications endorse TMDs for vibration‑comfort control in large‑span or tall structures.

A new explanatory paragraph has been inserted at page 11 lines 308–317. The inserted text is reproduced below for your convenience:

“Floor‑level TMDs were selected in preference to vibration isolators for four practical reasons: (1) the workshop is densely packed with machines that must be rigidly anchored, leaving no space or stiffness contrast for thick isolation pads; (2) isolators would only block local transmission, whereas a few well‑tuned TMDs can suppress the global modes excited by equipment distributed over several floors; (3) TMD parameters can be retuned after installation (by re‑weighting or hydraulic adjustment) to suit future equipment changes or structural ageing, a flexibility fixed‑stiffness isolators lack; and (4) both national (JGJ/T 478‑2020) and international (ISO 10137:2007) guidelines-supported by successful case studies-recommend TMDs for vibration‑comfort control in large‑span or tall buildings.”

We hope these additions clarify both the working mechanism and the rationale for employing TMDs in our study. Thank you again for your valuable feedback.

Comment 3:

The equipment-induced vibration generally involves many varying frequency contents, while TMD is always designed with a fixed natural frequency. Therefore, how to design TMD to satisfy the requirement.

Response 3:

Thank you for highlighting the apparent mismatch between broadband equipment vibrations and the fixed‑frequency nature of a TMD.

In our case, Section 3.2.3 (page 13 Lines 332–333) explains that the machine load was deliberately idealized as a steady 50 Hz sinusoid. Field power‑spectral‑density measurements (Fig. 18-20) confirm a sharp peak at 50 Hz, so the equipment behaves essentially as a single‑frequency source. This simplification allows us to concentrate on how a clearly identifiable wave propagates through the structure, which is the central objective of this study.

The TMD is therefore treated mainly as a representative disturbance‑mitigation device rather than as the focus of an optimization study. We fully recognize that many real‑world installations face multi‑tone or time‑varying excitations; future work will extend the methodology to adaptive or broadband TMD concepts so that the conclusions remain applicable when the vibration spectrum is less restricted.

Thank you again for your valuable feedback.

Comment 4:

The quality of Fig .21 should be improve with larger fonts.

Response 4:

Thank you for the suggestion.

We have enlarged the font sizes in Fig. 21-and, for consistency, in all other figures as well—to enhance clarity and overall readability. The revised high‑resolution images have been incorporated into the updated manuscript.

Reviewer 2 Report

Comments and Suggestions for Authors

This paper presents an analytical investigation of the cross-floor vibration wave propagation in high-rise industrial buildings under TMD control. The following major revisions should be performed before the paper can be accepted for publication:

1. Tsiavos et al. (2024) presented a low-cost and sustainable timber-based tuned mass damper system that can be installed in existing floor slabs for the response modification of the dynamic behavior of buildings. This publication could be cited in the paper:

         Tsiavos A, Kolyfetis D, Panzarasa G, Burgert I, Stojadinovic B (2023) Shaking table                         investigation of a low-cost and sustainable timber-based energy dissipation system                     with recentering ability. Bulletin of Earthquake Engineering 21, 3949–3968.                                   https://doi.org/10.1007/s10518-022-01464-2.

2. Page 3, Line 96: The authors define a simply supported rectangular plate as the selected model for the simulation of the vibration of the building. Please refer to the limitations arising from the adopted boundary conditions. How realistic are these boundary conditions for a high-rise building? Please discuss.
3. Page 3, Line 99: The authors define in eq. 1 the frequency of a pressure wave. Please mention the uncertainties related to the estimation of the frequency of this excitation. Moreover, such an excitation is typically characterized by a range of frequencies. Please discuss the limitations related to the adoption of the selected mathematical model.
4. Page 9, Line 253: Please present the equation quantifying mathematically the chosen excitation in this case. Please add a figure to illustrate the excitation.
5. Page 10, Line 268: How were the design parameters (mass, stiffness, frequency) of the tuned mass dampers selected in this case? Please discuss if the dampers are optimally tuned to mitigate the effects of the selected excitation for the chosen building.
6. Page 10, Line 280: How was the position of the dampers throughout the floor and the height of the building selected in this case? Please discuss and use an overview and a cross section of the building to illustrate this damper arrangement.
7. Section 3.2.2: What are the results of the model test conditions in terms of natural frequency and damping ratio of the building? Please quantify.
8. Page 12, Line 301: The authors mention that two vibration scenarios were designed in this case, both using a fixed excitation frequency of 50 Hz. What was the excitation type in this case? Please present the corresponding equation and figure illustrating the applied excitation. How was the fixed frequency of 50 Hz during the excitation maintained?
9. Figures 20-25: These figures are not clear. Please explain what each colour represents in these figures. Which colours represent the locked and the unlocked damping configuration? Why do some figures have responses indicated by different colours and some not? Please explain in detail.
10. General comment: It is not clear which of the presented test results correspond to the heel-drop tests (Fig. 13) and the sinusoidal excitation presented in Page 9. Please explain.

