Experimental investigation of vibration modulation induced by spatially distributed internal nonlinearities in a parallel thin-beam system.

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Learn more: PMC Disclaimer | PMC Copyright Notice iScience . 2026 May 28;29(6):116103. doi: 10.1016/j.isci.2026.116103 Experimental investigation of vibration modulation induced by spatially distributed internal nonlinearities in a parallel thin-beam system Tao Cao 1 , Haijian Cui 2, 3, ∗ , Qichen Wang 1 , Yuhao Zhao 1 Author information Article notes Copyright and License information 1 Key Laboratory of Advanced Manufacturing Technology of the Ministry of Education, Guizhou University, Guiyang 550025, P.R. China 2 Wuhan Second Ship Design and Research Institute, Wuhan 430064, P.R. China ∗ Corresponding author chj841524537@163.com 3 Lead contact Received 2026 Mar 9; Revised 2026 Mar 31; Accepted 2026 May 8; Collection date 2026 Jun 19. © 2026 The Author(s) This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/). PMC Copyright notice PMCID: PMC13233760  PMID: 42256289 Summary Broadband vibration regulation is important for lightweight beam-like structures and vibration-sensitive detection systems under complex excitations. However, the mechanisms of spatially distributed internal nonlinear interactions in parallel thin-beam systems (PBSs) remain unclear, particularly from experimental evidence. Here, we introduce controllable spatially distributed internal nonlinearities (SDINs) into a PBS and establish a corresponding Lagrange-based model. Combined theoretical and experimental studies are performed to examine the effects of nonlinear parameters, installation position, and distribution number on vibration modulation. The results identify two distinct SDIN vibration states and show that effective nonlinear activation enables broadband vibration suppression. Spatial distribution further produces synergistic nonlinear interactions, allowing attenuation across multiple resonance regions. These findings verify the feasibility of spatially distributed internal nonlinear modulation and provide guidance for designing vibration-regulated thin-beam structures for beam-like supporting systems and vibration-sensitive detection equipment. Subject areas: Applied sciences, Engineering Graphical abstract Open in a new tab Highlights • Spatially distributed nonlinearities are introduced into a parallel thin-beam system • A Lagrange-based model is developed for SDIN-induced vibration modulation • Two SDIN vibration states correspond to distinct nonlinear coupling mechanisms • Spatial distribution enables broadband suppression across multiple resonance regions Applied sciences; Engineering Introduction In many engineering systems, lightweight and high-slenderness structural components are widely used in beam-like supporting structures, plate-beam assemblies, and instrument-carrying frames. These structures are commonly composed of parallel thin-beam assemblies interconnected through mechanical joints or elastic connectors and often serve as load-bearing or vibration-sensitive supporting frameworks in engineering devices. Under complex excitations such as environmental disturbance, machinery operation, and impact loading, vibration energy can be transmitted and redistributed among coupled beams through internal connections, leading to complex dynamic interactions. Such vibration not only affects the service performance and structural stability of beam-like systems, but may also deteriorate the working accuracy of high-end detection instruments mounted on these structures. For example, when a structured-light scanner operates in a vibrating environment, even slight structural vibration may lead to degraded data quality, thereby reducing measurement accuracy and reliability. 1 , 2 , 3 Therefore, suppressing such internally coupled vibration in PBSs remains an important challenge in both structural vibration control and vibration-sensitive instrument applications. For analytical purposes, such frameworks are commonly simplified as coupled parallel elastic beams. 4 , 5 Consequently, suppressing interaction effects in parallel beam frameworks during vibration remains an ongoing research priority. Vibration control in dynamic systems fundamentally aims to dissipate or redirect undesirable vibrational energy. 6 Extensive studies on elastic beams 7 , 8 , 9 and beam systems 10 , 11 , 12 have provided the foundation for vibration suppression research. Building on this foundation, scholars have incorporated linear vibration absorbers into systems to dissipate vibrational energy. However, the effectiveness of linear absorbers strongly depends on precise frequency tuning, and their performance deteriorates significantly once the excitation deviates from the designed target frequency. 13 , 14 Consequently, linear absorbers are only effective within narrow operational bands. Their excessive sensitivity to frequency variations makes them unsuitable for broadband vibration suppression in complex engineering environments. To overcome these shortcomings, researchers introduced nonlinear elements into linear systems. 15 , 16 , 17 Subsequent studies identified vibration control devices characterized by purely nonlinear stiffness as nonlinear energy sinks (NESs), which possess the capability of broadband vibration suppression. 18 , 19 Early research primarily concentrated on investigating the intrinsic properties of nonlinear vibration absorbers. Vakakis and Gendelman 20 , 21 pioneered the analysis of targeted energy transfer mechanisms for nonlinear absorbers in vibrating systems. Gourdon and Lamarque 22 examined the high robustness of NES in systems with parametric uncertainties. Starosvetsky and Gendelman characterized the response of NES with nonlinear damping features in dynamic systems. Zang et al. 23 investigated parameter effects on transmissibility using harmonic balance solutions and numerical simulations. Wang and Ding 24 , 25 systematically reviewed and discussed optimization methodologies for parametric design in NES applications. Mohammad 26 coupled negative linear stiffness with nonlinear stiffness, significantly enhancing nonlinear oscillator performance in targeted energy transfer (TET). As understanding of nonlinear phenomena deepens, researchers have sought to further exploit nonlinear advantages in vibration suppression through optimized design strategies. 27 , 28 , 29 , 30 Building on extensive prior research, Ding and Chen 31 comprehensively summarized fundamental NES characteristics and design principles, providing pivotal guidance for future nonlinear absorber development. With advancements in nonlinear vibration suppression techniques, researchers have developed diverse nonlinear absorbers to address varied engineering requirements. Ahmadabadi and Khadem 32 integrated piezoelectric energy harvesting devices with NES, enabling energy recovery from dissipated vibrations. Starosvetsky and Gendelman 33 investigated the response characteristics of NES under strong nonlinear coupling with the primary system. Tripathi et al. 34 proposed a first-order perturbation analysis framework to derive the optimal parameters for improving the energy transfer efficiency of NES in multi-degree-of-freedom systems. Luo 35 and Wang 36 et al. designed track-based NESs for building structures subjected to seismic shocks. Zhang et al. 37 proposed an inertia-accumulator-enhanced NES for vibration mitigation in dynamic systems. Zang et al. 38 incorporated lever mechanisms into nonlinear absorbers, thereby establishing a lightweight vibration control strategy. Zhao and Cui 39 proposed a variable-stiffness NES and investigated its vibration suppression performance in elastic beams. Geng et al. 40 , 41 , 42 , 43 introduced piecewise springs into NES configurations to suppress large-amplitude oscillations of nonlinear oscillators. Recent studies have explored semi-analytical modeling approaches and the use of nonlinear absorbers in plate and beam structures. In particular, Wang et al. 44 investigated ABH-PSD composite plates using a semi-analytical model for underwater vibro-acoustic analysis, while Zhao et al. 45 analyzed fluid-loaded baffled rectangular plates equipped with nonlinear energy sinks using a Chebyshev spectral method. These works demonstrate the effectiveness of nonlinear devices and analytical modeling in predicting and controlling structural vibration responses. Beyond formal design variations, researchers have identified superior vibration control characteristics in parallel nonlinear absorber configurations compared to single-unit designs. Vaurigaud and Savadkoohi 46 , 47 theoretically and experimentally validated that parallel nonlinear oscillators maintain TET capabilities