Analytical analysis for optimizing mass ratio of nonlinear tuned mass dampers

To reduce the adverse vibrations of buildings, tuned mass dampers (TMDs), which are the most representative passive control devices, have been widely used and studied in aerospace, machinery, civil engineering, and other fields for many years. Most scholars used to treat the TMD as a linear damper, but they show some nonlinear characteristics owing to the use of limit devices and large displacements. It is necessary to consider the nonlinear coefficient of the TMD when designing its parameters. In this study, the mass ratio of the TMD was optimized by considering the nonlinear coefficient of the TMD. The complex variable average method and multiscale method were used for analysis. A mass ratio interval was found on the “ε-N2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon - N_{2}$$\end{document}” curve in which modulation response can occur, and then an analytical method for obtaining the optimal mass ratio of TMD was derived based on this phenomenon. The numerical results showed that taking the midpoint of this mass ratio interval as the optimal mass ratio can yield a better damping effect and robustness than using the traditional linear design method.


Introduction
Unpredictable natural disasters and human-induced attacks pose a great threat to the safety of buildings; therefore, research on vibration reduction in structures is important in the field of civil engineering. Vibration control is mainly divided into active, semi-active, and passive control. The main difference between them is whether they require an external energy input. Among the control devices, passive control, with the simplest structure and the lowest cost, has been widely studied for many years, and the tuned mass damper (TMD) is the most representative. Its mechanical model can be simplified as an auxiliary mass and a controlled primary structure, which are connected by a linear spring and damper. Energy is transferred between the TMD and primary structure through resonance. TMD originated in 1911, and Frahmn invented a type of undamped TMD called a dynamic vibration absorber (DVA) in the same year. However, the control bandwidth was too narrow owing to the lack of damping, even worse than the uncontrolled structure on the other frequency bands except tuning [1]. Based on the above defects, Den Harton introduced damping into the DVA, which greatly improved the damping effect [2], and the concept of TMD was also established. He optimized the frequency ratio and damping ratio of the TMD and provided their specific optimization formula [3]. Tsai  Den Harton's formula [4]. Setareh et al. obtained the optimal damping of a TMD by using the integrated modal method [5].
However, TMD has certain limitations. For example, the control effect of the TMD depends on whether it is precisely tuned. (Tuning refers to the frequency of the TMD being adjusted to be consistent with the first mode of vibration of the primary structure.) However, the frequency component of the excitation is very complex. The TMD will lose its efficiency or even make the vibration more intense when the system deviates from the 1:1 internal resonance. Therefore, scholars have begun to investigate this limitation of TMD and attempted to develop a more general and efficient TMD. For example, in 1988, Clark first extended a single-degree-of-freedom (DOF) TMD into a multi-degree-of-freedom one and proposed the concept of multiple tuned mass damper (MTMD) [6]. Alamnzan proposed a bidirectional homogeneous TMD (BH-TMD) [7]. Hujiu Zhan et al. used a distributed TMD to reduce the vibrations of a multimode cable-stayed bridge. The configuration optimization and parameter optimization of the TMD were produced by the H2 norm based on modal coordinates and the gradient optimization method based on H2 properties, respectively [8].
Later, some scholars suggested that it would produce nonlinearity owing to the use of a limitation device or large displacement when TMD works [9,10]. This implies that TMD is not really a purely linear system. Li and Cui analyzed the spontaneous nonlinearity in TMD and proposed that if this nonlinearity was ignored, it would cause adverse effects on vibration reduction. They simplified the system to one with a single-degree-of-freedom in which the force of the primary structure acting on the TMD was regarded as an equivalent load, and they optimized the parameters of the TMD designed by the linear method [11]. Li and Zhang used the complex variable average method and multiscale method to derive the optimal frequency formula of the TMD under the premise of considering the nonlinear coefficient of the TMD. The numerical results confirmed the superiority of the above formula [12].
