Abstract
Nonlinear energy sink (NES) absorbers excel compared to traditional tuned mass dampers (TMD) due to their superior efficiency over a wider range of frequencies. However, despite the advantages of NES over TMDs, their susceptibility to variations in system energy poses a challenge, particularly in applications where the main structure is subjected to random excitations. A modified NES, referred to as Asymmetric NES, significantly enhances NES robustness, expanding its potential uses. This study investigates the factors contributing to the superior robustness of asymmetric nonlinear energy sink (ANES) in comparison to the performance of traditional cubic NES. Through an analysis utilizing nonlinear normal modes (NNM) and frequency energy plots (FEP), this research shows that the improved effectiveness of asymmetric NESs is due to the intricate interaction between their NNM and the primary structure's kinetic energy FEP, leading to more robust targeted energy transfer (TET). In this work, the NNMs of tuned ANES and NES with similar mass are plotted alongside the kinetic energy FEP of a three-story shear building subjected to various earthquake ground accelerograms. The findings obtained here reveal that the NNM of the ANES absorber intersects more effectively with the system's energy FEP inducing a more effective modal energy redistribution, explaining the superior robustness of the ANES against variations in the input spectrum power.
1 Introduction
Nonlinear Energy Sinks (NESs) are recognized as effective passive vibration absorbers, resonating with any mode of the primary structure, unlike the traditional linear counterparts [1,2]. Despite their advantages over TMDs, NESs face challenges due to their susceptibility to energy fluctuations, notably in structures exposed to random excitations like seismic forces [3–5]. Previous works, as in Refs. [3–6], aim to optimize NES design to achieve a more robust absorber response.
Initially conceived as a lightweight device with smooth cubic nonlinear stiffness and low damping [7], the NES has evolved significantly, with researchers exploring diverse designs featuring distinct geometries, materials, and nonlinear characteristics [8]. While the original NES concept was simple, contemporary NES absorbers are more complex devices demanding advanced analysis techniques for effective design and optimization.
The NES employs an efficient targeted energy transfer (TET) process [9,10] for vibration reduction, involving the irreversible transfer of vibrating energy within the structure's modes [11,12]. Although this process is effective, its performance is sensitive to variations in excitation energy levels, which can make achieving optimal results challenging under varying operating conditions [13]. While one approach for improving NES robustness involves tuning the NES damping through extensive Monte Carlo-based simulations [5,14], other strategies focus on exploring alternative NES designs to further enhance NES robustness. These include incorporating negative linear stiffness [4,15,16], integrating bistable elements [17,18], utilizing magnets [19], exploring vibroimpact nonlinearities [20,21], using track-based nonlinearities [22], and incorporating asymmetric stiffness elements [23–25]. These studies suggest that incorporating diverse types of stiffness elements into the NES enhances TET as they further break the absorber symmetry [26].
In this work, a particular asymmetric nonlinear energy sink (ANES) is proposed. This absorber has shown significant improvements in the robustness of NES response to random excitation [25]. This research compares the performance and resilience of two tuned absorbers—one being a conventional NES, and the other, the proposed ANES. Both absorbers share identical mass and cubic stiffness properties, and their efficacy is evaluated when attached to a three-story structure subjected to seismic ground excitations. The assessment utilizes nonlinear normal modes (NNM) [27,28] and frequency energy plots (FEP) [29]. NNM analyzes nonlinear system response, surpassing linear approximations, while FEP visualizes vibrating energy distribution across frequencies. The combined tools predict effective absorber frequency ranges, identifying resonance conditions where NNM curves overlap the system's energy FEP [30].
This work's primary contribution is explaining why the ANES outperforms the NES. Integrated analyses of NNM and FEP reveal that both absorbers are most effective where their NNM curves intersect with the primary system's FEP plot, indicating resonance. This frequency matching is crucial for robust performance. The ANES's superior frequency matching with the FEPs enhances its alignment with kinetic energy distribution, demonstrating its improved robustness and performance.
The paper is structured as follows. Section 2 outlines the dynamic equations for three systems: a three-story frame without an absorber (No-NES), the same frame with an NES absorber on the top floor (NES), and the frame with an ANES absorber on the top floor (ANES). Section 3 compares the performance of both absorbers using various artificial Eurocode 8 [31] earthquakes, assessing effectiveness at different Arias energy intensities. Sections 4 and 5 analyze, respectively, the NES and ANES absorber's performance through its NNM and FEP plots. Section 6 conducts a similar analysis of data from the El Centro earthquake [32]. Finally, Sec. 7 presents the study's conclusions.
2 Dynamic Models
The natural frequencies in Hz are f1 = 1.42, f2 = 4.57, and f3 = 7.15 and the corresponding damping ratios are ξ1 = 0.007, ξ2 = 0.02, and ξ3 = 0.31.
Simulations of the model confirm that the oscillations of x3 are significantly higher than the oscillations of x2 and x1 which can be attributed to the dominance of the first mode of oscillation in the building's response. The upper floors typically experience larger displacements in this mode than the lower floors. This is because the first mode shape often results in more significant displacement at the top of the structure due to increased flexibility and larger amplitude of movement at higher levels. Consequently, x3, representing the displacement of the upper floor, is greater than x2 and x1 reflecting the structural behavior under the influence of the dominant first mode of vibration.
3 Nonlinear Energy Sink and ANES Performance
Initially, the NES parameters are tuned using an intensive numerical search in matlab, minimizing the index in Eq. (7), with the maximum absorber mass limited to 10% of the floor mass. This process finds the optimal values of , , and that minimize the structure vibrations during a specific seismic event. In this case, a 30-s artificial accelerogram, generated using the method in Ref. [35] to emulate a Eurocode 8, Soil Type E earthquake [31], with an Arias Intensity of IA = 6.5 m/s is used to excite the structures [36].
To explore if the robustness of a previously tuned NES can be enhanced by introducing an asymmetric stiffness with additional damping, the previously obtained NES parameters , , and remain unchanged for the ANES and only the ANES-specific parameters—cL, α, and KL—are adjusted. In this way, the effect of adding this additional asymmetric stiffness can be determined. These obtained parameters are summarized in Table 1.
Optimal parameters for both absorbers
Absorber | Mass (kg) | |||||
---|---|---|---|---|---|---|
NES | 1.324 × 106 | 1018 | — | — | — | |
ANES | 1.324 × 106 | 1018 | 11,800 | 20 | 500 |
Absorber | Mass (kg) | |||||
---|---|---|---|---|---|---|
NES | 1.324 × 106 | 1018 | — | — | — | |
ANES | 1.324 × 106 | 1018 | 11,800 | 20 | 500 |
Figure 4 shows the performance index of the absorbers for a Type E earthquake, scaled for Arias intensities from 3 m/s to 8 m/s. The ANES consistently outperforms the NES, achieving a high-performance index (PANES = 0.94), demonstrating resilience across varying energy levels. The rhombuses highlight a specific operating point (IA = 4.5 m/s). Figure 5 illustrates the corresponding time response, showing the third-floor displacement for no-NES, NES, and ANES systems at IA = 4.5 m/s.