during operation. Chen et al. 48 performed a comparative analysis of the vibration mitigation characteristics of parallel and coupled-parallel NES. Building on the demonstrated effectiveness of parallel configurations, Ding and Shao 49 advanced the concept of “NES cells” a distributed multi-unit design methodology. Subsequent research by Zheng et al. 50 and Wang et al. 51 , 52 implemented distributed NES in floating raft systems, demonstrating superior multi-modal vibration suppression. Li and Ding 53 applied distributed NES to systems under high-amplitude harmonic excitation, revealing their effectiveness in counteracting frequency island phenomena. Distributed configurations represent a promising research direction for nonlinear vibration control, with current investigations focused on elucidating the underlying mechanisms of distributed designs—a primary research focus in the field. For the vibration reduction studies in elastic beams, nonlinear vibration absorbers have also attracted considerable attention from scholars. Early studies mainly focused on single-beam systems. Georgiades and Vakakis 54 introduced an essentially nonlinear attachment into a linear elastic beam and demonstrated its capability for localized energy dissipation. Kani et al. 55 and Parseh et al. 56 , 57 investigated the vibration suppression performance of nonlinear absorbers in simply supported and cantilever beam structures, respectively. Zhang et al. 58 further assessed the effectiveness of a parallel NES in mitigating the dynamic response of axially moving beams. Subsequently, Chen et al. 59 revealed that the double NES can effectively suppress higher-order branch responses of simply supported elastic beams under harmonic excitation. For coupled parallel beam systems, Zhao et al. 60 , 61 investigated the vibration suppression effects of coupled nonlinear energy sink (CNES) on double-beam systems. Zhao 62 and Li 63 further examined the vibration attenuation capability of coupled nonlinear devices in parallel beam structures under single-source and dual-source excitations, respectively. Subsequently, Wang 64 , 65 and Zeng 66 explored the vibration control performance of single-point distributed CNES in double-beam systems. Zhan et al. 67 , 68 further investigated the dynamic behaviors of floating raft systems with nonlinear coupling connections, as well as those of serially connected dual oscillators embedded in elastic beams. Recent advances in distributed nonlinear absorber configurations have demonstrated promising multimodal vibration attenuation in plate and floating raft structures. However, in parallel thin-beam systems (PBSs), the fundamental mechanisms associated with spatially distributed internal nonlinear coupling remain insufficiently understood. Existing studies mainly focus on numerical simulations or parametric optimization of absorber performance, while the intrinsic nonlinear interaction behavior induced by spatial distribution along coupled beams has not been systematically investigated. In particular, how spatially distributed nonlinear units interact with the internal coupling dynamics of a PBS, and how their spatial arrangement influences global vibration modulation, have not yet been clarified through experimental validation. The lack of experimental evidence and mechanism-oriented analysis limits the understanding of distributed internal nonlinear regulation in coupled beam structures. Moreover, most existing parallel nonlinear absorbers are introduced as external attachment-type devices, with primary emphasis placed on absorber design and local energy dissipation performance. By contrast, the proposed spatially distributed internal nonlinearity (SDIN) is constructed as an internally embedded nonlinear interaction unit within the PBS, so that its regulation effect depends not only on the intrinsic nonlinear characteristics of the device, but also on its spatial coupling relationship with the host beams. This feature allows the SDIN to regulate vibration through spatially dependent internal nonlinear interaction and coordinated multi-resonance modulation, rather than through the isolated action of a conventional parallel absorber. Therefore, compared with existing parallel nonlinear absorbers, the proposed SDIN provides a more suitable framework for revealing the mechanism of distributed internal nonlinear regulation in coupled thin-beam structures, and offers greater potential for broadband vibration suppression and configuration-oriented design in practical engineering applications. Therefore, an experimental investigation specifically targeting spatially distributed internal nonlinear coupling in PBS is urgently needed. To experimentally clarify the vibration modulation mechanisms induced by SDINs in a PBS, this study introduces controllable SDIN units into a simplified PBS prototype and establishes a corresponding theoretical model for dynamic interpretation. The fundamental nonlinear characteristics and structural parameters of the SDIN are examined to ensure reliable internal interaction behavior. Through systematic experimental investigations, the effects of spatial distribution patterns, installation positions, and distribution numbers of SDIN on the global vibration responses of the PBS are evaluated. The results reveal distinct nonlinear interaction states arising from internal coupling dynamics and demonstrate that spatial distribution enables synergistic modulation across multiple resonance regions. These characteristics distinguish the present SDIN strategy from existing parallel nonlinear absorbers by emphasizing internal coupling regulation, spatial configurability, and broadband multi-resonance control capability. The findings provide experimental evidence for the underlying internal nonlinear coupling mechanisms in PBS and offer new insights into the design of spatially distributed nonlinear regulation strategies for thin-beam structural systems and vibration-sensitive detection equipment in engineering applications. Results and discussion Experimental setup and system description To experimentally investigate the vibration modulation behavior induced by SDIN in a PBS, it is first necessary to establish a consistent physical and analytical representation of the structural configuration. In this section, the simplified mechanical model corresponding to the experimental prototype is introduced, and the theoretical assumptions adopted for subsequent analysis are presented. Figure 1 presents the experimental setup of the PBS integrated with SDIN together with its corresponding idealized vibration model. For analytical formulation, the structural behavior of the PBS is established according to Euler-Bernoulli beam theory. The system comprises two elastic thin beams whose mechanical characteristics are defined by Young’s modulus ( E 1 / E 2 ), moment of inertia ( I 1 / I 2 ), material density ( ρ 1 / ρ 2 ), span length ( L 1 / L 2 ), cross-sectional area ( S 1 / S 2 ), and viscous damping coefficients ( C 1 / C 2 ). To approximate practical boundary constraints, linear and rotational springs are installed at both ends of each beam. The translational stiffness coefficients are denoted by k L1 , k L2 , k R1, and k R2 , whereas the corresponding rotational stiffness parameters are represented by K L1 , K L2 , K R1 , and K R2 . The two beams are interconnected by a pair of linear coupling springs with stiffness k E , positioned at x 1 n and x 2 n , forming the internal elastic linkage within the PBS. Harmonic excitation in the form F 0 sin( ωt ) is applied at the left boundary of beam 1, where F 0 and ω denote the excitation amplitude and angular frequency, respectively. Figure 1. Open in a new tab Actual configuration and simplified model of the PBS with SDIN The SDIN are embedded along the beam span as distributed nonlinear interaction units. Each unit is characterized by nonlinear stiffness k N , damping coefficient C N , and lumped mass m N . Their spatial locations are specified by x 1N d and x 2N d , where D denotes the total number of distributed units and d identifies an individual SDIN. In the dynamic formulation, the transverse responses of beam 1 and beam 2 are expressed as u 1 ( x 1 , t ) and u 2 ( x 2 , t ), while w N d ( t ) describes the internal nonlinear displacement of the d -th SDIN. In order to experimentally investigate the SDIN in PBS, a controllable laboratory-scale experimental platform is constructed. This section describes the physical configuration of the PBS, the SDIN implementation, and the experimental scheme adopted for subsequent investigation. Figure 2 presents the overall experimental platform, including the PBS structure and the control module. Two flat aluminum alloy strips arranged in parallel are