Analytical Analysis for Parameter Design of Attached Nonlinear Energy Sink

The dynamic responses of a linear primary structure coupled with a nonlinear energy sink (NES) are investigated under harmonic excitation in the 1 : 1 resonance regime. In civil engineering, initial conditions are usually zero or approximately zero. Therefore, in this study, only these conditions are considered. The strongly modulated response (SMR), whose occurrence is conditional, is the precondition for effective target energy transfer (TET) in this system. Therefore, this study aims to determine the parameter range in which the SMR can occur. The platform phenomenon and other related phenomena are observed while analyzing slow-varying equations. An excitation amplitude interval during which the SMR can occur is obtained, and an approximate analytical solution of the optimal nonlinear stiffness is found. The numerical results show that the NES based on the optimal stiffness performs better in terms of control performance.


Introduction
Natural disasters and man-made damages are considered in civil engineering. Hence, structural vibration control has become a research hotspot in recent years. Vibration control can be divided into active control, semiactive control, and passive control. e main differences between the three categories are the external energy requirement and the energy input method. Passive control devices have the simplest structure and lowest cost, as they do not require any external energy. Moreover, they have been widely investigated and applied for several years. e tuned mass damper (TMD) is the most representative type of passive control device that has been installed in several prominent buildings globally, e.g., the John Hancock Tower in Boston, Citicorp Center office building in New York, and Chiba Port Tower in Japan [1]. In China, a pendulum-type TMD coupled with a limiting stopper was installed in the Taipei Tower to control the vibration caused by winds and earthquakes [2].
In 1947, Den Hartog introduced damping in a dynamic vibration absorber and improved the control effect significantly. Den Hartog also provided optimization formulas for calculating the frequency and damping ratio of TMD [3].
Since then, the concept of TMD has been established. Tsai and Lin proposed an optimal parameter design method for TMDs based on Den Hartog's formulas and the fixed point theory [4]. Some scholars have proposed that in practical engineering, a TMD inherently exhibits nonlinear behavior owing to its large displacement or the application of a limiting stopper [5][6][7]. Li and Cui analyzed the nonlinearity in a TMD and proposed that the nonlinear behavior of the TMD should be considered during the design process to improve the control performance. ey simplified the system to a single-degree-of-freedom system for analytical analysis and optimized the traditional design method [8]. Li and Zhang derived an optimal frequency formula for designing TMD and considered the nonlinear coefficient. ey performed a numerical test on the optimized system and verified the superiority of the optimal method [9]. TMDs can function efficiently, depending on the tuning accuracy, and they absorb energy through the resonance of a primary structure. However, in practice, excitation frequencies are complex, and TMDs may lose their efficiency when the system deviates from the mode of resonance.
As linear dampers are very sensitive to tuning, more scholars are focusing on nonlinear dampers, which are also called nonlinear energy sinks (NESs). In 1952, Roberson pointed out that adding nonlinearity to a linear shock absorber can effectively increase the frequency bandwidth of vibration reduction [10]. e most widely used NESs are the particle dampers and nonlinear stiffness dampers. Particle dampers can be divided into four categories: one-unit singleparticle dampers [11], multiunit single particle dampers [12], one-unit multiparticle dampers [13], and multiunit multiparticle dampers [14]. Zheng et al. discussed in detail the development history, classification, structure, principle of particle dampers, and they also provided valuable references for future studies [15]. e 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. ey used a phase diagram for global analysis and proposed a one-dimensional map that could be regarded as the sufficient and necessary condition for the strongly modulated response (SMR) [16][17][18][19]. Liu et al. studied the cubic stiffness nonlinear energy sink system under ground harmonic excitation, and the necessary condition and sufficient condition of strongly modulated response (SMR) are obtained [20]. 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 nonconservative systems to obtain the parametric condition for the optimal target energy transfer (TET) [21][22][23][24][25][26]. Shudeifat et al. proposed a negative stiffness NES that could be determined using a precompressed spring or inverted pendulum [27][28][29]. Manevitch et al. applied the Laplace transform and performed a numerical analysis to obtain an approximate analytical solution for the energy of the NES system [30]. Niu et al. studied the nonlinear dynamics of the simply supported FGM conical panel with different forms of initial imperfections [31]. 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 [32]. Tan et al. studied a system consisting of a ring isolator. e harmonic balance analysis was used to derive the frequency response functions of the transmissibility, and the effects of nonlinear stiffness, damping ratio, and mass were examined [33]. Qiu et al. proposed design criteria for the vibroimpact damper. e method was extended to the case of multiple VI NESs. e principles of additivity and separate activities of VI NESs are analyzed theoretically [34,35].
NESs can resonate with the primary structure, but the mode will be destroyed instantly because nonlinear devices have no fixed frequency. erefore, the energy transferred to NESs cannot return to the primary structure, and such a transfer pattern is considered as the TET. is transfer mode is considered as an advantage of nonlinear dampers. e occurrence of the SMR implies that a TET exists in the system, which guarantees NES to work effectively. However, the occurrence of SMR is conditional. e purpose of this study is to determine the range of NES parameters with which the system can cause an SMR.
Owing to the difficulty of nonlinear system analysis, most researchers have performed numerical analyses. Even when an approximate analytical solution of the optimal nonlinear stiffness can be obtained, the solution is based on the global analysis of the initial conditions, and the initial conditions are need to be known [21,22]. However, the initial conditions are usually zero or approximately zero in civil engineering, and it cannot be measured in advance. erefore, an innovative local optimization method is proposed in this study. e platform phenomenon caused by the modulation response is proposed and analyzed for the first time in this study. Based on this phenomenon, the range of modulation response of the nonlinear system is derived, through which the optimal stiffness of the NES independent of initial conditions can be obtained. e proposed method can avoid the problem that the derivation of the optimal nonlinear stiffness depends on the initial conditions.
A two-degree-of-freedom (2-DOF) system under harmonic excitation composed of a linear primary structure and cubic stiffness NES is investigated in this study. e first section is the introduction of the research work. In the second section, the motion equation and analytical analysis of the system are determined. A platform phenomenon and other related phenomena are found while establishing the variation laws of the amplitude calculated using the slowly varying equation with excitation in the third section. In the fourth section, based on the above phenomenon, the formulas for the excitation range in which SMRs can occur are derived when other parameters are fixed. In the fifth section, an optimal approximate analytical solution of the nonlinear stiffness is obtained, and the optimization effect is verified through numerical analysis. Finally, the last section is the conclusion of this work.

