Design optimization and experimental study of flexible hinge-type nonlinear energy sink

Abstract

To address the difficulty of suppressing low-frequency line spectra in underwater vehicle radiated noise and the narrow operating bandwidth of conventional linear vibration absorbers with fixed tuning frequencies, this paper proposes a flexible hinge-type nonlinear energy sink (NES). First, a double-slotted flexible hinge-type NES structure is presented. Static mechanical analysis shows that the structure exhibits cubic stiffness characteristics near the static equilibrium position. Second, the IHB method is employed to characterize the periodic responses of the NES system. The effects of NES mass, stiffness, and damping on the amplitude-frequency response and vibration suppression performance are then investigated in detail. Subsequently, local optimization is performed to determine the optimal damping and stiffness parameters of the NES. The optimized NES shows strong robustness against variations in excitation frequency and amplitude. Finally, a prototype of the flexible hinge-type NES is fabricated, and a vibration test platform is established for experimental validation. Both simulation and experimental results show that the NES reduces the first-order resonance peak near 5.00 Hz by approximately 8.17 dB, decreases the average line-spectrum level over 4.43–5.82 Hz by about 3.74 dB, and broadens the vibration absorption bandwidth by approximately 27.8%. Compared with an equivalent linear dynamic vibration absorber, the NES provides better broadband vibration absorption performance. In addition, the NES can shift the resonance frequency of the coupled system and induce a strong modulation response.

1. Introduction

The low-frequency line spectra in the underwater radiated noise of ships are highly concentrated and can serve as acoustic fingerprints unique to individual vessels. These line spectra are key signatures used by modern passive sonar systems for target detection, tracking, and identification. Dynamic vibration absorbers are commonly used to suppress low-frequency line spectra. Dynamic vibration absorbers are widely used in ship hydraulic pumps, auxiliary cooling pumps, and floating isolation mats. These devices are typically designed based on linear theory and are effective mainly for suppressing a single stable line spectrum. However, they suffer from fixed tuning frequencies and narrow operating bandwidths. When the excitation frequency deviates from the absorber’s natural frequency, the suppression performance deteriorates significantly and may even amplify the vibration. Moreover, once the component parameters of a linear vibration absorber, such as mass, damping, and spring stiffness, are fixed, they are difficult to adjust, resulting in poor robustness under varying external excitations and limited adaptability to changing operating conditions of marine machinery.

By introducing nonlinear elements into dynamic vibration absorbers, nonlinear absorbers can be constructed that adaptively adjust the resonance frequency and broaden the absorption bandwidth without introducing new resonance peaks. Although the mechanisms used to realize nonlinear restoring forces vary, these nonlinear absorbers can generally be categorized as nonlinear energy sink (NES).^1,2 NES can transfer energy from the primary structure in a predominantly irreversible manner and dissipate it through damping in the attachment. ³ This phenomenon, known as targeted energy transfer, is the core mechanism underlying vibration suppression by an NES and constitutes its essential distinction from linear vibration absorbers. In addition, because an NES does not possess a linearized natural frequency, it can shift the resonance characteristics of the coupled system over a broad frequency range. As a result, high energy absorption and dissipation efficiency can be achieved over a wide frequency range. ⁴ This feature makes NESs promising for suppressing low-frequency line spectra under varying excitation conditions in marine machinery.