Third-floor displacement of No-NES/NES/ANES systems for Eurocode 8 Type E earthquake with IA = 4.5 m/s
The performance comparison of the tuned absorbers described in Table 1 is extended in Fig. 6 by using four additional numerically generated Eurocode 8 earthquakes (Types A, B, C, and D), each lasting 30 s. Scaled at varying Arias intensities, these earthquakes act as excitations for the examined structures. In every scenario, the ANES consistently outperforms the NES absorber, confirming the ANES system's superior vibration mitigation capabilities.

Performance index of NES and ANES absorbers at different Arias intensities for four 30-s Eurocode 8 earthquake types (A, B, C, and D)
Since the tuned absorbers (NES and ANES) have identical cubic stiffness and mass but differ in overall damping, for the NES and, for the ANES, in Fig. 7 the performance index of an NES with similar ANES damping is considered. In all scenarios, the ANES demonstrates superior performance.
To complement this analysis, Fig. 8 compares the performance of the NES and ANES absorbers (as described in Table 1) under nonparasitic conditions for four 30-s Eurocode 8 types (A, B, C, and D) earthquake, scaled at varying Arias intensities. The results demonstrate that under this condition, the ANES performance is also robust to changes in excitation energy.

Performance comparison for nonparasitic absorbers for four 30-s Eurocode 8 earthquakes (A, B, C, and D). Absorber parameters are defined in Table 1.