employed to represent the coupled beam system. Linear and rotational springs are installed at the beam ends to provide adjustable boundary conditions. The left end of beam 1 is connected to an electromagnetic exciter, whose built-in spring serves simultaneously as a boundary constraint and as the excitation input interface. The two beams are coupled through linear springs positioned 0.056 m from both ends, forming the elastic interaction between the beams. The observation position is set at 0.1 m on the right end of PBS, and the vibration amplitude of the beam structure is tested by an acceleration sensor. Figure 2. Open in a new tab Schematic design of the PBS vibration experimental platform with SDIN The control architecture integrates a controller, an National Instruments (NI)-based data acquisition module, and a power amplifier. Harmonic command signals generated by the controller are transmitted through the NI platform, converted into corresponding voltage outputs, and subsequently amplified to actuate the exciter. The vibration response signals collected by the accelerometer are transmitted back to the NI system for processing and storage. The stability and repeatability of the experimental setup are verified through preliminary calibration procedures. The design of the SDIN follows the work of Zhao et al. 62 , 63 on vibration energy harvesting-suppressing modules, while the present study focuses on the investigation from the perspective of spatially distributed design. The SDIN units are distributed along the span of the PBS to serve as internal nonlinear coupling elements. Each unit incorporates a nonlinear restoring mechanism together with viscous damping and a concentrated mass component. In terms of physical configuration, the SDIN consists of ring-shaped permanent magnets arranged along a guiding rod. The magnets positioned at the two ends remain fixed to the structure, while the intermediate magnet assembly is permitted to move in the vertical direction. The interaction between adjacent magnets produces a displacement-dependent repulsive force, which acts as the nonlinear restoring component and establishes internal nonlinear coupling within the PBS. Prior to the formal vibration tests, the nonlinear restoring characteristics of the SDIN are identified through static loading experiments to ensure parameter consistency among different units. Figure S1 illustrates the structural configuration of the SDIN and the corresponding stiffness identification setup. The measured restoring force-displacement relationship confirms the existence of pronounced nonlinear behavior within the operational range required for the present study. To examine the influence of spatial arrangement, a systematic installation configuration is established, as illustrated in Figure S2 . Because harmonic excitation is introduced at the left end of beam 1, vibrational energy is transmitted throughout the PBS via the inter-beam coupling elements. Accordingly, three representative regions along the span are identified for SDIN placement, covering two boundary zones and one mid-span zone. The specific coordinates are defined as follows: position 1 ( x 11 = 0.084 m and x 12 = 0.154 m), position 2 ( x 21 = 0.525 m and x 22 = 0.595 m), and position 3 ( x 31 = 0.765 m and x 32 = 0.835 m). The experimental investigation is organized into two main parts: assessment of installation location effects and assessment of distribution number effects. Theoretical formulation and model validation To comprehensively understand the vibration modulation characteristics induced by SDIN in the PBS, a theoretical framework is established to clarify the underlying dynamic mechanisms and provide a reference for subsequent analysis. Based on the physical configuration described in Figure 1 , the governing equations of the coupled PBS-SDIN system are derived, followed by model verification and numerical investigations to characterize the influence of spatial distribution parameters on system response. The theoretical results provide a foundation for interpreting the experimental observations presented later. The dynamic response of the vibration model is formulated through the Lagrange method (LM). To implement this approach, the corresponding Lagrangian of the PBS incorporating the SDIN must first be established. The governing Lagrangian expression of the system is therefore written as the following equation: L System = T System − V System (Equation 1) where T System is the kinetic energy of the system, and V System is the potential energy of the system. Accordingly, the energy formulation of the PBS with SDIN can be expressed as T System = T PBS + T SDIN (Equation 2) V System = V PBS + V SDIN (Equation 3) The kinetic energy associated with the PBS and the SDIN is formulated as follows: T PBS = ∑ i = 1 2 [ ∫ 0 L i 1 2 ρ i S i ( ∂ u i ∂ t ) 2 d x i ] (Equation 4) T SDIN = ∑ d = 1 D m N 2 D ( d w N d d t ) 2 (Equation 5) The corresponding potential energy of the system is defined by, V PBS = ∑ i = 1 2 [ ∫ 0 L i 1 2 E i I i ( ∂ 2 u i ∂ x i 2 ) 2 d x i ] + ∑ n = 1 4 [ ∫ 0 L 2 ∫ 0 L 1 1 2 G ( x 1 − x 1 n ) G ( x 2 − x 2 n ) k E ( u 1 − u 2 ) 2 d x 1 d x 2 ] + ∑ i = 1 2 [ ∫ 0 L i G ( x i ) 1 2 k L i u i 2 d x i ] + ∑ i = 1 2 [ ∫ 0 L i G ( x i − L i ) 1 2 k R i u i 2 d x i ] + ∑ i = 1 2 [ ∫ 0 L i G ( x i ) 1 2 K L i ( ∂ u i ∂ x i ) 2 d x i ] + ∑ i = 1 2 [ ∫ 0 L i G ( x i − L i ) 1 2 K R i ( ∂ u i ∂ x i ) 2 d x i ] (Equation 6) V SDIN = ∑ d = 1 D ( ∫ 0 L 1 1 4 k N [ w N d − G ( x 1 − x 1 N d ) u 1 ] 4 d x 1 + ∫ 0 L 2 1 4 k N [ w N d − G ( x 2 − x 2 N d ) u 2 ] 4 d x 2 ) (Equation 7) where G (.) denotes the Dirac function. In addition, the virtual work δW System related to the viscous damping of SDIN and PBS can be derived as δ W System = δ W V + δ W PBS + δ W SDIN (Equation 8) δ W V = ∫ 0 L 1 F 0 sin ( ω t ) δ u 1 G ( x 1 ) d x 1 (Equation 9) δ W PBS = − ∑ i = 1 2 ( ∫ 0 L 1 C i ∂ u i ∂ t δ u i d x i ) (Equation 10) δ W SDIN = − ∑ d = 1 D ( ∫ 0 L 1 C N [ d w N d d t − G ( x 1 − x 1 N d ) ∂ u 1 ∂ t ] δ ( w N d − u 1 ) d x 1 + ∫ 0 L 2 C N [ d w N d d t − G ( x 2 − x 2 N d ) ∂ u 2 ∂ t ] δ ( w N d − u 2 ) d x 2 ) (Equation 11) Within the Lagrangian framework, the transverse displacements of the flexible beams are represented through modal expansion, u 1 ( x 1 , t ) = ∑ z = 1 Z φ 1 z ( x 1 ) q 1 z ( t ) (Equation 12) and u 2 ( x 2 , t ) = ∑ m = 1 M φ 2 m ( x 2 ) q 2 m ( t ) (Equation 13) In the flexible displacement equation of PBS, φ 1z ( x 1 ) and φ 2 m ( x 2 ) represent the modal functions of PBS without SDIN, respectively, with Z and M as their truncations. q 1z ( t ) and q 2 m ( t ) are the unknown time terms coefficients in the equation. Subsequently, the flexible vibration displacement of the PBS can be expressed in matrix form, u 1 ( x 1 , t ) = φ 1 · q 1 (Equation 14) and u 2 ( x 2 , t ) = φ 2 · q 2 (Equation 15) The terms in Eqations 14 and 15 are listed below, φ 1 = [ φ 11 . . . φ 1 z . . . φ 1 Z ] (Equation 16) φ 2 = [ φ 21 . . . φ 2 m . . . φ 2 M ] (Equation 17) q 1 = [ q 11 . . . q 1 z . . . q 1 Z ] T (Equation 18) q 2 = [ q 21 . . . q 2 m . . . q 2 M ] T (Equation 19) To provide a unified description of the vibration system, the displacements of the SDIN can be expressed as follows: q 3 = [ w N 1 ⋯ w N d ⋯ w ND ] T (Equation 20) By substituting Equations 14 and 15 into Equation 1 and using the next step: ∂ ∂ t ( ∂ L System ∂ q ˙ i ) − ∂ L System ∂ q i = Q i (Equation 21) The dynamic equations are derived from the constructed Lagrangian of the PBS incorporating the SDIN. Solving these equations yields the time-dependent coefficients associated with the beam displacements. These coefficients are then substituted into Eqs. 12 and 13 to obtain the corresponding vibration responses of the coupled system. The overall computational procedure based on the Lagrange method is summarized in Figure S3 . The derived governing equations are implemented in MATLAB for numerical evaluation of the PBS integrated with SDIN, providing validation of the analytical formulation. Model parameters are selected to match the measured properties of the experimental setup, thereby maintaining consistency between simulation and physical testing. A harmonic load with amplitude F 0 = 30 N is applied, and the excitation frequency is varied from 1 to 120 Hz. For each frequency point, 500 loading cycles are simulated to extract steady-state responses. The remaining structural and boundary parameters are summarized in Table 1 , corresponding to the current experimental configuration and serving as the reference case for subsequent numerical studies. Table 1. Detailed structure and boundary parameters of PBS Symbol Value E 1 / E 2 (Pa) 6.89×10 10 /6.89×10 10 I 1 / I 2 (m 4 ) 1.667 × 10 −9 /1.667 × 10 −9 ρ 1 / ρ 2 (kg/m 3 ) 