Equations of Motion and Analytical Analysis
e model used in this study is a 2-DOF system under harmonic excitation, which consisted of a primary linear oscillator and cubic stiffness NES, as shown in Figure 1. e subsequent investigations are all based on the initial conditions of approximately zero. e equations of motion are expressed as follows: where x 1 and x 2 , c 1 and c 2 , and k 1 and k 2 are the absolute displacements, damping coefficients, and stiffness coefficients of the linear primary oscillator and NES, respectively. F is the amplitude of excitation. e following dimensionless coefficients are introduced to simplify equations (1) and (2) further.

Shock and Vibration
e above dimensionless parameters are substituted into equations (1) and (2), which transformed the motion equations into dimensionless forms, as shown in the following equations.
For the convenience of calculation, the absolute displacement coordinates of the primary structure and NES can be expressed using u and v, as follows.
e new coordinates are substituted into equations (4) and (5) to establish the motion equations.
For the convenience of calculation, the ω 1 in the above formula is set as 1, and the 0 (ε 2 ) term in the system is eliminated. en the equations (8) and (9) become e differential equations (9) and (10) are simplified using Taylor's formula, such that each equation contains only one second-derivative term. By some algebraic derivations, equations (9) and (10) become omitting the higher-order terms 0(ε 2 ) in equation (12) gives e next step is a coordinate transformation procedure based on the complex variable average method (CX-A) to eliminate the fast variables because they have a minimal significance in our study. erefore, the coordinates are converted into complex coordinates.
e complex variables are substituted into equations (11) and (13). e original complex variable equations are as follows: Figure 1: Simplified model of the 2-DOF system.