Extensive studies have been conducted on the vibration-reduction mechanisms, structural design, and engineering applications of NES. ⁵ Regarding vibration-reduction mechanisms, Lee et al. investigated the dynamics of a two-degree-of-freedom vibration-absorption system coupled with an ideal cubic-stiffness NES. They used the non-stationary transformation method to analyze the nonlinear modes of the undamped Hamiltonian system and presented a frequency-energy diagram, which represents the backbone curves and bifurcation phenomena of different periodic solutions. They pointed out that internal resonances, such as 1:1, 1:3, or even any frequency ratio, can occur between the two oscillators, and they also discovered that transient resonance capture can lead to localized vibration energy in the system. ⁶ Starosvetsky et al. employed the complex-variable averaging method and phase trajectory method to perform rigorous mathematical analysis on the equilibrium point types and bifurcation phenomena of the NES system. They indicated that, depending on the excitation amplitude, the system could exhibit three distinct responses: steady-state response, weak modulation, and strong modulation. ⁷ When energy transfer phenomena occur in the system, it corresponds to a strong modulation response mechanism, which was further used to suppress the limit cycle motion of the Van der Pol oscillator.^8,9 Alexander et al. used numerical continuation to identify a ring-shaped isolated branch in the amplitude-frequency response of the NES system. When the system response lies within the main resonance branch at high frequencies, the NES attenuates the vibration amplitude of the primary structure. However, when the system is on the solitary branch response at low frequencies, it does not exhibit vibration absorption performance. They also noted that when the isolated branch coexists with the main resonance response over a broad parameter range, the NES cannot outperform a linear vibration absorber. ¹⁰ In terms of structural design, Shudeifat et al. designed a NES structure using two slanted linear springs that simultaneously provide cubic stiffness and linear negative stiffness. The shock isolation efficiency of this structure can reach up to 99%. ¹¹ In another study, Al-Shudeifat and co-workers designed an NES based on two pairs of asymmetrically arranged ring-shaped permanent magnets, which also showed excellent vibration suppression performance. ¹² Lamarque et al. designed a bilateral collision oscillator with a non-smooth potential to suppress the vibration of a variable-mass single-degree-of-freedom primary structure. They used the multi-scale method to analyze the steady-state equilibrium points and bifurcation properties of the system. ¹³ Yao et al. designed a cubic stiffness NES structure by paralleling permanent magnets with coil springs, which was used to suppress the eccentric motion of a rotor. ¹⁴ In terms of engineering applications, Viguie et al. demonstrated the effectiveness of the NES in damping torsional vibrations of oil drilling columns, thereby enhancing the stability of the system. ¹⁵ Ahmadabadi and colleagues further validated through numerical calculations that this structure could suppress most of the limit cycle oscillations. ¹⁶ Yang et al. considered the vibration response of a cantilevered fluid-carrying pipeline with an added NES structure at both ends and provided the critical flow velocity at which the NES exhibits wide-frequency performance. ¹⁷ Mamaghani et al. further studied the vibration reduction effect of the NES on the above structure under harmonic excitation conditions. ¹⁸ The U.S. Air Force Research Laboratory, in collaboration with scholars from the University of Illinois at Urbana-Champaign, conducted research on the vibration control of large aircraft wing flutter using NES technology. ¹⁹ This technology was successfully applied in engineering practice, solving the issue of aerodynamic instability and limit cycle oscillations in two-dimensional rigid wings under subsonic conditions. ²⁰ Recent advances in the dynamic characterization and control of nonlinear systems have provided an important theoretical foundation for vibration suppression technologies. For instance, Wang et al.^21,22 conducted extensive studies on the dynamic characteristics of Duffing-Holmes systems with nonlinear damping under harmonic excitation, revealing complex bifurcation behaviors and stability boundaries that are critical to understanding nonlinear energy dissipation mechanisms. Furthermore, in the field of active and semi-active control, Wang et al.^23,24 investigated the dynamic characteristics and stability of electromagnetic SD-oscillator isolation systems under state-feedback control while explicitly accounting for the nonlinear characteristics of electromagnetic forces. These studies highlight the importance of accurately modeling nonlinear stiffness and damping and of employing robust analytical methods to predict system responses under various excitation conditions.