Performance comparison for nonparasitic absorbers for four 30-s Eurocode 8 earthquakes (A, B, C, and D). Absorber parameters are defined in Table 1.
4 Nonlinear Energy Sink Robustness Analyzed via Frequency Energy Plots and Nonlinear Normal Modes
Nonlinear normal modes offers valuable insights into the dynamic behavior of nonlinear systems, going beyond linear approximations and providing a deeper understanding of ANES's intricate nonlinear dynamics. Additionally, FEP facilitates the visualization of energy distribution across frequencies, enabling the assessment of how energy is managed throughout ANES's response spectrum. The integrated analyses of NNM and FEP lead to the conclusion that both ANES and the conventional NES absorber are anticipated to function effectively in frequency regions where their NNM curves intersect with the primary system's energy FEP plot. This intersection indicates resonance conditions, highlighting the significance of frequency matching for robust performance.
Figure 9 shows the performance index for the tuned NES and ANES absorbers obtained when a numerically generated 30-s Eurocode 8 earthquake, for soil Type C is applied to the systems. The earthquake is scaled at various Arias intensities and ranges from 1 m/s ≤ IA ≤ 20 m/s. As depicted, the NES system exhibits high sensitivity to energy variations, reaching a maximum performance index peak PNES = 0.7 at IA = 7.7 m/s and progressively decreasing at other values. In contrast, the ANES system demonstrates higher robustness to energy changes, maintaining a consistent performance index of PANES = 0.86 throughout the analyzed range.

NES and ANES performance index in a 30-s Eurocode 8 Type C earthquake at varied Arias intensities (three operational points)
Figure 10 shows the third-floor responses of the systems to a Eurocode 8 Type C Earthquake with Arias intensities of 2 m/s, 7.7 m/s, and 15 m/s. A comparison with Fig. 9 reveals that the NES performs poorly at lower Arias intensities, particularly at IA of 2 m/s where the performance is 0.34. The NES performance increases at higher Arias intensities, peaking at 0.7. Beyond an Arias intensity of 8 m/s, the NES performance gradually declines. In contrast, the ANES maintains constant performance within the considered range of Arias intensities (e.g., 0.86).

Third-floor displacement for No-NES, NES, and ANES systems under a 30-s Eurocode 8 Type C earthquake: three Arias intensity scenarios
Figure 11 illustrates the frequency distribution of kinetic energy in the NO-NES system subjected to a Eurocode 8 Type C earthquake. Utilizing a log–log FEP generated with matlab/simulink, the analysis considers three cases with excitation amplified at Arias intensities of 2 m/s, 7.7 m/s, and 15 m/s. The examination reveals the consistent presence of two distinct high-energy frequency bands across all scenarios, one at a low frequency (e.g., 0.12 Hz) and another at a higher frequency (e.g., 2.36 Hz). The confirmation of these high-energy frequency bands is further exemplified through the spectrograms in Fig. 12.

Kinetic energy frequency energy plot (FEP) for No-NES system under Eurocode Type C earthquake: three distinct Arias intensity scenarios

Kinetic energy spectrogram for No-NES system under Eurocode Type C earthquake: three distinct Arias intensity scenarios
At this stage, the NNM of the previously tuned cubic NES absorber is overlapped onto the FEPs corresponding to the kinetic energy of the No-NES structure. The NNM backbone curve provides a visual representation of the periodic or quasi-periodic motion of the nonlinear absorber when is subjected to various excitation conditions and precisely portrays its equilibrium paths concerning its energy or amplitude. The curve is obtained by analyzing the system's response to varying energy intensities, and it helps identify regions where the NES is expected to be effective in absorbing and dissipating energy.
The NNMs are computed using a matlab code by Peeters et al. [27] employing a mix of shooting and pseudo-arc length continuation methods. Starting from the low-energy Linear Normal Mode, the algorithm predicts NNM motions between different energy levels with a predictor step. The corrector step then refines predictions for the actual solution at a specific energy level. Input parameters specifying system characteristics (linear stiffness, inertia/mass, restoring force coefficients, and non-linearity order) are required. Optionally, known initial conditions like the system's initial displacement at equilibrium (zero velocity) can be provided.
Figure 13 overlaps the NNM curves of the tuned grounded NES onto the kinetic energy FEPs of the No-NES system. This kinetic energy FEP was shown previously in Fig. 11. The analysis considers three Arias intensities (2 m/s, 7.7 m/s, and 15 m/s). The NNM curves capture the nonlinear dynamic behavior of the optimized NES, highlighting its key characteristics. As noted by Haris et al. [30], the NES operates most effectively at frequencies where its NNM curves intersect the system's energy FEPs. Red ellipses highlight these overlapped regions for the three scenarios and their extent varies with the energy levels. In the initial case (i.e., IA = 2 m/s), the NNM backbone only marginally overlaps a small region of the FEP, resulting in a low-performance index (i.e., PNES = 0.34). Conversely, for the second scenario (i.e., IA = 7.7 m/s), the NNM curve encompasses a larger portion of the FEP, intersecting frequencies associated with higher energy levels, leading to an increased performance (i.e., PNES = 0.70). However, in the third case (i.e., IA = 15 m/s), the elevated energy shifts the FEP, reducing the overlapped section and consequently decreasing its performance (i.e., PNES = 0.52). These results align with the observations in Figs. 5 and 10, highlighting the considerable energy sensitivity inherent in the cubic NES performance.