Shock and Vibration 3
Considering the tuning parameter between the primary structure and the excitation as σ, the frequency ω can be written as After performing the average process and omitting higher-order terms, the following equations are obtained.
e time-history curves are obtained using exact equations (7) and (8) and complex variable equations (18) and (19) to verify the accuracy of the complex variable motions.
It can be observed from Figures 2 and 3 that the amplitudes calculated using the original equations and the complex variable equations are almost consistent, although the phase deviates slightly. However, this study focuses mainly on the amplitude. erefore, the complex variable equations (18) and (19) are sufficiently precise to be investigated further. e response form shown in Figures 2 and 3 is called SMR, which indicates that the steady-state response is unstable, the amplitude of the steady-state response is not constant, and the modulation intensity is deep (it is called weakly modulated resonant (WMR) when the modulation intensity is weak). e appearance of the SMR indicates that effective TET occurs between the primary structure and NES. erefore, the SMR is the response mode that should be achieved in this study. Subsequent investigations are aimed at adjusting the parameters to obtain an SMR.
Based on a multiscale method, the equation can be divided into two dimensions as fast and slow. e fast and slow dimensions are relative concepts that can be subdivided infinitely in theory. e time variable is divided into two scales according to the accuracy requirements of this study.
us, the derivative can be written as follows.
Equations (22) and (23) are considered as polynomials of ε, and the coefficients of the first and second powers of ε are deduced as follows.
As the slow variables of the steady-state response are analyzed mainly, the derivatives of the large time scale τ 0 are ignored.
e complex variables are written in the form of polar coordinates, where the slowly varying amplitude and phase of NES are represented, as follows.
By substituting from equations (28)-(31) into (27) and separating the real and imaginary parts, the following equation can be obtained.
erefore, the slow invariant manifold (SIM) can be obtained by (32) as the thick line in Figure 4, and the thin line is the actual response of complex variable equations (18) and (19). It can be observed that the steady-state part of the system always operated according to the SIM. However, when the response reaches the jump point, the system jumps at the jump points, and the steady-state response is no longer stable, jumping between the two steady-state branches is inconsistent with the SIM. It can be inferred that when the fast-scale derivative is set to zero, and only the results of the stable steady-state response is accurate. However, jumping is a rapidly varying behavior whose response path may not be adequately simulated. is deduction, which is also the theoretical basis of the following equation derivation, is confirmed later while deriving the slow-varying equations. Considering the derivative of the above equation with respect to Z 2 to determine the jump points on the SIM and set this derivative as zero to obtain the following expression.
Hence, the two local extremum points (jump points) can be solved as follows:

Shock and Vibration
It can be observed from equations (34) and (35) that the two extreme points can appear if only the damping condition satisfies λ 2 ≤ � 3 √ /3. If λ 2 does not satisfy this inequality, then the jump phenomena will not occur in the system regardless of how the other parameters change. On the basis that damping satisfies the above conditions, the SMR occurrence depends on the value of nonlinear stiffness kn. However, critical stiffness is related to excitation amplitude.
If the jump occurs in the system, N 21 , N 22 are the two jumping points, and they are known as the saddle junction bifurcation (SN bifurcation). In the actual response, where the jump occurs depends on the initial conditions. e initial conditions are usually approximately zero in civil engineering, and it can be inferred from the SIM that the jumping point should be N 21 in this case. erefore, N 21 was directly considered as the SN bifurcation point later.
Substituting equation (27) into equation (25), a slowly varying equation for ψ 2 is obtained as follows: By solving the above equations, the following equation is obtained: Among them, e variable ψ 2 was written as an exponential form N 2 e iδ 2 and substituted into the above equations. By separating the real and imaginary parts, the following equations can be obtained.
Among them,  Shock and Vibration Equations (39) and (40) can be expressed as follows: Among them, e denominator M will result in a singularity at the jumping points; hence, the system cannot be simulated normally. In order to overcome this problem, rescaling the time by the term M [9] yields the following equations.
where ′ is the derivative with respect to the rescaled time.

