The provided paper investigates the nonlinear vibration characteristics of wind turbine blades based on virtual mass match via the theoretical analysis and numerical simulation. An equivalent dynamic model of the blade-virtual masses test system is established using the Lagrange method. The nonlinear effects of blade amplitude and installation parameters of virtual masses on the test system are obtained by numerical methods. The nonlinear amplitude-frequency characteristics of the test system are analyzed theoretically based on the nonlinear vibration theory. Two blades over 80m are analyzed under the dynamic simulation environment.

This study is very interesting and deals with the nonlinear vibration characteristics of wind turbine blades based on virtual mass match. However, following issues should be addressed and the decision should be made depending on the author's response.

Comment 1: Grammar errors should be noted (e.g. in abstract Lines 10-11: the nonlinear amplitude-frequency characteristics of the test system IS analyzed...). Please check.

Comment 2: In Lines 43-46, authors state: “Therefore, IWES conducted further research, designed a device to convert virtual masses from translation to rotation……” Where is the corresponding reference of this research? Please check it.

Comment 3: In Lines 46-47, authors state: “the inertia force generated by rotating virtual masses is different from that generated by translational virtual masses.” Please explicitly illustrate the difference between these two setups and explain its effects on inertia force. What is the motivation of studying nonlinear vibration characteristics of wind turbine blades based on Virtual mass match.

Comment 4: In Fig. 1(b), the setup of virtual masses in is different from those reported in previous works (White et al., 2004; Greaves et al., 2012; Snowberg et al., 2014; Hughes et al). It is noted that this setup introduces nonlinear terms to the test system resulting in a more complex scenario. Please explain the mechanism of the device and illustrate advantages of this device comparing with previous setups.

Comment 5: In Lines 75-79, authors state: “the inertial force of the virtual masses also affects the flap-wise direction of the blade……since the frequency of the inertial force is close to the first order modal frequency in edge-wise direction, the perturbation to the flap-wise direction is relatively small……”. Is there any evidence (reference or theoretical analysis) supporting that the perturbation to the flap-wise direction is relatively small?

Comment 6: In section 2.1, the equivalent dynamic model of the blade-virtual masses test system is established with only edge-wise degree of freedom considered. Considering that this kind of device is designed for biaxial fatigue test, why is the flap-wise degree of freedom not included?

Comment 7: The amplitude-frequency curves are incomplete with their peak points missing. From this figure, it can be observed that saddle node bifurcation occurs. Does the existence of saddle node bifurcation have effects on the results of biaxial fatigue test when the dynamic characteristics of such a system differ from those of the linear system?

Comment 8: In Lines 200-202, authors state: “modal analysis is carried out and compared with the transfer-matrix method (TMM) and test data……” But there is no description about transfer-matrix method or the test. Please check. 

In this provided paper, the effect of virtual masses on blade vibration characteristics is studied, and the nonlinear effect of virtual masses on blade is verified by theoretical and simulation methods.

This study is very interesting and can be used to analyze the nonlinear characteristics of blade- virtual masses systems based on practical application scenarios. However, some minor problems need further explanation.

Comment 1: In line 200-202, authors should describe the transfer-matrix method (TMM) to let the reader better understand.

Comment 2: In section 4.3, effects of virtual masses on biaxial test are considered and described in Figure 11. Authors should add a figure to describe the biaxial trajectory of the blade when the virtual masses are translational. The comparison of the two results (translation and rotation of virtual masses) can better illustrate the effect of virtual masses on blade biaxial test.

Comment 3: What effect does the nonlinear effect introduced by virtual masses have on the actual test? Authors need to add further explanations.

Comment 4: In Figure 8(d), there is (a) in this figure. Please check.

Content summary:

The provided manuscript investigates the nonlinear vibration characteristics of rotating virtual mass systems used in wind turbine rotor blade fatigue tests and their influence on the overall test behavior. First the analytical dynamic model of a virtual mass attached to a blade substitute system and its nonlinearity is established. Then the theory of nonlinear amplitude-frequency characteristics for general nonlinear vibration systems is presented. Further, a numerical simulation model including the rotor blade is shown which is used to identify resonance frequencies and loads of the test system for different configurations. Based on results of the simulation the amplitude-frequency, the radius-frequency and the radius-load relations are examined mainly for flap-wise and edge-wise load cases of the rotor blade fatigue test. The novelty of the presented research lies in the numerical investigation of nonlinear properties of virtual mass systems, instead of assuming linear behavior.