As linear dampers are very sensitive to tuning, in 1952, Roberson pointed out that adding nonlinearity to linear shock absorber can effectively increase the frequency bandwidth of vibration reduction [13]. The most widely used NESs are the particle dampers and nonlinear stiffness dampers. Lu et al. discussed in detail, the development history, classification, structure, principle of particle dampers, and they also provided valuable references for future research [14]. The nonlinear stiffness energy sink is another widely used damper. Gendelman et al. investigated the response regimes of a linear oscillator attached to an NES under external harmonic forcing. They used a phase diagram for global analysis and proposed a onedimensional map that could be regarded as the sufficient and necessary condition for the strongly modulated response (SMR) [15][16][17][18][19]. Zhang et al. derived the relationship between the initial conditions and nonlinear stiffness for conservative systems using the Hamiltonian theory when the energy transfer was complete and extended the results to non-conservative systems to obtain the parametric condition for the optimal target energy transfer (TET) [20][21][22][23][24][25]. Young S.Lee et al. researched nonlinear system identification by HHT, complex variable average method and wavelet analysis method, formulated a reduced-order modeling methodology that can be widely used in the study of nonlinear modal interaction. And they developed a time-domain nonlinear system identification (NSI) technique [26][27][28]. J.Prawin et al. studied nonlinear damage identification; they obtained the slowly varying amplitude and phase by complex averaging method, and presented a nonlinear system identification methodology using empirical slow flow model [29,30]. ZQ Lu et al. explored the connection between the resonance response interaction and bubble-shaped response curve that may appear in the forced response of a nonlinear magnetoelectric coupled system and studied the resonance response interaction to enhance the vibratory energy harvesting bandwidth [31].
The model studied in this paper is a 2-DOF system composed of a linear main structure under harmonic excitation and a TMD mass, which is connected by springs and dampers in parallel. This study focuses on the optimization of TMD's mass ratio. The paper is structured as follows. Section 1 introduces the research background of TMDs and the main content of this paper. Section 2 introduces the mechanical model of the system and its dimensionless form. In Sect. 3, the complex variable average method and multiscale method are used to remove the fast-varying part of the variable, the exact equation is transformed into the complex variable equation, and the slow-varying equation of the system is derived. Section 4 introduces the process and principle of the mass ratio optimization method, and a range of TMD mass ratios in which the modulation response can occur is obtained. In Sect. 5, by taking the mass ratio at the starting point and at the midpoint of the above interval as the mass ratio of TMD, respectively, and by comparing the control effects of these two cases, we concluded that the midpoint of the interval would provide stronger robustness as the optimal mass ratio of TMD. Finally, Sect. 6 summarizes the main content of this paper.

Motion equations of the system
The simplified model of the system considering the nonlinear coefficient of the TMD is shown in Fig. 1. It should be noted that this nonlinear coefficient is not a design parameter, but an inevitable attribute of TMD due to the use of a limitation device or large displacement in the vibration process. In this section, we first establish a mechanical model for the system.
The equations of motion are expressed as follows: To facilitate calculations, Eqs. (1) and (2) can be made dimensionless.
By substituting the dimensionless parameters into Eqs. (1) and (2), the dimensionless forms of the equation of motion are obtained as follows: The absolute displacements x 1 and x 2 of the primary structure and the TMD are transformed into relative displacements u and v, respectively. u Substituting Eq. (6) into Eqs. (4) and (5) allows us to establish the motion equations with relative coordinates.
To smoothly carry out the next steps, the Taylor formula is used to change the equations into a form that only contains one second derivative in each equation. Equations (7) and (8) are regarded as polynomials of e. Expanding Eqs. (7) and (8) near e ¼ 0 yields the following: From Eq. (9), we can rewrite Eqs. (7) and (8) as follows: Thus far, the equations of motion expressed in relative coordinates have been obtained. In later calculations, Eqs. (10) and (11) will be referred to as being the exact equations of the system.