Platform Phenomenon and Effective Range of Excitation Amplitude
e accuracy of equations (43) and (44) is verified by comparing their solutions with the result of the original equation. Consider the SIMs, time-history curves, and phase diagrams of the system. In Figures 5 and 6, the solid bold lines and dashed bold lines are obtained using complex variable equations (18) and (19) and slow-varying equations (43) and (44), respectively. Since it is not easy to adjust the time scale for the slow-varying equations, only the steadystate response of the slow-varying equations is plotted for comparison purpose. e solid thin lines in Figures 5 and 6 are obtained using the slow invariant manifold equation (32). e excitation amplitude of Figures 5-7 is 0.25; at this moment, the system amplitude do not reach the jump point yet; hence, no jump occurred. e steady-state response of the system remained stable in the region of the low amplitude, as shown in Figure 5. e excitation amplitude in Figures 8-10 is f � 1. In Figures 8 and 9, the solid bold lines and dashed bold lines are obtained using complex variable equations (18) and (19) and slow-varying equations (43) and (44), respectively. e solid thin line shown in Figures 8 and 9 is obtained using the slow invariant manifold equation (32). Here, the system amplitude has experienced the jump point. Only one jump appears, and then, the steady-state response of the system, which always stays within the region of the high amplitude (Figure 8), becomes stable again.
As the jump corresponded to the fast behavior, equations (43) and (44), in which rapid-varying variables are ignored, cannot describe the jump accurately. As shown in Figures 11  and 12, the solid bold lines and dashed bold lines are obtained using complex variable equations (18) and (19) and slow-varying equations (43) and (44), respectively. e solid thin lines in Figures 11 and 12 are obtained using the slow invariant manifold equation (32). e amplitude N 2 obtains using equations (43) and (44), experiences a rapid increase, and finally stays at point N 21 (the jump point). e excitation amplitude of Figures 11-13 is 0.5.
In this case, it appears impossible to use the slow-varying equations (43) and (44) to analyze the SMR. However, the parameter region of SMR can be solved by using the limitation of slowly varying equations. In the three sets of figures, the stiffness kn is constant, but the excitation amplitude f is different.
e steady-state response is stable when f � 0.25; f � 0.8, and the steady-state response is the SMR when f � 0.5. e steady-state response is stable again when f � 1. erefore, it is speculated that when the nonlinear stiffness kn is fixed, an interval of excitation amplitude exists, and the SMR occurs only in this interval. e upper and lower limits of the interval could be related to kn. e nonlinear stiffness is considered as kn � 3, 5, 8, 11 to verify this inference. e steady-state amplitude N 2 is calculated using equations (43) and (44), as shown in Figure 14. Figure 14 shows the above conjecture regarding a particular interval in which the SMR can occur. Moreover, in Figure 14, a unique platform region can be observed, and the value of the vertical axis corresponding to the platform is precisely the SN bifurcation point N 21 . e SMR occurred when f is in this platform. Furthermore, when the platform scope is exceeded, the steady-state response of the system is stable again. e jump behavior will not occur in the monotonically increasing region in front of the platform, and the amplitude of the steady-state response remains in the low-amplitude region. e monotonically increasing region behind the platform is attracted by a stable attractor in the high-amplitude region after a few jumps, and the steady-state response is also stable. However, neither of these cases has sustained SMRs. erefore, they are regarded as invalid.
Consider parameters λ 1 � 0.8, λ 2 � 0.3, kn � 5, along with f in front of the platform and behind the platform, i.e., f � 0.2, f � 0.4, f � 0.6, f � 0.8, and verify the above phenomenon in Figures 15-18 . In Figures 15-18, the solid thin lines are obtained using equations (7) and (8), solid bold lines using complex variable equations (18) and (19), and dashed bold lines using slow-varying equations (43) and (44). e cause of the platform is similar to that mentioned above. When the excitation amplitude is small, the steadystate amplitude do not reach the jump point. erefore, the system presents a stable steady-state response in the region of the low amplitude. When the SMR occurs, equations (43) and (44), in which the fast-varying parts are ignored, cannot be used to simulate the jumping process accurately. After experiencing a rapid increase within the first few seconds, the response finally stays at the jump point. e value of the jump point determined using equation (34) is related to kn and λ 2 and independent of f. erefore, in the SMR process, the rate of change of the amplitude N 2 with the excitation amplitude is zero. When the amplitude of excitation continues to increase, the system will jump to the Shock and Vibration 7 high-amplitude branch on the SIM. However, no SMR occurs. erefore, the stable steady-state response will appear, and the slowly varying equation is able to simulate the amplitude accurately again. Moreover, when the other parameters are fixed, the x-coordinates of the starting and ending points of the platform corresponded to the upper and lower limits of the SMR excitation amplitude, respectively. is interval is set as [f 1 , f 2 ], within which an effective TET occurs. It can be observed from Figure 14 that the value of [f 1 , f 2 ] is related to kn. e change in the phase δ 2 with f is shown in Figure 19. It is found that δ 2 decreases monotonically only in the platform regions. Another phenomenon that can be observed is that the ordinates corresponding to the monotonically decreasing interval are independent of kn, and the phases at the starting and ending points of the  Shock and Vibration monotonically decreasing region can be adopted as negative. By analyzing the time-history curves obtained using the slow-varying equations (43) and (44), it can be observed that N 2 ′ � δ 2 ′ � 0 is always precise during the steady-state response. erefore, it can be inferred that the numerators of h 1 and h 2 are always zero in this case.
For convenience, equations (39) and (40) can be rewritten as follows.
As N 2 ′ � δ 2 ′ � 0, it can be deduced that C � D � 0. is assumption is realistic when the steady-state response is stable. However, when the parameters are in the platform region, as shown in Figure 14, it becomes impossible to obtain C � D � 0. N 2 calculated from equations (43) and (44) is fixed, and the value of cos δ 2 increases monotonically with the change in f. erefore, it is impossible that C and D are unchanged when f increased. e time-history curves in  e above expressions and conditions could be used to obtain an approximate analytical solution for the range of excitation amplitude for the SMR. e following expressions are assumed: At the starting point of the platform, C, D is still equal to zero. erefore, equation (50) can be derived from equations (46) and (47).
From equation (50), the lower limit f 1 of the platform is obtained.
e upper limit f 2 is calculated by setting the values of C and D at the starting and ending points as C 1 , C 2 , D 1 , D 2 , respectively.
e starting point phase can be obtained from equation (51).
From equations (51)-(54), the following expressions are derived.   10 Shock and Vibration By using equations (42) and (43) at the starting and ending points of the platform, respectively, equation (58) can be obtained.
Based on trigonometric identities, the following expression can be obtained.
By substituting equations (56) and (58) into equation (59), the abscissa of the ending point f 2 can be found.
erefore, the excitation amplitude region of the SMR is given as follows.
Substituting f 1 and f 2 into the above inequality yields the following equation.
Equation (63) is the equation for the excitation amplitude region of the SMR derived using the platform phenomenon. e accuracy of equation (63) is verified.  rough calculations, the average errors of f 1 and f 2 are 0.0032 and 0.0322, respectively. It can be noticed that the error of f 1 is significantly smaller than that of f 2 . However, the error is within the acceptable range and have a minimal effect on the results of the subsequent optimizing stiffness kn b . Furthermore, the larger the kn value, the smaller the error. e values of the nonlinear stiffness kn listed in Tables 1 and 2 and the kn values shown in Figure 21 are the same.
rough the analysis in this section, the following conclusions can be drawn for a two-degree-freedom system which is composed of a linear primary structure and a cubic stiffness NES under harmonic excitation.