NES, a class of energy-dissipating vibration mitigation devices characterized by inherent stiffness nonlinearity, have attracted considerable attention in civil, aerospace, and mechanical engineering because of their light weight, simple structure, robustness, and broad absorption bandwidth. ²⁵ However, most existing NES configurations are laterally arranged with respect to the primary system, which may introduce large frictional damping and pronounced gravitational effects of the attached mass. Furthermore, the application of NES technology to ship vibration and noise reduction remains limited.^26–29 Meanwhile, existing cubic-stiffness NES configurations, such as magnetic, inclined-spring, and buckled-beam types, still exhibit significant limitations under the complex operating conditions associated with controlling low-frequency line spectra in marine machinery and equipment: magnetic NESs are susceptible to environmental magnetic-field interference, permanent magnets are prone to demagnetization, and activation typically requires a high excitation threshold; inclined-spring NESs rely on installation geometry to realize nonlinear stiffness, resulting in dispersed layouts and pronounced gravitational effects, which make them difficult to accommodate within the compact installation space of marine equipment; and traditional buckled-beam NESs lack a design that prevents permanent buckling, which can lead to plastic deformation under large-amplitude vibrations. In addition, their stiffness is not readily adjustable and they lack a gravity-compensation mechanism, causing the operating point to deviate easily from the static equilibrium position. In addition, most of the aforementioned configurations adopt laterally coupled designs, lack vertical load-bearing capacity, and exhibit relatively high friction damping, making them difficult to match the vibration characteristics and operating requirements of marine equipment. To address these issues, this paper proposes a novel dual-slotted flexible hinge-type nonlinear energy sink. The dual-slotted structural design effectively prevents permanent buckling deformation caused by large-amplitude vibrations, thereby significantly enhancing the operational stability and service life of the device. In addition, it integrates a preload-adjustment mechanism and a helical-spring height-adjustment mechanism, enabling precise control of nonlinear stiffness and efficient gravity compensation for different added masses and thereby overcoming the traditional difficulties of cumbersome parameter tuning and operating-point drift. The design combines structural compactness with adjustability, thereby preserving the cubic-stiffness characteristics and the efficiency of targeted energy transfer. In engineering applications, it is compatible with the vertical vibration characteristics and compact installation requirements of marine equipment, offering improved robustness and practicality. This design provides a more practical engineering solution for the broadband and efficient control of low-frequency line spectra in marine machinery and equipment.

Therefore, there is a need for an NES with adjustable stiffness, a compact structure, and adequate vertical load-bearing capacity. In this context, this paper proposes a dual-grooved flexible hinge-type cubic-stiffness NES based on the parallel combination of positive and negative stiffness. This paper investigates the vibration suppression performance of the NES under harmonic excitation and optimizes its parameters using a local optimization algorithm. Frequency-sweep and fixed-frequency experiments are conducted to verify the vibration suppression performance of the NES over a broad frequency range. The goal is to achieve broadband suppression of low-frequency line spectra and to overcome the performance limitations of linear vibration absorbers used in ship applications.

2. Static mechanical analysis of the flexible hinge and design of the corresponding nonlinear energy sink structure

2.1. Static mechanical characteristics analysis of a single flexible hinge

The loading and deformation of a single flexible hinge are illustrated in Figure 1. The middle section can be considered as a rigid beam, while the grooved sections at both ends can be viewed as elastic beams. The elastic beams undergo buckling deformation under the axial preload F _X and longitudinal load F _Y , with its static characteristics are governed by the bending deformation of the elastic beam, the load-transfer behavior of the rigid beam, and the regulating effect of the preload.

Figure 1.

Force and deformation diagram of a single flexible hinge.

This subsection derives an analytical expression for the longitudinal stiffness by mechanically modeling the buckling deformation process of the flexible hinges (see the Appendix for the detailed derivation) and simplifies the stiffness characteristics by means of a Taylor-series expansion. The results show that the longitudinal restoring force of a single flexible hinge is strongly linearly correlated with the preload. When the stiffness expression is expanded in a Taylor series about the static equilibrium position (y = 0), retaining only the first two terms is sufficient to satisfy engineering accuracy requirements.