Analytical analysis
In this section, we will use the complex variable average method and the multiscale method to analyze Eqs. (10) and (11) to eliminate the fast-varying and higher-order terms because they have minimal significance in the current study. Only the necessary slowvarying terms that we can deduce directly are retained.
As the amplitude is mainly studied in this paper, using the method of complex variable average can change the real variables to the slowly varying complex variables which can also be regarded as the envelope of the time-history curves of the fast variables. The complex variable average method is a general analytical method which can be easily used to deal with both strong and weak nonlinear systems. Then, the time is divided into two scales by multiscale method. This can help us eliminate the part of the equation that changes on a fast time scale. It can also be understood as eliminating the redundant terms in the later calculation.
Because this study only focuses on the response near 1:1internal resonance, the natural frequency of the primary structure is defined as x 1 ¼ 1, and the tuning parameter is defined as r to facilitate calculations. Then, the excitation frequency can be expressed as follows: The real variables u and v are replaced by the complex variables u 1 and u 2 , respectively, and u 1 and u 2 are expressed as: After substituting Eq. (13) into Eqs. (10) and (11), performing the average process, and omitting higherorder terms, the following equations are obtained: There is still a fast-varying term in Eq. (14), that is, the exponential term e iert . To eliminate it, we can set w 1 and w 2 as follows: The derivation is expressed as follows: D ¼ d dt . Using the multiscale method, we can divide the time variable t into two new independent time variables: The derivatives can now be written as The new time variables can be substituted into Eqs. (16) and (17).
Consider Eqs. (19) and (20) as polynomial equations about e, and then the coefficient of each term can be expressed as follows: To verify that the analytical method selected in this study is sufficiently accurate, Figs. 2, 3, 4, 5, 6, 7 show the time-history figures of the primary structure and TMD, which were obtained by simulating Eqs. (10), (11), (16), (17), (22), and (23) with the parameters a ¼ 0:003; r ¼ 0; F ¼ 0:5; k 1 ¼ 5; k 2 ¼ 0:3; k 21 ¼ 0:0194, and the mass ratios of TMD are e ¼ 0:02; 0:03; 0:04. As shown in the figures, the analytical method used in this study is effective because the curves from the three equations are in agreement. Although there are some minor deviations in the phase between complex variable equations and exact equations when the modulation response occurred, those can be ignored as the focus of this study is amplitude.
Because we only studied the steady-state part of the response and the steady-state response was assumed to be stable on a fast-varying scale, s 0 , the derivative of Eq. (23) can be assumed to be 0. Thus, the slow invariant manifold (SIM) of this system can be obtained as follows: The complex variables in Eq. (24) can be transformed into an exponential form. Among them, N 1 and N 2 represent the slow-varying amplitude of the primary structure and the TMD, respectively, whereas d 1 and d 2 represent the phases of the primary structure and the TMD, respectively.
Let the square of the amplitude be.
By substituting Eqs. (25) and (26) into Eq. (24) and separating the equation into real and imaginary parts, the following equation is obtained: To find the extreme points of the SIM, we derive Z 2 from Eq. (21) and take its derivative as 0.  Two local extreme points are obtained, which can also be called saddle-node bifurcation (SN bifurcation) points or jump points.