Optimal Stiffness of NES
As shown in Figures 14 and 21, the values of f 1 , f 2 are related to kn. e smaller the kn value, the broader the platform. It is a good phenomenon if the platform becomes wider. However, the steady-state amplitude of the primary structure increases simultaneously. A trade-off relationship exists between the platform width and steady-state amplitude. us, it can be inferred that the nonlinear stiffness is not better when it becomes larger or smaller. A range of kn must exist within which the valid TET will occur.
Because f 1 and f 2 are analytical expressions of λ 1 , λ 2 , and kn, a range of effective stiffness can be obtained using equation (63) when λ1 and λ 2 are determined, and the effective stiffness range of the response can be set as [kn 1 , kn 2 ]. As can be observed from Figure 14, when the excitation amplitude f is determined, finding the platform with f as the abscissa of the endpoint and the corresponding stiffness of this platform is regarded as the optimal stiffness appears to be reasonable. However, as the platform range obtained using equation (63) is only an approximate solution, it can be observed from Tables 1 and 2 that some inescapable errors occur, especially for f 2 . us, a significant error must occur to consider kn 2 as the optimal stiffness. erefore, in this study, the average value of the sum of kn 1 and kn 2 is considered as the optimization stiffness kn b , which not only ensures that the steady-state response of the system is SMR but also ensures that the system can cope with the fluctuation of the excitation amplitude within a small range and improve robustness. Several amplitudes of f are considered, and equation (63) is used to obtain kn 1 , kn 2 , and kn b under these working conditions. e results are listed in Table 3.