NNM of tuned grounded NES overlapping kinetic energy FEP for the No-NE system under Eurocode Type-C earthquake: three distinct scenarios
Examining spectrograms in Figs. 14 and 12 for each energy-intensity case, a clear trend is revealed: the NES absorber effectively damps the kinetic energy in frequency bands that directly correlate with the specific FEP region intersected by the corresponding NNM curve. This correlation between NNM-FEP intersections and the vibrating energy absorbed highlights the importance of NNM in NES absorber design.

Kinetic energy spectrogram for NES system under Eurocode Type C earthquake: three distinct Arias intensity scenarios
5 ANES Robustness Analyzed via Frequency Energy Plots and Nonlinear Normal Modes
This section extends the NNM-FEP analysis to the asymmetric NES absorber. Figure 15 illustrates the NNM branches of the tuned grounded asymmetric ANES absorber. In Fig. 16, these branches are overlapped with the kinetic energy of the No-NES system when this structure is subject to the Eurocode 8 Type C earthquake. The NNMs are also computed using a matlab code by Peeters et al. [27] but this time using the unilateral nonlinear class object to represent the asymmetric stiffness. Similarly to the analysis conducted for the NES, this investigation encompasses three Arias intensities (2 m/s, 7.7 m/s, and 15 m/s).

NNM ANES overlapping kinetic energy FEP for No-NES system under Eurocode Type C: three distinct scenarios
The nonlinear nature and intricate dynamics of the ANES nonlinear normal modes (shown in Figs. 15 and 16) are characterized by multiple NNM branches, which can be obtained in the continuation software by Peeters and coauthors [27], through the selection of proper initial conditions. The ANES is a two degrees-of-freedom system and grounded ANES exhibits a plethora of NNM branches. As shown in Fig. 16, remarkably, across all three Arias intensities FEPs, these branches consistently interact at different frequency energy ranges. Some branches intersect the lower energy region, while a second set of branches targets the higher energy region. This consistent overlap behavior across various energy levels explains the low-energy content observed in the spectrograms of Fig. 17, highlighting the impressive robustness of the ANES (Fig. 9) and its capacity to effectively resonate and absorb energy under diverse frequency and excitation conditions.

Kinetic energy spectrogram for ANES system under Eurocode Type C earthquake across three Arias intensity scenarios
A final analysis in this section delves into finding numerical evidence of the robust and enhanced TET induced by the ANES. For this purpose, the three structures (NES/ANES/No-NES) are excited using a Eurocode 8 type E accelerogram with an Arias intensity of 7.0 m/s. To check parameter robustness, it is assumed low NES damping (e.g., ). The resultant third-floor displacement of the structures is shown in Fig. 18. The NES performance is low (e.g., PNES = 0.22) while the ANES performance is substantially higher (e.g., PANES = 0.85). The low NES performance is reflected in the scarce vibration attenuation achieved when compared to the No-NES structure.

Third-floor displacement for No-NES, NES, and ANES systems under Eurocode 8 Type E earthquake excitation (IA = 7.0 m/s, )
Figure 19 shows spectrograms that compare the modal energy of the No-NES, NES, and ANES systems. The NES fails to transfer irreversibly the energy from mode 1 to mode 3, leaving a significant amount of energy in mode 1, which explains the high structural vibration obtained. Conversely, the ANES irreversibly transfers energy from mode 1 to mode 3, resulting in lower energy levels in mode 1 and achieving significant vibration reduction. As shown in Fig. 20, the NES performs poorly due to bidirectional energy flow between modes 1 and 3, resulting in alternating oscillations of energy between the two modes.