In practice, jumping occurs when the points depend on the initial condition. The parameters a¼ 0.003,k 21 ¼ 0:0194; c 1 ¼ 0:1; c 2 ¼ 0:0028; F ¼ 0:5; r ¼ 0 were kept constant, and the mass ratios were taken as e¼ 0.020.030.04, respectively. The SIMs obtained from Eq. (27) are shown in Figs. 8, 9, and 10. By observing and comparing the three SIMs, we can see that there is a jump phenomenon in the weakly nonlinear TMD. However, it is clear that the occurrence of jumping is conditional, and the response mode changes with the change in e. As shown in Figs. 8, 9, 10, the jump phenomenon can only be observed in Fig. 9, and the response mode is the    Figures 1, 2, 3, 4, 5, 6 show the corresponding time-history curves of the three SIMs, from which the response modes can be observed more clearly. This shows that the response mode does not change linearly with an increase in the mass ratio, and the modulation response only occurs in a certain range of mass ratios. To further analyze the change law of the response mode with the mass ratio, slow-varying equations should be deduced. Substituting Eq. (24) into Eq. (22) yields the following: Equation (31) can be rewritten more conveniently as follows: A, B, and G are defined as follows: Writing w 2 in exponential form, substituting it into Eq. (32), and separating the real part from the imaginary part yields the following: C, D, and M are defined as follows: Equations (40) and (41) can be rewritten as follows: When the steady-state response is stable, the following is true: Equation (43) can be rewritten as: The variables P and Q can be defined as follows: Then, the coefficient matrix in Eq. (44) can be written as: The SIMs obtained from Equation (27) The numerical solution from Equations (16) and (17)  The SIMs obtained from Equation (27) The numerical solution from Equations (16) and (17) The solution of Eq. (44) can be written as follows: Substituting Eqs. (48) and (49) into the trigonometric identity, the following frequency response equation can be obtained: To analyze the frequency response of Eq. (50), the tuning parameter e is first approximated as being 0. Then, Eq. (50) can be rewritten as follows: In the later analysis of the slow-varying equations, because the denominator M is equal to the derivative of the SIM (i.e., M¼ oN 1 oN 2 ), it will result in a singularity at the jumping point, causing the system to not be simulated normally. To address this problem, rescaling the time by the term M without affecting the analysis results [13][14][15][16][17] yields the following equations: where 0 is the derivative with respect to the rescaled time. The SIMs obtained from Equation (27) The numerical solution from Equations (16) and (17)  In reference [12], the optimal frequency function of the TMD was successfully derived and applied to the design of the TMD (please refer to [12] for the full derivation). However, it is necessary to adjust the natural frequency of the TMD by adjusting its mass ratio. This section will attempt to analyze the mass ratio of the TMD directly and observe the change in amplitude with variations in the mass ratio of TMD.
First, we use exact Eqs. (10) and (11), complex variable Eqs. (16) and (17) (51) N2 from equation (10) and (11) N2 from equation (16) and (17) N2 from equation (52) and (53) N1 from equation (10) and (11) (51) N2 from equation (10) and (11) N2 from equation (16) and (17) N2 from equation (52) and (53) N1 from equation (10) and (11) Fig. 12 Relationship between mass ratio and amplitudes for the case of a¼ 0.004 As shown in these figures, the ''e À N 2 2 '' curve has a minimum point. The abscissa corresponding to this point is the TMD mass ratio, which minimizes the steady-state amplitude of the primary structure. It seems that this mass ratio corresponding to the minimum point should be regarded as the optimal mass ratio of the TMD. However, it should be noted that Figs. 11,12,13,14 are only for the case of x¼ 1.
In fact, the excitation frequency fluctuates and cannot be constant. Therefore, the analysis in the frequency domain is necessary.
As shown in Figs. 11,12,13,14, the abscissa of the maximum point of N 2 2 and the minimum point of N 1 almost coincide, which can also be understood from the perspective of energy conservation. Therefore, we can transform the goal of finding the minimum point of  (51) N2 from equation (10) and (11) N2 from equation (16) and (17) N2 from equation (52) and (53) N1 from equation (10) and (11) (51) N2 from equation (10) and (11) N2 from equation (16) and (17) N2 from equation (52) and (53) N1 from equation (10) and (11) Fig. 13 Relationship between mass ratio and amplitudes for the case of a¼ 0.005 N 2 1 to finding the maximum point of N 2 2 . We can observe from Figs. 10, 11, 12, 13 that the ''curve'' simulated by the exact, complex variable, and super slow-varying equations basically coincides with the curve drawn by the continuous Eq. (51), except for small convex or concave ranges, and the abscissa of these parts almost coincide. This is evidently not an accidental calculation error. Equation (51) is based on the stable steady-state response, but when the modulation response occurs, the steady-state response is unstable. Thus, the derivative is not equal to 0, and the calculation result of Eq. (51) is no longer accurate at this time. Therefore, it can be speculated that the abscissa regions corresponding to the convex or concave parts are in the TMD mass ratio range where the modulation response can occur. This conjecture will be verified by the numerical method.