General comments:

The results of this work are interesting and can be relevant to the progress of wind turbine rotor blade testing. However, the manuscript requires major review to be reconsidered for publication, as the key method used to derive the results is not sufficiently described and the connections between the different described methods are unclear. Overall, it is not clear which methods and theories are developed by the authors and which are considered from the literature. A clear description of origin and use within the work is mandatory. The connection between the presented analytical model (Chapter 2.1) the nonlinear vibration theory (Chapter 2.2) and the numerical simulation (Chapter 3) is unclear. It seems that the analytical differential equation (line 97 ff) could be transformed into the nonlinear vibration theory (line 140) but there is no description on how they are transformed and if this was done. The connection of the parameters in the nonlinear vibration theory to the investigated virtual mass system are unclear. Therefore, the applicability of the characteristics shown in Fig. 4 is questionable. Also, it is not stated how the simulation is connected to either of the analytical approaches of if any of the analytical results are used for the simulation. As the discussed results seem to be based only on the numerical simulation, the purpose of presenting the analytical methods is unclear.  Additionally, the origin, functionality and modelling methods of the numerical simulation software, were not stated. The mentioned “numerical modal and harmonic analysis” (lines 187-188), which seem to be the key methods in this work, are generally linear simulation methods, and it is not explained how these were adapted to consider the described nonlinear characteristics and how the results were derived. Details on finite modelling of the test system were not given either. Also, it seems that a lot of model simplifications and assumptions were made compared to the behavior of real wind turbine rotor blades, but these assumptions are not clearly stated or explained (for details see comments 3-5 below). Furthermore, the presented results (chapter 4) for amplitude-frequency and radius-frequency are not put into context with realistic testing conditions. All these issues need to be addressed and the manuscript must be corrected accordingly. Further details and issues are described below.

Major comments:

• Comment 1 (lines 1-2): The title of the manuscript “Nonlinear Vibration Characteristics of Wind Turbine Blades Based on Virtual Mass Match” does not represent the presented work. Not the characteristics of the blade, but the characteristics of the virtual mass are examined. Also, the phrase “virtual mass match” is confusing as there is no “matching” presented in this work. Please change accordingly. (Proposed new title: “nonlinear vibration characteristics of virtual mass systems for wind turbine blade fatigue testing”)
• Comment 2 (lines 7-20): The abstract only lists the separate steps of the method and a few results taken out of context. It is not clearly conveying the essence of the paper and it is missing an introduction into the topic and an interpretation of the result in the context of advancing the wind energy. Please correct accordingly.
• Comment 3 (line 72): “The aim of virtual masses is to decouple the test load in the biaxial fatigue test.” But most of the presented investigations are only concerning uniaxial blade testing. This suggests that the biaxial test can be derived by superimposing two uniaxial tests. As this assumption may be reasonable for linear behavior, when investigating nonlinear characteristics this assumption would not be valid. Please clarify the taken assumptions.
• Comment 4 (lines 72-74): “… inertial force that is transmitted to the blades through push rods, thereby adjusting the load distribution in the main vibration direction”. This suggests that the blade displacement is assumed perfectly in line with the push rod. But as the blade is moving in biaxial testing the orientation of the push rod is constantly changing and not in line with either of the blades main directions (flap-wise, edge-wise). Please state and explain the taken assumptions.
• Comment 5 (lines 76-77): “…frequency in edge-wise direction, the perturbation to the flap-wise direction is relatively small.” This assumption is only valid if the edge-wise mode shape would be perfectly in line with the edge-wise blade direction. But typically, the first edge-wise mode shape of rotor blades also has a component in the flap-wise direction. Therefore, the push rod and virtual mass would need to be tilted accordingly to be perfectly in line with the motion, which would increase the effect on the flap-wise direction. Was this behavior considered in the presented work? Even, if the mode-shape is perfectly in line with the blade main direction, this assumption would only be valid for small angles beta. But the results in Chapter 4 suggest larger angles. Please clarify the used assumptions and state the chosen orientation of the push rod.
• Comment 6 (lines 82-92): The formulas are stated without any introduction, explanation or reference of origin. Also, no initial conditions are specified. Please correct accordingly.
• Comment 7 (line 106-108): The authors state “… it is difficult to obtain the corresponding analytical expression. Therefore, the numerical analysis methods are used to solve the equation.” But it is not clearly explained which methods were used and how they were applied to the investigated system. Please clarify.
• Comment 8 (line 112): “To illustrate this, numerical analysis is performed …” which numerical analysis was performed and how? Were the differential equations above used for this simulation?
• Comment 9 (line 115 and Fig. 3): The authors state “the resonance frequency of the test system decreases nonlinearly” How were these resonance frequencies computed?
• Comment 10 (lines 140-141): “𝑓(𝑦) = 1 + 𝜀_1𝑦 + 𝜀_2𝑦^2 + 𝜀_3 𝑦^3 + 𝜀_4𝑦^4”: Please explain how these formulas were derived for the investigated system, what they represent and add reference. What do the individual values for the small parameters represent and how can they be derived? If they cannot be derived, why is this relevant?
• Comment 11 (line 168): “the free vibration frequency of the nonlinear system with respect to the amplitude when there is no external excitation.” How can there be vibration without external excitation, especially outside the natural frequency? Does this behavior apply to virtual mass systems? Please elaborate.
• Comment 12 (Fig. 5): The shown curves suggest that on the left side of the backbone for the same frequency and the same excitation there are three different states of vibration. How can this be possible? Does this behavior apply to the investigated virtual mass systems? If not, why is this relevant?
• Comment 13 (lines 186-188): What is the name and origin of the “blade motion simulation software” and how does it work? How can the “harmonic analysis” consider nonlinear characteristics, as harmonic analyses are generally linear simulations?
• Comment 14 (line 192): The authors state “The equivalent damping ratio of the blade changes during vibration, resulting in a change in the resonance frequency of the test system.” How can the damping change if only the geometry of the system is different, and no damping elements are present? Is this structural damping? Please elaborate on the physics behind this.
• Comment 15 (line 229) “sweep-frequency analysis is performed … to obtain the resonance frequencies…” What was the resolution of the frequency sweep? How was the resonance frequency identified? What are the vibrations amplitudes at different frequencies outside the resonance? Pleas add a plot like Fig. 5 for the behavior of the actual system.
• Comment 16 (lines 262-263): The authors state “… effectively simulating additional masses that are directly fixed onto the blade.” This is only a valid assumption if the displacement is exactly in line with the push rod (see comments 4 and 5). Please check.
• Comment 17 (lines 271-273): the authors state “frequency sweep analysis are used to obtain the frequencies at which specific excitations are applied to the test system … The spatial coupling trajectory of the blade can be obtained…” How exactly were these frequencies and spatial trajectories derived? Was this done separately for flap-wise and edge-wise? If geometric nonlinearities of virtual masses are to be considered, the oscillations in flap-wise and edge-wise direction must not be evaluated separately as they influence each other. (see also comment 4) Please clarify.
• Comment 18 (line 284-301): The conclusion provides an incomplete summary of the results which must be taken into context. Also, the conclusion is missing an interpretation of the results and an evaluation of usability for future wind energy technology. Please review.