Modal energy spectrogram for No-NES, NES, and ANES systems under Eurocode 8 Type E Earthquake excitation (IA = 7.0 m/s, )

Normalized modal energy for NES/ANES systems in modes 1 and 3 under Eurocode 8 Type E Earthquake (IA = 7.0 m/s, ) showing alternating bidirectional energy flow between modes
6 Performance Comparison of ANES and Nonlinear Energy Sink for El Centro Earthquake
In this section, the NNM analysis is conducted by exciting both the NES and ANES using a recorded accelerogram from the El Centro earthquake [32]. The absorber parameters for both systems remained consistent with those included previously in Table 1. Figure 21 illustrates the performance of the absorbers as the El Centro earthquake is scaled at various Arias intensities within the range of 1 m/s ≤ IA ≤ 20 m/s. The NES system demonstrated substantial sensitivity to energy variations, peaking at a maximum and subsequently diminishing with an increase in earthquake intensity. In contrast, the ANES system exhibited greater robustness, maintaining a steady performance index of PANES = 0.85 throughout the entire analyzed range of Arias intensity.
The findings are further corroborated in Fig. 22, which illustrates the third-floor time responses of the No-NES, NES, and ANES systems. In this scenario, the structures undergo excitation at three distinct Arias intensities. Notably, the results demonstrate a substantial variation in the performance index for NES. In contrast, the ANES consistently maintains high performance across all three considered cases.

Third-floor displacement for No-NES, NES, and ANES systems under El Centro earthquake: three Arias intensity scenarios
In Fig. 23, the NNM curves of the grounded NES are superimposed onto the kinetic energy FEPs of the NO-NES system under the excitation of the El Centro accelerogram. Similarly, in Fig. 24, the NNM curves of the ANES overlap with the corresponding system's FEPs. Both analyses encompass three Arias intensities (2 m/s, 7.7 m/s, and 15 m/s). The NES nonlinear normal mode branch overlaps occurs within a narrow frequency region around 2; Hz, resulting in not considerable reduction in vibration. In contrast, the NNM of the ANES absorber exhibits multiple branches interacting at various frequencies, inducing a more substantial vibration reduction. As illustrated in Fig. 24, across all three Arias intensities FEPs, these branches consistently engage at different frequency energy ranges. This enduring overlapping behavior across diverse energy levels serves to further underscore the remarkable robustness of the ANES.

NNM of tuned grounded NES overlapping kinetic energy FEP for the No-NES system under El Centro earthquake: three distinct scenarios

NNM of tuned grounded ANES overlapping kinetic energy FEP for the No-NE system under El Centro earthquake: three distinct scenarios
7 Conclusions
This study examined the performance robustness against random excitations of a special class of ANES and compared its performance with traditional NES using NNM and FEP. The proposed ANES absorber builds upon a standard NES design by incorporating additional elements to enhance its performance and robustness, including a rope, two linear springs, and a viscous damper. A key innovation lies in the asymmetric stiffness provided by the linear springs and rope. Unlike a conventional spring, which can exert both tension and compression, a spring connected on one side to a rope can only generate tension. This asymmetry is crucial for the ANES's functionality. To isolate the impact of this added asymmetry, the cubic stiffness, absorber mass, and NES component damping remain unchanged. By keeping these elements constant, the study focuses solely on evaluating the performance improvement brought about by the asymmetric stiffness in the ANES design.
Significant factors contributing to ANES's superior robustness were identified through comprehensive NNM and FEP analyses. Key findings emphasized the critical importance of frequency alignment between NNM and kinetic energy FEP plots. Eurocode 8 earthquake simulations consistently demonstrated ANES's superior performance across a spectrum of excitation intensities. The analysis further illustrated ANES's ability to sustain effective energy absorption and redistribution across diverse energy levels, showcasing its exceptional resilience under varying frequency and energy excitation conditions. Numerical evidence unequivocally supports ANES's capability for robust targeted energy transfer, effectively sending the vibration energy from lower to higher damped modes. These findings underscore the substantial advantages of integrating ANES into structural vibration applications, highlighting its enhanced performance and resilience in mitigating the effects of random excitations.
While the cable-based ANES offers significant advantages in robustness compared to traditional NES, its manufacturing introduces a higher level of complexity. This increased complexity affects both theoretical analysis and practical application. Specifically, the ANES requires meticulous tuning and calibration of its asymmetric stiffness and additional damping component, which adds to the design and implementation extra challenges compared to conventional NES.
Future work will concentrate on developing an experimental rig to validate our numerical results, which is essential for confirming the theoretical advantages of the ANES. This validation will be crucial for demonstrating ANES's robustness compared to traditional NES when subjected to random excitations with varying energy levels and will provide fundamental insights into its real-world performance. In addition, we plan to establish a fabrication methodology that enables the assembly of ANES in practical systems. This includes addressing potential challenges related to achieving the desired asymmetric stiffness and absorber damping, thereby informing future refinements and ensuring effective integration of ANES into practical applications.
Conflict of Interest
There are no conflicts of interest.
Data Availability Statement
The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.