Taking the parameters as being k 21 ¼ 0:0194; c 1 ¼ 0:1; c 2 ¼ 0:0028; F ¼ 0:5; r ¼ 0; a ¼ 0:003, the time-history curves of displacement are obtained using Eqs. (10) and (11), complex variable Eqs. (16) and (17), and slow-varying Eqs. (52) and (53), respectively, as shown in Figs. 15, 16, 17. Figure 15 shows that when e ¼ 0:02, the steadystate response of the system is stable, and the timehistory curves obtained from the three equations are in good agreement. As shown in Fig. 7, there is no jump point at this time, so there is no modulation response. When e ¼ 0:03, the steady-state response of the system is a modulation response, which can also be observed from the corresponding SIM (Fig. 8).
Although there are some errors in their convex or concave parts between the three equations, they do not affect the analysis results and, thus, can be ignored. In the concave part, N 2 is only equal to the amplitude at the jump point of the SIM, rather than the real steadystate amplitude of TMD. This is because the modulation response is generated by the continuous jump, but the jump belongs to the instantaneous fast-varying behavior. Hence, the slow-varying equations that ignore the fast-varying variable are no longer accurate when the modulation response occurs. It is also the case that the slow-varying equations are based on the premise that the derivative of N 2 with respect to s 0 is 0. However, N 2 cannot be fixed when the modulation response occurs, which implies that the derivative oN 2 os 0 is no longer equal to 0 at this time. However, the slowvarying equations are still accurate before the jump occurs. Therefore, the SIM made by the slow-varying equations will stay at the jumping point N 21 (or N 22 ) after experiencing instantaneous growth. Figure 10 shows that when the mass ratio is 0.04, there is no jump in the system, the steady-state response from Fig. 17 is stable, and the three curves are in good agreement again. The above analysis confirmed the  conjecture that the steady-state response mode in the inconsistent interval is the modulation response, and the steady-state response is stable in the other coinciding part.
To further verify the above inference, another larger nonlinear coefficient can be taken as 0.005, and let e ¼ 0:025; 0:035; 0:045. The same conclusion can be drawn from Figs. 18, 19, 20. It has been proved that the inconsistent interval of the ''eÀN 2 2 '' curves is the interval where modulation response occurs, which implies that the starting points and ending points of this interval are the starting points and ending points of the jump, respectively. It can be inferred that the values of N 2 at these two points are equal to the   amplitudes at the jump points on the SIMs. Then, we attempt to find the range of e where the modulation response occurs according to the above analysis. Equation (51) can be transformed into a form expressed by e using the following substitutions: Set the parameter Y as follows: By substituting Eqs. (54) and (55) into Eq. (51), it can be rewritten as: The derivative of Eq. (56) with respect to e is as follows (this step is to find the extremum of ''e À N 2 '' curve): The relationship between e and N 2 at the extreme point of the ''e À N 2 2 '' curve can be obtained by simplifying Eq. (57) The value of N 2 at the extreme point can be obtained by substituting Eq. (58) into Eq. (56). Then, the mass ratio of TMD at the extreme point can be obtained by substituting this N 2 into Eq. (58).