Further analysis of the axial deformation of the elastic beam reveals that it consists of both motion-induced deformation and compressive deformation. After the axial-deformation expression is expanded in a Taylor series about the static equilibrium position and the first three terms are retained, the higher-order coefficients (A ₄ , A ₅ , and A ₆ ) can be neglected in comparison with the third-order coefficient A ₃ . Therefore, the axial deformation of the flexible hinge can be regarded as being dominated by the third-order term, which constitutes the fundamental physical basis for realizing cubic nonlinear stiffness in the subsequent flexible hinge-type NES.

The shape coefficient γ (the length ratio of the rigid beam to the elastic beam) is a key structural parameter affecting the stiffness characteristics of the flexible hinge. Analysis of the relationship between the coefficients of each order and the shape coefficient shows that γ directly determines the magnitude of the third-order coefficient A ₃ , thereby governing the strength of the nonlinear stiffness of the flexible hinge and providing an important parametric basis for the future design and optimization of the cubic stiffness of the nonlinear energy sink.

2.2. Overall static mechanical characteristics analysis of the structure

Flexible hinges are generally used in pairs in the present design. Under the action of the preload adjustment mechanism, the preload applied to the right end of the flexible hinge can be expressed as:

$p=\frac{k_h\left(x_0-2\delta x\right)}{s}$

(1)

where $k_h$ is the preload adjustment spring stiffness, $x_0$ is the initial compression amount, and $s$ is the structural logarithm of the flexible hinge. Additionally, $k_h=K_h{L_n}^3/EI$ , and $x_0=X_0/L_n$ After the vertical helical spring is added in parallel, the overall restoring characteristics of the NES structure can be expressed as:

$f_m=k_v\delta y+f_y=k_v\delta y+2s\times \left(A_1-A_2\frac{k_h\left(x_0-2A_3\delta y^2\right)}{s}\right)\delta y={\sigma }_1\delta y+{\sigma }_2\delta y^3$

(2)

where, ${\sigma }_1=k_v+2s{A}_1-2A_2{k}_h{x}_0$ , ${\sigma }_2=4A_2{A}_3{k}_h$ , Here, $k_v$ is the stiffness of the vertical helical spring, and $f_m$ is the total longitudinal restoring force of the NES structure. If ${\sigma }_1=0$ , the critical preload amount of the preload adjustment spring is given by:

${x_0}_{cr}=\frac{k_v+2s{A}_1}{2k_h{A}_2}$

(3)

For the parameters $\gamma =1$ , $s=2$ , $k_v=5$ and $k_h=10$ , as shown in Figure 2, when the initial preload of the adjustment spring equals the critical value, the linear term in Eq. (2) can be neglected, and the NES structure exhibits approximately cubic stiffness. Additionally, as shown in Figures 3 and 4, the cubic stiffness characteristics are mainly influenced by the shape factor $\gamma$ and the stiffness of the adjustment spring $k_h.$ Decreasing $\gamma$ or increasing $k_h$ can effectively enhance the nonlinear stiffness. The core regulating mechanism of this structure is to adjust the equivalent linear stiffness near the static equilibrium position by changing the preload of the horizontal spring. When the central mass block undergoes a vertical displacement, the connecting rod forms an angle with the horizontal direction, causing the preloaded spring to generate a negative stiffness component in the vertical direction. This component increases with increasing preload and offsets the positive stiffness provided by the vertical support and the torsion spring. As a key parameter controlling the potential topology and steady-state regime of the system, the preload enables continuous switching among positive-stiffness, zero-stiffness, and negative-stiffness configurations. As the preload increases, the negative-stiffness effect of the horizontal preload spring gradually strengthens and can eventually compete with, or even exceed, the positive-stiffness component, thereby becoming the dominant contribution to the equivalent stiffness.

Figure 2.

Relationship curve between longitudinal restoring force and displacement.

Figure 3.

Influence of preload adjustment spring stiffness on the cubic stiffness characteristics of the nonlinear energy sink.

Figure 4.

Influence of flexible hinge shape factor on the cubic stiffness characteristics of the nonlinear energy sink.