Numerical Analysis of Stiffness
Optimization Results e optimization effect of kn b was verified using the numerical method by selecting the excitation amplitude as f � 0.5 and considering four values of kn b , i.e., kn � 1 < kn 1 , kn 1 < kn b � 6.1691 < kn 2 , kn � kn 2 � 10.2650, and kn � 14, 18 > kn 2 . As no fixed amplitude response of the SMR occurs, the average value of the total energy is used as a measure of the vibration reduction effect. e transient total energy of the system can be expressed as (64) Figure 22 shows the result obtained by changing the tuning parameters σ and testing the average total energy near the 1 : 1 internal resonance.
It can be observed from Figure 19 that the total energy is significantly lower when the optimal stiffness value is considered. e optimized frequency-energy curve is plotted separately in Figure 23. e smooth areas on both sides are the stable steady-state responses, and the location near the    middle represents the SMR; this response mode is beneficial for reducing vibration. In the case of ω � 1, the total energy of kn b is relatively small, but it is not the minimum value. Under actual conditions, the excitation frequency is probably not exactly equal to 1.
us, the robustness in the frequency domain is critical. Selecting kn � kn b achieves a balance between the effects of the vibration reduction and robustness on frequency.
By considering an excitation amplitude of f � 0.8 and selecting the corresponding five values of kn to verify the vibration reduction effect of kn b , the conclusion inferred from Figures 24 and 25

Conclusion
is study presents an analysis of a 2-DOF system under harmonic excitation composed of a linear primary structure and a NES using the CX-A and multiscale method. e initial conditions of this system are always approximately zero, and the fast variables that are not useful for analytical analysis are eliminated. While studying the variation trends of the NES amplitude N 2 calculated using the slowly varying equation of the system with the excitation amplitude f, a platform phenomenon and other related phenomena are determined. In the platform, the steady-state amplitude N 2 obtained using the slow-varying equations is inconsistent with the actual response and do not change with f. Moreover, it is always fixed at the SN bifurcation point, and the phase δ 2 at the starting and ending points of the platform can be regarded as negative. Based on the platform phenomenon, the region [f 1 , f 2 ] of the excitation amplitude in which the SMR can occur is obtained, and the accuracy of this range is verified numerically.

18
Shock and Vibration About the platform phenomenon, the following conclusions can be drawn: (1) When the system parameters are determined, there exists an interval [f 1 , f 2 ] of excitation amplitude f, and the modulation response can occur in the steady-state phase if and only if f is in this interval. e range of the region is related to the nonlinear stiffness kn of the NES. After the excitation range of SMR is obtained, the approximate analytical solution of the optimal nonlinear stiffness kn b can be determined when other parameters are fixed. Finally, the average value of the total energy is adopted as the measurement, and the superior vibration reduction effect of the optimized stiffness kn b is verified using the numerical method.
It is the first time that the platform phenomenon is put forward. Some laws related to this phenomenon have great research and application value and play a crucial role in the parameter design of the NES. More laws about this phenomenon and the deeper physical meaning leading to this phenomenon still need to be explored.

Data Availability
e data, models, or codes that support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.