Author Response

Comment 1:

Tsiavos et al. (2024) presented a low-cost and sustainable timber-based tuned mass damper system that can be installed in existing floor slabs for the response modification of the dynamic behavior of buildings. This publication could be cited in the paper:

Tsiavos A, Kolyfetis D, Panzarasa G, Burgert I, Stojadinovic B (2023) Shaking table investigation of a low-cost and sustainable timber-based energy dissipation system with recentering ability. Bulletin of Earthquake Engineering 21, 3949–3968.

Response 1:

Thank you for pointing out the need to cite recent literature. In the revised manuscript we have now included the recommended references at page 2 lines 70–73, where they are highlighted for easy review. The inserted text is reproduced below for your convenience:

“Additionally, Tsiavos et al. [8] presented a low-cost and sustainable timber-based tuned mass damper system that can be installed in existing floor slabs for the response modifi-cation of the dynamic behavior of buildings.”

Thank you again for your valuable feedback.

Comment 2:

Page 3, Line 96: The authors define a simply supported rectangular plate as the selected model for the simulation of the vibration of the building. Please refer to the limitations arising from the adopted boundary conditions. How realistic are these boundary conditions for a high-rise building? Please discuss.

Response 2:

Thank you for raising this point.

We have inserted a concise discussion at page 25 lines 564–572 to explain the realism and limitations of the simply supported plate assumption.

In the studied building, each floor slab is framed by deep steel girders that provide vertical support but only moderate rotational restraint; the actual condition therefore lies between “fully clamped” and “simply supported.” Representing the slab as simply supported is conservative, because it slightly under‑predicts natural frequencies and thus yields upper‑bound vibration amplitudes. More importantly, our study focuses on the relative propagation of vibration waves between floors rather than on the exact values of local modal parameters, and those propagation characteristics are governed primarily by the global stiffness and mass distribution, not by small variations in edge fixity.

We have nevertheless noted in the revised manuscript that future work will incorporate semi‑rigid boundary springs to further verify the present conclusions.

Comment 3:

Page 3, Line 99: The authors define in eq. 1 the frequency of a pressure wave. Please mention the uncertainties related to the estimation of the frequency of this excitation. Moreover, such an excitation is typically characterized by a range of frequencies. Please discuss the limitations related to the adoption of the selected mathematical model.

Response 3:

Thank you for highlighting the potential uncertainty associated with the excitation frequency introduced in Eq. (1).

In practice we determined the nominal circular frequency ? from repeated field measurements: the power‑spectral‑density curve exhibits an isolated, sharp peak at 50 Hz, while all harmonics are at least one order of magnitude lower. The equipment therefore behaves, for the purpose of this study, as an almost pure single‑frequency source; variations caused by instrumentation accuracy, temperature drift, or slight load changes are within ±2 %, and this margin is already covered by the damping adopted for the TMD. Because our primary objective is to track how a clearly identifiable wave propagates through the structure, representing the excitation by a single tone is both convenient and conservative. A brief note to this effect has been added immediately after Eq. (1) as follow (page 3 lines 114-116):

“We nevertheless acknowledge that a broadband or time‑varying load would require a spectral input model and, in that case, the analytical framework presented here would need to be extended to account for multiple frequencies.”

Thank you again for your valuable feedback.

Comment 4:

Page 9, Line 253: Please present the equation quantifying mathematically the chosen excitation in this case. Please add a figure to illustrate the excitation.

Response 4:

Thank you for requesting a clearer mathematical description and a visual illustration of the excitation device.

We have addressed this by inserting a new equation and a supporting figure in the revised manuscript (page 10 lines 270-272).

These additions quantify the excitation unambiguously and give the reader an intuitive view of its waveform and frequency content.

Comment 5:

Page 10, Line 268: How were the design parameters (mass, stiffness, frequency) of the tuned mass dampers selected in this case? Please discuss if the dampers are optimally tuned to mitigate the effects of the selected excitation for the chosen building.

Response 5:

Thank you for requesting clarification on the TMD design parameters.