2.3. Structural design of the flexible hinge-type nonlinear energy sink

He structural design of the flexible hinge-type NES is shown in Figure 5. In this configuration, the symmetrically distributed flexible hinges are arranged in parallel with vertical helical springs. A preload is applied at both ends of the flexible hinge, while the additional NES mass is attached at its center. The structure includes two adjustment mechanisms: a preload adjustment mechanism and a helical-spring height adjustment mechanism. The former is used to tune the stiffness characteristics of the flexible hinge under buckling, whereas the latter compensates for the weight of different attached masses so that the NES structure remains close to its static equilibrium position. In addition, the flexible hinge adopts a double-groove design and can be divided into rigid-beam and flexible-beam segments. The main purpose of this design is to prevent permanent buckling deformation caused by impacts or large-amplitude vibrations, thereby improving the operational stability and service life of the NES.

Figure 5.

Schematic of flexible hinge-type nonlinear energy sink structure.

Figure 6 shows the configuration of the primary system coupled with the NES. The NES consists of a pair of rigid-link hinges and an internal mass $m_2$ , while vertical springs are used to compensate for gravity. Both ends of the link hinge are fixed, and an axial preload, and an axial force $P=k_h(x_1+x_0)$ is applied at one end, where $M_0=M_C+F_X{Y}_C+F_Y{X}_C$ denotes the stiffness of the horizontal spring installed between the preloaded slider and the moving slider, $M_C=F_X\frac{L_s}{2}\sin \hspace{0.17em}\theta +F_Y\frac{L_s}{2}\cos \hspace{0.17em}\theta$ denotes the horizontal displacement of the preloaded slider, and $X_C=\frac{2L_n+L_s-\delta X-L_s\hspace{0.17em}\cos \hspace{0.17em}\theta }{2}$ denotes the inward horizontal displacement of the slider induced by the inclination of the connecting rod. By varying the axial force $P,$ the NES can be configured as either a bistable NES or a cubic-stiffness NES. For convenience, the main vibrating structure is referred to as the primary system, and the NES is referred to as the secondary system.

Figure 6.

Schematic diagram of the vibration absorption system with coupled nonlinear energy sink.

The equations of motion of the coupled NES system are written as follows:

$\{\begin{array}{l}{m}_1{\ddot{y}}_1+c_1{\dot{y}}_1+k_1{y}_1+c_v\left({\dot{y}}_1-{\dot{y}}_2\right)+k_2{\left(y_1-y_2\right)}^3=F\hspace{0.17em}\cos \left(\mathrm{\Omega }T\right)\\ m_2{\ddot{y}}_2+c_v\left({\dot{y}}_2-{\dot{y}}_1\right)+k_2{\left(y_2-y_1\right)}^3=0\end{array}$

(4)

The NES is attached to the primary system, which consists of mass $m_1$ , linear spring $k_1$ , and damper $c_1$ The NES is coupled to the primary system and includes mass $m_2$ stiffness $k_2$ and damping $c_v$ . The external excitation signal is $F\hspace{0.17em}\cos \left(\mathrm{\Omega }T\right)$ , with the vertical displacements generated by the primary system and the nonlinear energy sink being $y_1$ and $y_2$ respectively. The restoring force of the NES is $f_{nes}=k_2{\left(y_2-y_1\right)}^3$ , and $l_0$ is the static displacement of the linear spring $k_1$ under the action of mass $m_1$ . Introducing the following variable substitutions, Eq. (16) can be nondimensionalized as follows:

$\{\begin{array}{l}\epsilon =\frac{m_2}{m_1},{\mathrm{\Omega }}_1=\sqrt{\frac{k_1}{m_1}},\alpha =\frac{k_2}{m_1},{\xi }_1=\frac{c_1}{m_1{\mathrm{\Omega }}_1},{\xi }_2=\frac{c_v}{m_1{\mathrm{\Omega }}_1}\\ \omega =\frac{\mathrm{\Omega }}{{\mathrm{\Omega }}_1},f=\frac{F}{l_0{k}_1},\beta =\frac{k_2{l}_0^2}{k_1},z_1=\frac{y_1}{l_0},z_2=\frac{y_2}{l_0},t={\mathrm{\Omega }}_{1}T\end{array}$