2.8×10 3 /2.8×10 3 L 1 / L 2 (m) 1/1 C 1 / C 2 (Ns/m) 12/12 S 1 / S 2 (m 2 ) 1.2 × 10 −4 /1.2 × 10 −4 k E (N/m) 3,660 k L1 / k L2 / k R1 / k R2 (N/m) 3,500/7,435/7,435/7,435 K L1 / K L2 / K R1 / K R2 (Nm/rad) 0/0/0/0 x i 1 / x i 2 (m) 0.056/0.944 Open in a new tab To enable a systematic investigation of the effects of SDIN installation position and parameter variation, the PBS is discretized into 11 candidate installation positions along the beam span in the present study. The corresponding installation coordinates on beam 1 and beam 2 for each position are summarized in Table 2 . On the basis of this discretized spatial arrangement, the subsequent numerical analysis is carried out to further examine the influences of installation position, parameter variation, and spatial distribution on the vibration response characteristics of the system. Table 2. The installation positions of the SDIN on PBS Position Beam 1 Beam 2 Installation position 1 0.065 m 0.135 m Installation position 2 0.145 m 0.215 m Installation position 3 0.225 m 0.295 m Installation position 4 0.305 m 0.375 m Installation position 5 0.385 m 0.455 m Installation position 6 0.465 m 0.535 m Installation position 7 0.545 m 0.615 m Installation position 8 0.625 m 0.695 m Installation position 9 0.705 m 0.775 m Installation position 10 0.785 m 0.855 m Installation position 11 0.865 m 0.935 m Open in a new tab Prior to performing the simulations, the reliability and robustness of the computational framework are first assessed. For this verification case, one SDIN unit is installed at position 8, with parameters k N = 1 × 10 8 N/m 3 , C N = 5 N/m, and m N = 0.2 kg. To ensure accuracy within the investigated frequency range while maintaining computational efficiency, different truncation numbers of 2, 3, 4, 5, and 6 are tested. As shown in Figure S4 , the magnitude frequency response gradually converges when the truncation number reaches 5, indicating stable numerical results. Therefore, a truncation number of 5 is adopted in subsequent analyses. Figure S5 presents an investigation into the accuracy of the predicted dynamic behavior of the vibration system. In this case, three SDIN units are considered, and their installation positions are set at positions 2, 6, and 10. The Galerkin method is employed to verify the dynamic model of the vibration system established using the LM. The Galerkin method derives directly from the governing equations and solves them through a weighted residual approach, whereas the LM constructs the model based on the system’s energy, incorporating both kinetic and potential energy. The comparison shows that the dynamic responses obtained from the two methods are in close agreement, confirming that the LM can effectively predict the dynamic characteristics of the vibration system in this study. Core parameters analysis To further characterize the dynamic influence of SDIN on the PBS, a numerical parametric analysis is conducted based on the validated theoretical model. The parameter settings remain consistent with those defined in previous section to ensure coherence with the experimental configuration. The following analysis focuses on the effects of key SDIN parameters, spatial installation positions, and distribution numbers, as well as the coupling relationships between distribution quantity and location. These results provide a theoretical reference for understanding the spatial nonlinear interaction behavior observed in the experiments. For clarity, the main parameters of SDIN considered in this article are as follows: the nonlinear stiffness k N varies from 1×10 7 to 4 × 10 8 N/m 3 , the viscous damping C N varies from 3 to 5 N/m, and the lumped mass m N varies from 0.01 to 0.07 kg. Figure 3 illustrates the influence of SDIN nonlinear stiffness on the vibration response of the PBS under three different installation positions. In this case, the other SDIN parameters are fixed at C N = 5 N/m and m N = 0.2 kg. As shown in Figure 3 , with increasing nonlinear stiffness, the response magnitude of the PBS generally decreases in most resonance regions, indicating that a larger nonlinear stiffness enhances the restoring-force effect of the SDIN and improves the vibration regulation capability of the system. The most pronounced suppression is observed in the 2 nd and 3 rd resonance regions, while complex vibration responses further emerge near the 3 rd resonance region, suggesting that the system enters a stronger nonlinear coupling state in this frequency range. Figure 3. Open in a new tab Influence of k N on the vibration response of the PBS under three different SDIN installation positions (A) Vibration response of the PBS when the SDIN is installed at position 3. (B) Vibration response of the PBS when the SDIN is installed at position 8. (C) Vibration response of the PBS when the SDIN is installed at position 10. In contrast, the 1 st , 4 th , and part of the 5 th resonance regions remain weakly sensitive to stiffness variation, exhibiting non-working resonance characteristics. This indicates that the effectiveness of the SDIN depends not only on the parameter magnitude, but also on the local relative motion between the two beams at the installation location. When the relative displacement is insufficient, increasing stiffness alone cannot generate a strong enough nonlinear restoring force to significantly regulate the PBS response. Overall, although the detailed peak amplitudes and the frequency ranges of complex responses vary with installation position, the fundamental regulation law of nonlinear stiffness remains qualitatively consistent, indicating reasonable stability of this parametric trend under different spatial arrangements. To further explain the complex vibration responses and the non-working resonance regions observed in Figure 3 and Figure 4 presents the phase trajectories and the maximum nonlinear restoring force of the SDIN oscillator for representative cases. From the perspective of complex vibration behavior, Figures 4 A and 4B show the phase trajectories at two representative frequencies. It can be seen that the trajectories remain bounded and form stable closed-loop bands, rather than diverging irregularly. This suggests that the complex vibration responses in Figure 3 are not caused by random errors or fully disordered chaos, but are more closely associated with a quasiperiodic vibration state. Figure 4. Open in a new tab Phase trajectories and nonlinear restoring force of the SDIN for representative cases (A) Phase trajectory at 68 Hz in the third resonance region in Figure 3 A. (B) Phase trajectory at 69 Hz in the third resonance region in Figure 3 B. (C) Maximum nonlinear restoring force of the SDIN oscillator in the magnitude-frequency response results of Figure 3 A. (D) Maximum nonlinear restoring force of the SDIN oscillator in the magnitude-frequency response results of Figure 3 B. From the perspective of the SDIN activation mechanism, Figures 4 C and 4D further present the maximum nonlinear restoring force of the SDIN oscillator under the corresponding cases. The vibration mitigation effect of the SDIN fundamentally depends on whether sufficient relative displacement can develop between the two beams at the installation location. Only when a noticeable displacement difference develops can the internal nonlinear oscillator be effectively activated, thereby generating a strong nonlinear restoring force and enabling vibration energy redistribution and dissipation. As shown in Figures 4 C and 4D, the nonlinear restoring force remains consistently small near the 1 st and 4 th resonance regions. This indicates that the relative displacement between the two beams is limited in these frequency ranges, resulting in insufficient activation of the internal oscillator. Consequently, the vibration suppression effect in the corresponding regions of Figure 3 remains weak, and these regions behave as typical non-working resonance regions. Therefore, the complex vibration responses and the non-working resonance regions identified in Figure 3 are not independent phenomena, but are both closely related to the activation level of the SDIN. Figure S6 illustrates the influence of lumped mass on the vibration response of the PBS under three different installation positions. In this case, the other SDIN parameters are fixed at C N = 5 N/m and k N = 3 × 10 8 N/m 3 . As shown in Figure S6 , increasing the lumped mass generally reduces the vibration response in the 2 nd , 3 rd , and part of the 5 th resonance regions, while the 1 st and 4 th resonance regions remain weakly sensitive to mass variation. Meanwhile, under some installation positions and frequency intervals, a larger lumped mass also induces local branches, multivalued peaks, and complex