Detail comments:

• Comment 19 (line 14): The authors state: “… the resonance frequency of the test system decreases by approximately 2%. …” Please clarify, which change in amplitude does this correspond to and what that means in the context of blade testing?
• Comment 20 (lines 17-18): The authors state: “The decrease of the resonance frequency will also reduce the area of interest for blade load verification, …” How is the area of interest affected by the frequency decrease? Please clarify. The “area of interest” is usually part of the test requirements and cannot change due to test conditions as it defines the area that must be fully tested. Only the area which is actually fully tested can change, so please clarify definition.
• Comment 21 (line 26): “… with over one million damage-equivalent loads cycles…” Even though, usually the cycles do not fall below 1 million it is not a requirement of the industry standards (IEC). Please review wording.
• Comment 22 (line 26): “… are performed at the 1st and 2nd natural frequency of the blade.” It is required to test the blade in the flap-wise and edge-wise direction, which can be done utilizing the corresponding mode shapes. But this is not necessarily corresponding to the 1st and 2nd natural frequency. Please clarify.
• Comment 23 (line 58): The author state “the modal characteristics of the testing system are basically determined, as shown in Fig. 1”, but there is no process of determination shown in Fig. 1. Please clarify.
• Comment 24 (line 61-62): “…the inertia force generated by the virtual mass only acts in the direction of an individual blade mode.” Please clarify which individual blade mode is concerned.
• Comment 25 (line 71): Please explain the “Lagrange method”, how it is applied and state a reference.
• Comment 26 (Fig. 2): The mass of the seesaw beam and the push rod were neglected in the modelling. Was this assumption investigated? Please state the assumption in the manuscript.
• Comment 27 (lines 109-110): “… length of the push rod typically remains unchanged due to space limitations…” How is this “typical”? Is the later assumed length of 4m representative for all flap-wise, edge-wise and biaxial testing?
• Comment 28 (lines 115-126): These results seem as they are not generated from the Lagrange method and would therefore be in the wrong chapter. Please check.
• Comment 29 (line 117): why was the influence of k investigated but not the influence of M?
• Comment 30 (Fig. 3): Using “/” as separator between measure and unit (e.g. “Y / m”) is not advisable as it suggests division. Also, the measures in the legends are missing units (e.g. “m=1000” instead of "m=1000kg").
• Comment 31 (lines 124-126): How where the values for k and M derived and what do they represent?
• Comment 32 (line 135): Please explain what “linear vibration theory” means in the presented context and add references.
• Comment 33 (line 138): “Thus, the weakly nonlinear dynamic equation…” Please clarify what “weakly” mean in this context?
• Comment 34 (line 142): Is the natural frequency 𝜔 _n different from the resonance frequency? Please elaborate.
• Comment 35 (line 157): Please add references for “triangle transform” and “harmonic balance method”
• Comment 36 (line 166): “Similar to forced vibrations in linear systems, nonlinear systems also exhibit similar amplitude-frequency characteristic curves.” What is the source of this information? Please add reference.
• Comment 37 (line 169): “By setting B = 1 and ζ = 0…” What do these measures and the used values represent?
• Comment 38 (lines 193-194): The authors state “In order to accurately assess the influence of virtual mass on the characteristics of the testing system, aerodynamic damping is not considered in the simulation model”. But if the aerodynamic damping has an influence on the characteristics of the testing system it must be considered, as it is part of this system. Has it been confirmed that the aerodynamic damping does not influence the test system? Please elaborate.
• Comment 39 (Figure 6): what are the degrees of freedom of the test system? Can the blade move in any direction? How was the model discretized?
• Comment 40 (line 200): What is the transfer-matrix method (TMM)? What is the relevance of comparing the TMM to the test besides the simulation?