(5)

Substituting Eq. (5) into the governing equations yields:

$\{\begin{array}{l}{\ddot{z}}_1+{\xi }_1{\dot{z}}_1+z_1+{\xi }_2\left({\dot{z}}_1-{\dot{z}}_2\right)+\beta {\left(z_1-z_2\right)}^3=f\hspace{0.17em}\cos \hspace{0.17em}\omega t\\ {\ddot{z}}_2+\frac{{\xi }_2}{\epsilon }\left({\dot{z}}_2-{\dot{z}}_1\right)+\frac{\beta }{\epsilon }{\left(z_2-z_1\right)}^3=0\end{array}$

(6)

3. Dynamic characteristics analysis of the flexible hinge-type nonlinear energy sink system

3.1. Calculation method for the periodic solution of the system

The Incremental Harmonic Balance (IHB) method is an efficient semi-analytical approach for solving periodic motions in strongly nonlinear systems. It is particularly advantageous for analyzing bifurcation points and boundaries, periodic and subharmonic orbits, chaotic parameter domains, and routes to chaos. ³⁰ Accordingly, the IHB method is employed here to analyze the periodic solutions of the NES system. Considering the dynamic system in Eq. (6), where $\tau =\omega t$ , we can rewrite it in matrix form as follows:

${\omega }^2\mathbf{M}\ddot{\mathbf{q}}+\omega \mathbf{C}\dot{\mathbf{q}}+\left(\mathbf{K}+{\mathbf{K}}_{\mathbf{N}}\left(\mathbf{q}\right)\right)\mathbf{q}=\widehat{\mathbf{F}}\cos \hspace{0.17em}\tau$

(7)

In this system, the unknown variables are $\mathbf{q}={\left[q_1,q_2\right]}^{\mathrm{T}}={\left[z_1,z_2\right]}^{\mathrm{T}}$ , and the excitation condition is $\widehat{\mathbf{F}}={\left[f,0\right]}^{\mathrm{T}}$ . $\mathbf{M}$ , $\mathbf{C}$ , and $\mathbf{K}$ represent the system’s mass matrix, damping matrix, and linear stiffness matrix, respectively. ${\mathbf{K}}_{\mathbf{N}}\left(\mathbf{q}\right)$ is the nonlinear stiffness matrix, and for equation (7), the matrices satisfy the following conditions:

$\mathbf{M}=\left[\begin{array}{ll}1& 0\\ 0& 1\end{array}\right],\mathbf{C}=\left[\begin{array}{ll}{\mathrm{\xi }}_1+{\mathrm{\xi }}_2& -{\mathrm{\xi }}_2\\ -\frac{{\mathrm{\xi }}_2}{\mathrm{\epsilon }}& \frac{{\mathrm{\xi }}_2}{\mathrm{\epsilon }}\end{array}\right],\mathbf{K}=\left[\begin{array}{ll}1& 0\\ 0& 0\end{array}\right]$

${\mathbf{K}}_{\mathbf{N}}\left(\mathbf{q}\right)=\left[\begin{array}{ll}\beta q_1^2-2\beta q_1{q}_2+\beta q_2^2& -\beta q_1^2+2\beta q_1{q}_2-\beta q_2^2\\ -\frac{\beta }{\epsilon }{q}_1^2+\frac{2\beta }{\epsilon }{q}_1{q}_2-\frac{\beta }{\epsilon }{q}_2^2& \frac{\beta }{\epsilon }{q}_1^2-\frac{2\beta }{\epsilon }{q}_1{q}_2+\frac{\beta }{\epsilon }{q}_2^2\end{array}\right]$