The TMD parameters were derived directly from the heel‑drop test results in Section 3.3.1 (page 15 lines 365-372) which revealed a dominant floor about frequency of 5 Hz; so we set the absorber to 5 Hz. Its mass was fixed at 0.5 % of the participating floor mass (page 11, line 286), and the corresponding spring stiffness followed from k=(2πf)²m.

Our primary objective is to study how vibration waves propagate through the structural system. Because no production equipment had been installed in the workshop at the time of testing, we introduced a self‑owned shaker and deliberately selected a 50 Hz excitation: its higher frequency produces larger, more easily measured responses, which facilitates observation of propagation patterns. TMDs were included because such slabs are commonly fitted with them for comfort control; however, strict “optimal tuning” was not the focus here. Besides natural frequency, damper mass plays a decisive role—larger masses generally provide better attenuation, yet economic constraints precluded fabricating a heavier TMD for this exploratory study. Importantly, the analytical framework and the regression formula developed in the paper do not rely on the absorber being perfectly optimised, so the findings remain applicable.

We nevertheless appreciate the reviewer’s suggestion and will consider testing heavier TMDs in future work to examine propagation under fully optimised conditions.

Comment 6:

Page 10, Line 280: How was the position of the dampers throughout the floor and the height of the building selected in this case? Please discuss and use an overview and a cross section of the building to illustrate this damper arrangement.

Response 6:

Thank you for requesting clarification.

As explained on page 11 lines 291–307, the damper locations were chosen in two steps. First, floor‑level heel‑drop tests identified the beam intersection nodes with the highest modal accelerations; placing each TMD at these nodes maximises energy coupling in plan. Second, a modal participation analysis indicated that floors 12, 10, and 8 exhibit the largest vertical amplitudes, so dampers were installed on those three levels to control the upper, middle, and lower vibration‑sensitive zones simultaneously. An overview of the floor layout and a cross‑section illustrating this three‑tier arrangement are provided in Fig. 15 (page 13) for the reviewer’s reference.

Thank you again for your valuable feedback.

Comment 7:

Section 3.2.2: What are the results of the model test conditions in terms of natural frequency and damping ratio of the building? Please quantify.

Response 7:

Thank you for the query.

The quantified dynamic properties derived from the model tests have been added to Section 3.3.1: the natural‑frequency results appear in Table 1 on page 15, and the associated modal damping ratios are listed in Table 2 on the same page.

Thank you again for your valuable feedback.

Comment 8:

Page 12, Line 301: The authors mention that two vibration scenarios were designed in this case, both using a fixed excitation frequency of 50 Hz. What was the excitation type in this case? Please present the corresponding equation and figure illustrating the applied excitation. How was the fixed frequency of 50 Hz during the excitation maintained?

Response 8:

Thank you for highlighting this point.

The excitation is a mechanical shaker that produces a sinusoidal force, fully described on page 10 lines 267–272.

The fixed 50 Hz frequency was maintained by setting the shaker’s built‑in inverter controller to 50 Hz and continuously verifying the output with a real‑time FFT of the accelerometer signal during each test run, ensuring no drift occurred throughout the experiments.

Thank you again for your valuable feedback.

Comment 9:

Figures 20-25: These figures are not clear. Please explain what each colour represents in these figures. Which colours represent the locked and the unlocked damping configuration? Why do some figures have responses indicated by different colours and some not? Please explain in detail.

Response 9:

Thank you for your valuable remarks on Figures 20–25.

We have addressed the clarity issue by re‑exporting all panels as vector graphics and enlarging fonts throughout.

Regarding colour usage: when a figure presents time‑ or frequency‑history curves from numerous measurement points (sometimes more than ten), a distinct colour is assigned to each point so that overlapping traces remain distinguishable. In contrast, comparison sets containing only a few sensors are plotted in a single colour, which is sufficient for visual separation and avoids unnecessary clutter.

The locked versus unlocked TMD configurations are not indicated by colour but are stated explicitly in each caption and legend (e.g., “TMD‑inactive” and “TMD‑active”).

We hope these revisions make the figures self‑explanatory and easier to interpret.

Comment 10:

General comment: It is not clear which of the presented test results correspond to the heel-drop tests (Fig. 13) and the sinusoidal excitation presented in Page 9. Please explain.

Response 10:

Thank you for pointing out the possible confusion.

The heel‑drop tests shown in Fig. 13 belong to the modal test conditions described in Section 3.2.2; their outcomes are presented in Section 3.3.1 (Modal Test Results).

By contrast, the sinusoidal excitation introduced on page 9 falls under Section 3.1 (Vibration Wave Propagation Test), and its corresponding data and discussion appear in Section 3.3.2 (Vibration Test Results).

We trust this clarifies which results are associated with each test type.