By substituting the jump point amplitude from Eq. (27) into Eq. (58), two solutions of e are obtained (because the initial conditions in civil engineering are mostly zero or minimal, the jumping point considered in this study is N 21 ), which are the mass ratios of the starting point and the ending point of the inconsistent interval. Although there are some small errors due to the approximate calculation among the original, complex variable, and slow-varying equations, the inconsistent interval is not completely coincident with each other, but it can be ignored because it does not affect the optimization effect. Table 1 shows the modulation response interval calculated by Eqs. (56)-(58), and Table 2 shows the modulation response interval displayed on the curve when the nonlinear coefficients are between 0.003 and 0.007. Comparing the data in the two tables, it can be observed that although there is a small error, the modulation response interval calculated by Eqs. (56)-(58) is reliable. Now, what needs to be considered is which point on the ''e À N 2 2 '' curve should be regarded as the optimal mass ratio when the nonlinear coefficient is determined. Two schemes are considered: the first scheme takes the starting point of the modulation response, whereas the other one takes the midpoint of the modulation response.
The reason for taking the starting point is that it almost coincides with the minimum of N 1 . However, as shown in Figs. 11, 12, 13, 14, the inconsistent intervals of the exact, complex variable, and slowvarying equations are not completely coincident with each other. In other words, there will be some errors, which lead to the poor robustness of the starting point. When the input condition is slightly changed, the response mode may significantly change. Based on this, considering the midpoint of the inconsistent interval provides higher robustness. On the other hand, the modulation response stimulates the occurrence of target energy transfer, and the TMD can be regarded as a nonlinear energy sink. The energy almost does not return to the primary structure after it is transferred to the TMD because of the instantaneous resonance caused by the nonlinear characteristics of the TMD. This mode can improve the energy-dissipation efficiency.  In summary, the process of calculating the optimal mass ratio of TMD is shown step by step as following. In the next section, the optimization effects of the above two schemes are compared using a numerical method.

Comparison of the control effect
In this section, a numerical method is used to test the control effect of the above two optimal mass ratio schemes. The parameters in Figs. 21, 22, 23 were k 21 ¼ 0:0194; c 1 ¼ 0:1; c 2 ¼ 0:0028; F ¼ 0:5. From these, the frequency response curve of the primary structure when the nonlinear coefficient are a ¼ 0:003; 0:005; 0:007, respectively. The control effect of the TMD when the mass ratio is at the starting point or the midpoint of the inconsistent interval on the ''eÀN 2 2 '' curve can be compared. The linear method used in this study was the fixed-point theory [4] .
From the results in the frequency domain, it can be observed that the amplitudes are smaller when the starting points are x¼ 1. However, when x 6 ¼ 1, it is better to take the midpoint because N 1 is more stable near x¼ 1 when taking the midpoint, and the amplitude fluctuation of the primary is relatively small. This is because the response mode is always modulated around the midpoint and the amplitude is more robust. However, the input condition is substantially complex in practice, and the frequency of excitation cannot always be equal to 1, so the robustness in the frequency domain is crucial. According to the above analysis, the mass ratio at the midpoint of the modulation response interval is regarded as the optimal mass ratio of the TMD.

Conclusion
This paper presents an analysis of a 2-DOF system under harmonic excitation comprising a linear primary structure and a TMD using the complex variable average and multiscale methods. An analytical method for optimizing the mass ratio of the TMD was proposed on the premise of considering the nonlinear characteristics of the TMD. The effectiveness of the method was verified using a numerical method. This study can be divided into the following three stages: 1. The mechanical model of the system was established, and the slow-varying equations of the system were derived, which provided the possibility for the following optimization process. The correctness of the equation was verified by a numerical method. The analytical methods used in this study were proven to be sufficiently accurate. 2. When studying the ''eÀN 2 2 '' curve, it was found that the analytical and numerical solutions were partially inconsistent owing to the occurrence of the modulation response. Based on this, the interval of the TMD mass ratio that can produce the modulation response was deduced, and two schemes for selecting the optimal mass ratio of TMD were preliminarily proposed. 3. The numerical results showed that the above two schemes were much better than the linear method in terms of the control effect. Further, e at the midpoint of the interval at which the modulation response occurred with higher robustness was finally determined as the optimal quality ratio of the TMD through a comparison of the frequency domain.