vibration responses. This behavior indicates that lumped mass mainly affects the system response by changing the inertial coupling level of the SDIN. When the mass is small, the energy exchange between the oscillator and the coupled beams is limited. As the mass increases, the inertial effect is strengthened, which is beneficial for improving vibration suppression in the working resonance regions. However, when the mass becomes excessively large, the system is more likely to enter a strongly nonlinear coupled state, thereby inducing local complex responses. Overall, the fundamental regulation law of lumped mass remains consistent under different installation positions, although the specific regulation intensity still exhibits some spatial dependence. Figure S7 illustrates the influence of viscous damping on the vibration response of the PBS under three different installation positions. In this case, the other SDIN parameters are fixed at m N = 0.2 kg and k N = 3 × 10 8 N/m 3 . As shown in Figure S7 , with increasing damping, the peak responses in the 2 nd , 3 rd , and part of the 5 th resonance regions gradually decrease, and the local abnormal peaks and complex responses are also significantly weakened. In contrast, the 1 st resonance region remains insensitive, while the improvement in the 4 th resonance region is relatively limited. This indicates that viscous damping mainly improves response stability by enhancing the energy dissipation capacity of the system, and its effect is more pronounced in the working resonance regions where the SDIN is effectively activated. By contrast, in the non-working resonance regions, the damping effect cannot be fully developed because the local relative motion is insufficient. Therefore, although the local peak values and response amplitudes vary with installation position, the overall regulation trend of damping remains qualitatively consistent. Based on the above analysis of nonlinear stiffness, lumped mass, and viscous damping, Figure S8 further considers installation position as an independent spatial variable and investigates its effect on the vibration response of the PBS, with k N = 3 × 10 8 N/m 3 , C N = 6 N/m, and m N = 0.1 kg fixed. The results show that vibration suppression performance varies significantly with installation position and differs among resonance regions. In some regions, specific positions lead to pronounced attenuation, whereas in others the response is less sensitive to positional change. In addition, the positional effect does not follow a simple monotonic trend along the beam span, indicating that the installation location is a critical design variable for effective SDIN deployment. These results provide a direct basis for the subsequent analysis of spatially distributed configurations. Spatially distributed analysis Building upon the preceding parametric analysis, this section further investigates the influence of spatial distribution characteristics, including installation positions and distribution number of SDIN, on the vibration response of the PBS. The previous results have demonstrated that installation position plays a decisive role in nonlinear coupling intensity, providing the theoretical basis for spatially distributed design schemes. In this section, both the installation positions and the number of SDIN are treated as primary structural variables. Two and three SDIN configurations are considered to evaluate the combined spatial interaction effects. The intrinsic parameters of each SDIN are fixed as k N = 3 × 10 8 N/m 3 , C N = 6 N/m, and m N = 0.1 kg. Figure S9 presents the magnitude-frequency responses of the PBS with two spatially distributed SDIN under different installation combinations. The results indicate that vibration regulation performance is strongly influenced by the spatial arrangement relative to both the excitation location and the predefined observation point. When the SDIN are installed at positions 1 and 3, which are located near the excitation end of the beam, improved vibration suppression is observed in the resonance regions of 3 rd and 5 th . Physically, placing nonlinear units close to the excitation source enhances their interaction with high modal participation components excited at the input boundary, thereby strengthening nonlinear energy modulation for higher-order resonance modes. In contrast, when the SDIN are installed at position 11, which is close to the predefined observation point, enhanced suppression is observed in the 1 st and 2 nd resonance regions. In this configuration, the nonlinear unit interacts more directly with the dominant displacement response at the observation point, effectively regulating lower-order modal responses. More importantly, when the SDIN are distributed across both the excitation side and the observation side (position 1&11 or 3&11), the vibration reduction performance becomes more balanced across multiple resonance regions. This configuration coordinates nonlinear interaction at different spatial locations, allowing energy redistribution to occur both near the input boundary and near the response-dominant region. As a result, the overall vibration response of the PBS exhibits a more globally regulated behavior rather than selective suppression confined to specific resonance regions. To facilitate a direct comparison among different installation scenarios, a quantitative metric, namely the peak suppression ratio, is introduced for the experimental results. It is defined as η = A u - A c A u × 100 % (Equation 22) where A u and A c denote the peak amplitudes of the uncontrolled and controlled cases, respectively. To enable a concise quantitative comparison of different two-unit installation scenarios, the comprehensive peak suppression ratios in the main resonance regions are summarized in Table 3 . Here, the suppression ratios of beam 1 and beam 2 are first calculated separately for each resonance region and then averaged to obtain a comprehensive suppression ratio. An overall evaluation value is further defined as the average of the comprehensive suppression ratios over the selected resonance regions. Table 3. Comprehensive suppression ratios for two-unit SDIN configurations Installation scenario Resonance region 2 (%) Resonance region 3 (%) Resonance region 4 (%) Resonance region 5 (%) Overall evaluation (%) Position 1 9.74 63.41 34.09 53.84 40.27 position 3 37.07 57.94 23.92 17.66 34.15 position 11 58.89 57.57 37.40 18.85 43.18 position 1&11 41.63 70.50 50.05 65.59 56.94 a position 3&11 55.56 61.55 42.35 29.65 47.28 Open in a new tab a Bold values indicate the maximum peak suppression ratio for each SDIN distribution scenario. As shown in Table 3 , the two-unit installation scenarios exhibit noticeable differences in overall control performance. In particular, position 1&11 achieves the highest overall evaluation value, indicating that this configuration provides the most balanced and effective vibration suppression over multiple resonance regions. By contrast, single-position installation may lead to good attenuation in specific regions, but its overall regulation capability remains relatively limited. These quantitative results are consistent with the trends observed in Figure S9 and further confirm the importance of coordinated spatial placement. Figure S10 illustrates the vibration suppression effect of the PBS with three spatially distributed SDIN under different installation combinations. When the distribution number of SDIN increases from 2 to 3, further vibration reduction can be observed in certain resonance regions. However, this improvement is not uniform across all frequency bands. Instead of exhibiting a simple cumulative enhancement, the regulation performance depends strongly on the specific spatial configuration of the nonlinear units. This behavior suggests that when multiple SDIN are introduced at different locations, complex spatial energy transfer and nonlinear coupling interactions occur within the structure. The nonlinear oscillators do not act independently; rather, their mutual interaction through the host beam modifies the overall energy redistribution pathways. As a result, although some resonance regions benefit from additional nonlinear units, others exhibit only marginal improvement. Nevertheless, in general, the configuration with three distributed SDIN maintains relatively effective vibration suppression across most resonance regions, indicating that increasing distribution number enhances global regulation robustness rather than producing strictly additive attenuation. Figure S10 C further summarizes the vibration control performance in each resonance region under four different installation combinations when the distribution number is three. It can be observed that no