The first step of the IHB method is the Newton–Raphson incremental procedure. Let $q_{j0}$ and ${\omega }_0$ be the solution of Eq. (7) at a given time. The neighboring state of the solution can be expressed in incremental form as follows:

$\omega ={\omega }_0+\mathrm{\Delta }\omega ,q_j=q_{j0}+\mathrm{\Delta }{q}_j,j=1,2$

(8)

Substituting Eq. (8) into Eq. (7), performing a Taylor expansion, and neglecting higher-order terms in the increment yields the incremental equation:

${\omega }_0^2\mathbf{M}\mathrm{\Delta }\ddot{\mathbf{q}}+{\omega }_0\mathbf{C}\mathrm{\Delta }\dot{\mathbf{q}}+\left(\mathbf{K}+\mathbf{3}{\mathbf{K}}_{\mathbf{N}}\right)\mathrm{\Delta }\mathbf{q}=\mathbf{R}-\left(2{\omega }_0\mathbf{M}{\ddot{\mathbf{q}}}_0+\mathbf{C}{\dot{\mathbf{q}}}_0\right)\mathrm{\Delta }\omega$

(9)

$\mathbf{R}=\mathbf{F}\hspace{0.17em}\cos \hspace{0.17em}\tau -\left({\mathrm{\omega }}_0^2\mathbf{M}{\ddot{\mathbf{q}}}_0+{\mathrm{\omega }}_0\mathbf{C}{\ddot{\mathbf{q}}}_0+\mathbf{K}{\mathbf{q}}_0+{\mathbf{K}}_{\mathbf{N}}{\mathbf{q}}_0\right){\mathbf{q}}_0=\left[q_{10},q_{20}\right],\mathrm{\Delta }\mathbf{q}=\left[\mathrm{\Delta }{q}_1,\mathrm{\Delta }{q}_2\right]$

where $\mathbf{R}$ is referred to as the residual matrix of the incremental equation. When $q_{j0}$ and ${\omega }_0$ are the exact solution of the system, there exists $\mathbf{R}=\mathbf{0}$ .

The second step of the IHB method is the Ritz–Galerkin harmonic-balance procedure. Because Eq. (7) contains only cubic nonlinear stiffness terms, the steady-state periodic solution can be approximated by a finite superposition of odd-order harmonics in the following Fourier-series form:

$q_{j0}={\displaystyle \sum }_{k=1}^{N_c}{a}_{j\left(2k-1\right)}\hspace{0.17em}\cos \left(2k-1\right)\tau +{\displaystyle \sum }_{k=1}^{N_s}{b}_{j\left(2k-1\right)}\sin \left(2k-1\right)\tau ={\mathbf{C}}_s{\mathbf{A}}_j,j=1,2$

(10)

$\mathrm{\Delta }{q}_j={\displaystyle \sum }_{k=1}^{N_c}\mathrm{\Delta }{a}_{j\left(2k-1\right)}\hspace{0.17em}\cos \left(2k-1\right) \tau +{\displaystyle \sum }_{k=1}^{N_s}\mathrm{\Delta }{b}_{j\left(2k-1\right)}\sin \left(2k-1\right)\tau ={\mathbf{C}}_s\mathrm{\Delta }{\mathbf{A}}_j,j=1,2$

where $N_c$ and $N_s$ represent the truncation orders for the cosine and sine harmonics, respectively. ${\mathbf{C}}_s$ is the harmonic matrix, ${\mathbf{A}}_j$ is the Fourier coefficient matrix for the harmonics, and $\mathrm{\Delta }{\mathbf{A}}_j$ is the Fourier coefficient matrix for the harmonic increment, satisfying the following conditions:

${\mathbf{C}}_s=\left[\cos \hspace{0.17em}\tau ,\cos \hspace{0.17em}3\tau ,\cdots \cos \left(2N_c-1\right)\tau ,\sin \hspace{0.17em}\tau ,\sin \hspace{0.17em}3\tau ,\cdots ,\sin \left(2N_s-1\right)\tau \right]$