• Comment 41 (line 202): The authors state “… a high level of accuracy, with an error … less than 4%” But in the results a maximum of 2% deviation is visible which seems to be significant as they are repeated in the conclusion. Please comment on this discrepancy and correct accordingly.
• Comment 42 (Table 1): Why was only the 1st modal frequency investigated? How much deviation does the 2nd modal frequency show? How is the deviation of the 102m blade relevant to the work as it is not investigated further? How were the test data derived? Please correct manuscript to answer these questions.
• Comment 43 (line 207, Figure 7): “The values of the additional masses are shown in Table 2 and the Section properties of the blades are shown in Fig. 7” Why and how were the shown positions and magnitudes for the additional masses chosen? Are the flap stiffness, edge stiffness and density the only section properties considered in the simulation? What about torsional stiffness and coupling terms?
• Comment 44 (line 226) “simulation results are fitted using relevant functions to verify the relationship” Which functions were used? How were they fitted? How well do they fit (coefficient of determination)? Was the relationship verified? Please elaborate and correct accordingly.
• Comment 45 (lines 232-235): Which position along the blade correspond to the stated amplitudes (blade tip or position of virtual mass)? Why are amplitudes of 2.6m for flap and 2.2m for edge significant? Are these amplitudes representative for real blade test of this size? Are the Amplitudes close to the geometrical limit where the push rod is parallel to the seesaw (beta - theta = 90°)? Would this be a realistic scenario where 2% frequency drop occurs? Please elaborate more on the interpretation of the results within the context of realistic blade testing.
• Comment 46 (Fig. 8): The data seems to fit poorly to the fitted line. What can be causes for this? Especially in Fig. 8(c) the frequency seems to rise for low amplitudes before it drops? How can this be explained?
• Comment 47 (lines 248-249): Why are radii of 3m for flap and 2m for edge significant? Are these realistic for blade test of this size? Please correct as described in comment 45.
• Comment 48 (Figure 10): How were these load distributions derived? Was constant force or constant displacement used for excitation? At what position along the blade was the excitation placed? What were the excitation frequencies? Please change manuscript to answer these questions.
• Comment 49 (line 263): “As R decreases, the amplitude of blade loads reduces rapidly”. The word “rapidly” seems inappropriate for a load amplitude drop of only 3% with a change in R from infinity to 3m. Please check.
• Comment 50 (lines 264-265): “…resulting in a reduction in the area of interest.” See comment 20.
• Comment 51 (Figure 11): Please show, how would these trajectories change with different values for R?
• Comment 52 (line 288-289): “The square of the resonance frequency is inversely proportional to the polynomial steady-state response of the system.” What does “polynomial steady-state response” mean? Which consequences can be taken from this? Please clarify.
• Comment 53 (line 290): “decreases by approximately 2%” Please state the reference from which it has decreased and the corresponding conditions of R=L=4m.
• Comment 54 (line 294): “in the edge-wise direction, the radius of the seesaw has minimal impact on the resonance frequency” Why is 1.8% (flap-wise) significant enough to be mentioned but 1.0% (edge-wise) considered “minimal impact”? Also see comment 53. Please clarify.
• Comment 55 (lines 296-297): “… descreases by nearly 3% in the flap-wise direction under the given operating conditions” please clarify these operating conditions in terms of excitation force, displacement, and frequency.
• Comment 56 (lines 302-341): The reference are numbered in order of appearance but in the manuscript they are not referenced by number. Please either sort references by author of use numbers in text.
• Comment 57 (lines 305-306): The doi-link of reference [2] Liao et al. is not valid. Please check.
• Comment 58 (lines 325, 327 and 337): There are three different references corresponding to “Melcher et al., 2020”. Please clarify with refences are meant at the respective positions within the manuscript.
• Comment 59 (lines 332-334): The doi-link of reference [14] Zhang et al. is not valid. Please check.