clear monotonic or deterministic pattern emerges among these spatial arrangements. All configurations demonstrate relatively good vibration control capability, but the degree of suppression varies across resonance regions without a simple positional rule. This observation is consistent with the earlier position-parameter study, where the variation of installation location did not lead to a regular suppression trend. Such behavior confirms that the vibration regulation mechanism is governed by complex spatial-modal coupling rather than by purely geometric proximity. More importantly, compared with the previous parameter study, the 4 th resonance region-previously identified as a non-working resonance region under certain conditions-exhibits noticeable vibration attenuation after adjusting the installation positions. This phenomenon indicates that the working or non-working state of a resonance region is not an inherent modal property, but a spatially dependent nonlinear interaction state. By modifying spatial distribution, the nonlinear activation level in specific frequency bands can be altered, thereby expanding the effective working bandwidth of the SDIN. Similarly, to compare the vibration suppression performance of different three-unit installation scenarios, the comprehensive peak suppression ratios are summarized in Table 4 . Among the three-unit configurations, position 1&3&11 yields the highest overall evaluation value, followed by position 1&7&11, whereas position 3&6&9 shows relatively low comprehensive suppression ratios. These results indicate that increasing the distribution number does not necessarily lead to a simple additive improvement; instead, an appropriate spatial combination is more critical for achieving robust multimodal vibration attenuation. Table 4. Comprehensive suppression ratios for three-unit SDIN configurations Installation scenario Resonance region 2 (%) Resonance region 3 (%) Resonance region 4 (%) Resonance region 5 (%) Overall evaluation (%) Position 1&3&11 36.26 66.11 54.97 71.65 57.25 a Position 1&7&10 26.80 60.62 48.07 65.96 50.36 Position 1&2&3 28.70 59.58 48.07 65.67 50.50 Position 3&6&9 36.25 43.47 32.34 43.48 38.89 Position 9&10&11 45.33 52.80 46.06 52.08 49.07 Position 1&7&11 27.37 66.41 52.27 68.30 53.59 Open in a new tab a Bold values indicate the maximum peak suppression ratio for each SDIN distribution scenario. Overall, the simulation results in this section demonstrate that vibration regulation in the PBS is jointly determined by intrinsic nonlinear parameters and spatial distribution characteristics. While increasing the number of SDIN does not guarantee proportional improvement, appropriate spatial configuration provides an effective means of broadening frequency coverage and improving vibration mitigation stability. These findings highlight the engineering significance of spatially distributed nonlinear design strategies for practical broadband vibration control applications. Experimental verification of installation position effects To further verify the dynamic response characteristics predicted in the numerical analysis, an experimental investigation is conducted in this section. Considering that nonlinear activation and spatial coupling behavior may be influenced by structural uncertainties and assembly conditions, experimental validation is necessary to evaluate the physical performance of the proposed SDIN-equipped PBS. An experimental platform is established to examine the vibration response of the system under different working conditions. Special attention is given to the identification of distinct nonlinear working states observed during the experimental process and their relation to the spatial configuration of the SDIN. Based on the experimental preparation described in previous section, this section investigates the influence of different installation positions of the SDIN on the vibration response of the PBS. The experiments were conducted using a frequency sweep excitation method. The excitation frequency range was set from 10 to 80 Hz, and the power amplifier voltage was maintained at 2 V. During the experiments, the sampling frequency of the data acquisition system was set to 1,000 Hz, and 30 sampling points were recorded at each excitation frequency. For each frequency step, the response signal was collected after the transient vibration had sufficiently decayed, and the steady-state amplitude was then extracted for subsequent analysis. Since the introduction of the SDIN mass may alter the dynamic characteristics of the PBS, equivalent counterweights were placed at the corresponding installation positions to obtain the reference vibration response of the uncontrolled structure, ensuring experimental consistency. Figure 5 presents the experimental vibration responses of the PBS when the SDIN is installed at positions 1–3. According to the experimental response characteristics, beam 1 exhibits four main resonance regions, whereas beam 2 exhibits three main resonance regions within the investigated frequency range. After introducing the SDIN, the overall resonance distribution of the PBS remains generally unchanged, but the vibration amplitudes in different resonance regions vary significantly with the installation position. Figure 5. Open in a new tab Experimental study on the vibration suppression effects of SDIN (one distribution numbers) (A) Magnitude-frequency response results of the PBS when the SDIN is installed at position 1. (B) Magnitude-frequency response results of the PBS when the SDIN is installed at position 2. (C) Magnitude-frequency response results of the PBS when the SDIN is installed at position 3. When the SDIN is installed at position 1, the suppression effect in the 1 st resonance region is relatively limited for both beams. More noticeable vibration reduction is observed in the 2 nd and 3 rd resonance regions, especially around the dominant peaks of beam 2. In the 4 th resonance region of beam 1, the reduction effect is still present, but it is less pronounced than that in the 2 nd and 3 rd resonance regions. For position 2, the SDIN exhibits a stronger overall suppression effect than that at position 1. Beam 1 shows obvious attenuation in the 1 st and 3 rd resonance regions, and beam 2 also exhibits clear amplitude reduction in the 1 st and 3 rd resonance regions. In contrast, the suppression effect in the 2 nd resonance region of beam 2 is comparatively limited. In addition, the response peaks associated with the 2 nd and 3 rd resonance regions become less sharply separated, implying a certain degree of resonance reorganization induced by the SDIN. When the SDIN is installed at position 3, effective suppression is still observed in the 3 rd and 4 th resonance regions of beam 1 and in the 2 nd and 3 rd resonance regions of beam 2, whereas the response variations in the 1 st resonance region become less uniform. In particular, beam 1 exhibits local response amplification in the 1 st resonance region, while beam 2 still maintains attenuation in the corresponding resonance region. This opposite tendency indicates that the SDIN does not simply function as an independent absorber at this location, but may act as a nonlinear coupling bridge that redistributes vibration energy between the two beams. Moreover, for beam 1, the 2 nd and 3 rd resonance regions become less clearly separated, suggesting that the SDIN modifies the local modal interaction and leads to partial merging or weakening of adjacent resonance peaks. It is noteworthy that, beyond the theoretical analysis, two distinct vibration states of the SDIN were observed during the experimental process, as illustrated in Figure 6 . This phenomenon is closely related to the practical structural design of the magnetic nonlinear device. In the present experimental setup, magnetic elements are employed to generate the nonlinear restoring force. To prevent direct magnetic interaction between adjacent nonlinear units installed on the sub-beam, an interlaced installation configuration is adopted. As a result, the two attachment points of the SDIN on beam 1 and beam 2 are not vertically aligned, but exhibit a certain spatial offset. This intentionally staggered arrangement avoids unintended magnetic coupling while simultaneously introducing a positional difference between the two installation points. Figure 6. Open in a new tab Two different operational modes of SDIN During operation, vibration transmission along the beams leads to a frequency-dependent phase difference between these two attachment points. Owing to the spatial offset, this phase difference gives rise to different relative-motion patterns of the SDIN. When the phase difference is small, the two attachment points move nearly in phase, and the SDIN mainly exhibits a vertical vibration state. Under this condition, the relative motion is concentrated along the intended working direction of the magnetic nonlinear unit, so that the nonlinear restoring force can be effectively activated and the vibration suppression capability is enhanced. In contrast, when the phase difference becomes pronounced, synchronous vertical motion cannot be maintained, and the SDIN enters a swinging vibration state with additional lateral or rotational motion components. In this state, the effective vertical compression-extension process of the magnetic nonlinear unit is weakened, and the nonlinear restoring interaction cannot be fully developed. These two vibration states therefore correspond to two different nonlinear coupling mechanisms. In the vertical vibration state, the SDIN mainly functions as an effective nonlinear regulator, because the nonlinear restoring interaction is effectively developed along its designed working direction. By contrast, in the swinging vibration state, the SDIN tends to act more as a nonlinear coupling bridge that redistributes vibration energy between beam 1 and beam 2, rather than as an efficient nonlinear vibration suppressor. Therefore, the observed swinging vibration state and vertical vibration state are not merely different motion patterns, but represent two distinct nonlinear working regimes governed by the phase relationship of the beam responses and the spatial installation geometry of the SDIN. This mechanism also explains why different installation scenarios may result in markedly different vibration regulation performance. Experimental verification of distribution number effects Building on the vibration suppression results for different installation positions, this section examines how the quantity of SDIN units affects the PBS dynamic response. The experimental parameters are kept the same as before, while two and three SDIN units are deployed to explore the mechanism of spatially distributed vibration regulation. Figure 7 presents the experimental magnitude-frequency responses of the PBS with two distributed SDIN under different installation combinations. Compared with the single-position cases in Figure 5 , the distributed configurations provide broader vibration regulation over multiple resonance regions, but the control performance still depends strongly on the specific spatial arrangement. Figure 7. Open in a new tab Experimental study on the vibration suppression effects of SDIN (two distribution numbers) (A) Magnitude-frequency response results when the SDIN is installed at positions 1 and 2 with a distribution number of 2. (B) Magnitude-frequency response results when the SDIN is installed at positions 1 and 3 with a distribution number of 2. (C) Magnitude-frequency response results when the SDIN is installed at positions 2 and 3 with a distribution number of 2. For the combination of positions 1 and 2, the distributed SDIN provide noticeable suppression in the 2 nd , 3 rd , and 4 th resonance regions of beam 1, while the suppression of beam 2 is more pronounced in the 2 nd and 3 rd resonance regions. However, in the 1 st resonance region, the attenuation remains relatively limited, and a non-working resonance behavior can still be observed locally. For the combination of positions 1 and 3, the suppression effect becomes more spatially selective. Compared with the uncontrolled case, beam 2 exhibits obvious attenuation in the 2 nd and 3 rd resonance regions, whereas beam 1 shows local amplification or weakened suppression in the 1 st and 3 rd resonance regions. Meanwhile, the response peaks in the 2 nd and 3 rd resonance regions become less clearly separated, reflecting stronger modal interaction restructuring under distributed nonlinear coupling. For the combination of positions 2 and 3, the distributed configuration exhibits relatively balanced suppression over most resonance regions. Both beam 1 and beam 2 show clear vibration reduction in the 2 nd and 3 rd resonance regions, and beam 1 also exhibits attenuation in the 4 th resonance region. The corresponding response curves become smoother than those of the other two two-unit configurations. Figure 8 shows the experimental vibration response of the PBS with three distributed SDIN installed at positions 1, 2, and 3. Compared with the single- and two-unit configurations, the overall response profile becomes smoother, and the vibration amplitudes in several resonance regions are further reduced. For beam 1, the introduction of three SDIN leads to clear attenuation in the 2 nd , 3 rd , and 4 th resonance regions, while the 1st resonance region still exhibits relatively weak suppression and even local amplification. For beam 2, significant vibration reduction is observed in the 1 st , 2 nd , and 3 rd resonance regions, especially in the 2 nd and 3 rd resonance regions. Compared with the two-unit cases, the three-unit configuration provides stronger global regulation by introducing more spatially distributed nonlinear coupling paths. The coordinated action of multiple SDIN promotes more uniform energy redistribution and weakens sharp resonance transitions, thereby making the overall response of the PBS smoother. Figure 8. Open in a new tab Experimental study on the vibration suppression effects of SDIN (three distribution numbers) To enable a more direct comparison among different installation scenarios, the corresponding quantitative results are further summarized in Table 5 using the same peak-suppression metric introduced previously in previous section. For each resonance region, the suppression ratios of beam 1 and beam 2 are 1st calculated separately and then averaged to obtain a comprehensive suppression ratio. In the 4 th resonance region, only the result of beam 1 is considered because no corresponding dominant peak is observed for beam 2. An overall evaluation value is further defined as the arithmetic mean of the comprehensive suppression ratios over all considered resonance regions. Table 5. Comprehensive peak suppression ratios of different installation scenarios obtained from the experimental results Installation scenario Resonance region 1 (%) Resonance region 2 (%) Resonance region 3 (%) Resonance region 4 (%) Overall evaluation (%) Position 1 19.16 37.84 31.41 21.80 27.55 Position 2 63.93 10.91 51.04 19.65 36.38 Position 3 0.45 30.56 46.14 16.79 23.49 Position 1&2 −0.90 27.57 53.52 61.77 35.49 Position 1&3 −100.99 57.60 −75.65 42.69 −19.09 Position 2&3 37.31 31.83 48.71 23.12 35.24 Position 1&2&3 −14.98 36.32 60.31 57.46 34.78 Open in a new tab As shown in Table 5 , the vibration suppression performance depends strongly on the installation scenario and varies across different resonance regions. Among the single-position cases, position 2 yields the highest overall evaluation value of 36.38%, indicating the best global control effectiveness among the three single-installation configurations. By contrast, position 1 and position 3 show lower overall evaluation values of 27.55% and 23.49%, respectively. For the distributed two-position cases, positions 1&2 and 2&3 exhibit relatively balanced suppression over multiple resonance regions, with overall evaluation values of 35.49% and 35.24%, respectively. In particular, position 1&2 achieves the strongest attenuation in the 4 th resonance region, reaching 61.77%, whereas position 2&3 maintains comparatively stable suppression in the 1 st , 2 nd , and 3 rd resonance regions. For the three-position configuration, position 1&2&3 provides stronger attenuation in the 3 rd and 4 th resonance regions, with suppression ratios of 60.31% and 57.46%, respectively. These results show that increasing the distribution number can improve vibration regulation in specific resonance regions, but the improvement is not uniformly distributed over all resonance regions. It is also noted that some installation scenarios produce negative suppression ratios in certain resonance regions. For example, position 1&3 leads to response amplification in the 1 st and 3 rd resonance regions, resulting in a negative overall evaluation value. Similarly, position 1&2&3 still show a negative suppression ratio in the 1 st resonance region. These results indicate that spatial distribution does not always produce a cumulative improvement, and that an improper combination of installation positions may weaken the overall vibration control effect. Combining the frequency-response results in Figures 5 , 7 , and 8 with the quantitative comparison in Table 5 , it can be seen that the vibration regulation effect of the PBS is highly sensitive to the spatial arrangement of the SDIN. The distributed configuration does not merely increase the number of nonlinear units, but changes the inter-beam coupling relationship and the coordination of nonlinear working states, thereby affecting the suppression characteristics in different resonance regions. In this sense, the experimentally observed swinging and vertical oscillation states provide a physical explanation for why some installation scenarios achieve strong suppression, whereas others exhibit weak suppression or even local amplification. Therefore, the experimental results further confirm that spatially distributed nonlinear regulation in the PBS is essentially a configuration-dependent vibration control mechanism rather than a simple superposition effect of multiple nonlinear devices. Conclusion This study experimentally investigated the vibration control performance of the SDIN in a PBS. A simplified experimental model was established, and the Lagrange method was employed to predict the dynamic behavior of the PBS. Through combined numerical analysis and experimental validation, the influence of core parameters and spatial configuration of the SDIN on vibration regulation was systematically examined, clarifying the fundamental mechanism of spatially dependent nonlinear coupling under practical operating conditions. Based on the comprehensive investigation, the following conclusions are drawn. 1. Theoretical analysis indicates that intrinsic parameters and spatial configuration of the SDIN significantly influence PBS vibration control by altering modal participation and nonlinear coupling intensity. 2. Experimental results demonstrate that the SDIN effectively suppress vibrations in multiple resonance regions. Increasing distribution number enhances global energy redistribution; however, vibration control performance depends on working state coordination rather than solely on device quantity. 3. Two operational modes of the SDIN were identified: swinging and vertical vibration. These modes are governed by installation position and excitation frequency, reflecting spatially dependent nonlinear coupling and directly determining vibration mitigation effectiveness. 4. Spatial configuration plays a decisive role in broadband vibration regulation. Optimizing installation layout, rather than relying solely on parameter tuning, provides a practical and flexible strategy for nonlinear vibration control. Future work may explore the combined influence of nonlinear stiffness, concentrated mass, and spatial layout to further enhance robustness and expand applicability. Limitations of the study This study has several limitations. 1. The experiments were conducted on a simplified laboratory-scale PBS-SDIN prototype, and the theoretical model was established under idealized assumptions. Therefore, the present results mainly reveal the fundamental mechanisms of spatially distributed internal nonlinear regulation, whereas their direct quantitative extension to more complex engineering structures still requires further verification. 2. The experimental investigation focused mainly on harmonic excitation applied at the left end of beam 1, together with representative installation positions and distribution numbers of the SDIN. The vibration-regulation behavior under other excitation types, broader operating conditions, and more complex spatial layouts remains to be explored. Resource availability Lead contact Further information and requests for resources should be directed to and will be fulfilled by the lead contact, Haijian Cui ( chj841524537@163.com ). Materials availability This study did not generate new unique reagents. Data and code availability • This study does not report original code. • All data reported in this study will be shared by the lead contact upon request. • Any additional information required to reanalyze the data reported in this study is available from the lead contact upon request. Acknowledgments This work was supported by the National Natural Science Foundation of China (grant no. 52465057); The Guizhou Provincial Basic Research Program (Natural Science) Youth Guidance Project (no. Qiankehe Basic-[2024] Youth 172). Author contributions All authors contributed to the conception and design of the study; T.C.: conceptualization, methodology, investigation, formal analysis, writing – original draft, and writing – review and editing; H.C.: conceptualization, methodology, investigation, formal analysis, writing – original draft, and writing – review and editing; Q.W. and Y.Z.: formal analysis, data curation, and writing – review and editing. Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this study. The authors declare no conflict of interest in preparing this article. STAR★Methods Key resources table REAGENT or RESOURCE SOURCE IDENTIFIER Software and algorithms MATLAB 2023a MathWorks https://ww2.mathworks.cn/products/matlab.html Open in a new tab Method details The coupled PBS-SDIN equations were derived using the Lagrange method and implemented in MATLAB. The PBS was modeled according to Euler-Bernoulli beam theory, and the SDIN units were represented as distributed nonlinear interaction elements characterized by nonlinear stiffness, damping, and lumped mass. Harmonic frequency-sweep simulations from 1 to 120 Hz were performed, with 500 loading cycles at each frequency point to extract steady-state responses. The model was then used to examine the effects of parameter variation, installation position, and spatial distribution on the vibration response of the system. In the experiments, the PBS-SDIN platform was excited by single-frequency harmonic input, and the vibration responses were acquired through the NI-based control and acquisition system for subsequent magnitude-frequency analysis. Experimental model and study participant details The experiments were conducted on a laboratory-scale PBS equipped with SDIN. Two parallel aluminum alloy beams were coupled by linear springs and constrained by adjustable boundary springs. Harmonic excitation was applied at the left end of Beam 1 through an electromagnetic exciter, and the vibration response was measured by an accelerometer. The SDIN units were embedded along the beam span and physically realized using ring-shaped permanent magnets arranged on a guiding rod, with a movable intermediate magnet assembly providing displacement-dependent nonlinear restoring interaction. The experiments were designed to investigate the effects of installation position and distribution number on the vibration response of the PBS. Main experimental equipment Equipment Model Electromagnetic exciter JZK-20 Power amplifier YE2706A Accelerometer PCB 352C33 Data acquisition module NI 9230 Chassis controller NI cDAQ-9174 Signal output module NI 9263 Computer – Open in a new tab Quantitation and statistical analysis The vibration responses were quantified using steady-state magnitude-frequency curves obtained from numerical simulations and experiments. To compare different installation scenarios, the peak suppression ratio was defined as η = A u − A c A u × 100 % (Equation 22) where A u and A c denote the peak amplitudes of the uncontrolled and controlled cases, respectively. For each resonance region, the suppression ratios of Beam 1 and Beam 2 were first evaluated separately and then averaged to obtain a comprehensive suppression ratio. In the fourth resonance region of the experimental study, only the result of Beam 1 was considered because no corresponding dominant peak was observed for Beam 2. An overall evaluation value was further defined as the arithmetic mean of the comprehensive suppression ratios over all considered resonance regions. Footnotes Supplemental information can be found online at https://doi.org/10.1016/j.isci.2026.116103 . Supplemental information Document S1. Figures S1–S10 mmc1.pdf (2MB, pdf) References 1. Ren C., Xu Y., Liu X., Liu H. 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Figures S1–S10 mmc1.pdf (2MB, pdf) Data Availability Statement • This study does not report original code. • All data reported in this study will be shared by the lead contact upon request. • Any additional information required to reanalyze the data reported in this study is available from the lead contact upon request. Articles from iScience are provided here courtesy of Elsevier ACTIONS View on publisher site PDF (8.8 MB) Cite Collections Permalink RESOURCES Similar articles Cited by other articles Links to NCBI Databases Follow NCBI NCBI on X (formerly known as Twitter) NCBI on Facebook NCBI on LinkedIn NCBI on GitHub NCBI RSS feed Connect with NLM NLM on X (formerly known as Twitter) NLM on Facebook NLM on YouTube National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894 Web Policies FOIA HHS Vulnerability Disclosure Help Accessibility Careers NLM NIH HHS USA.gov Back to Top

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