${\mathbf{A}}_j={\left[a_{j1},a_{j3},\cdots ,a_{j\left(2N_c-1\right)},b_{j1},b_{j3},\cdots ,b_{j\left(2N_s-1\right)}\right]}^{\mathrm{T}}$

$\mathrm{\Delta }{\mathbf{A}}_j={\left[\mathrm{\Delta } a_{j1},\mathrm{\Delta } a_{j3},\cdots ,\mathrm{\Delta }{a}_{j\left(2N_c-1\right)},\mathrm{\Delta } b_{j1},\mathrm{\Delta } b_{j3},\cdots ,\mathrm{\Delta } b_{j\left(2N_s-1\right)}\right]}^{\mathrm{T}}$

Let $\mathbf{A}={\left[{\mathbf{A}}_1,{\mathbf{A}}_2\right]}^{\mathrm{T}}$ , $\mathrm{\Delta }\mathbf{A}={\left[\mathrm{\Delta }{\mathbf{A}}_1,\mathrm{\Delta }{\mathbf{A}}_2\right]}^{\mathrm{T}}$ , $\mathbf{S}=\text{diag}\left[{\mathbf{C}}_s,{\mathbf{C}}_s,\cdots ,{\mathbf{C}}_s\right]$ , then the system’s response and its increment can be represented as:

${\mathbf{q}}_0=\mathbf{SA},\mathrm{\Delta }\mathbf{q}=\mathbf{S}\mathrm{\Delta }\mathbf{A}$

(11)

Substituting equation (11) into the first equation of the incremental equation (9), and applying the Galerkin process over one time period, we obtain a system of linear algebraic equations for the increments $\mathrm{\Delta }\mathbf{A}$ and the frequency increment $\mathrm{\Delta }\omega$ :

${\overline{\mathbf{K}}}_{\mathbf{mc}}\mathrm{\Delta }\mathbf{A}=\overline{\mathbf{R}}-{\overline{\mathbf{R}}}_{\mathbf{mc}}\mathrm{\Delta }\omega$

(12)

where:

${\overline{\mathbf{K}}}_{\mathbf{mc}}={\int }_0^{2\pi }{\mathbf{S}}^{\mathrm{T}}\left[{\omega }_0^2\mathbf{M}\ddot{\mathbf{S}}+{\omega }_0\mathbf{C}\dot{\mathbf{S}}+\left(\mathbf{K}+3{\mathbf{K}}_{\mathbf{N}}\right)\mathbf{S}\right] \mathrm{d}\tau ,$

$\overline{\mathbf{R}}={\int }_0^{2\pi }{\mathbf{S}}^{\mathrm{T}}\left\{\widehat{\mathbf{F}}\cos \hspace{0.17em}\tau -\left[{\omega }_0^2\mathbf{M}\ddot{\mathbf{S}}+{\omega }_0\mathbf{C}\dot{\mathbf{S}}+\left(\mathbf{K}+{\mathbf{K}}_{\mathbf{N}}\right)\mathbf{S}\right]\right\}\mathrm{d}\tau \mathbf{A},{\overline{\mathbf{R}}}_{\mathbf{mc}}={\int }_0^{2\pi }{\mathbf{S}}^{\mathrm{T}}\left(2{\omega }_0\mathbf{M}\ddot{\mathbf{S}}+\mathbf{C}\dot{\mathbf{S}}\right)\mathrm{d}\tau \mathbf{A}$

The computational accuracy of the IHB method depends on the prescribed number of harmonic terms used in the periodic solution. When the truncation orders for the harmonics are set to $N_c=N_s=2$ , the solution expression for equation (12) can be written as:

$\{\begin{array}{l}{z}_1=q_1=a_{11}\hspace{0.17em}\cos \hspace{0.17em}\tau +a_{13}\hspace{0.17em}\cos \hspace{0.17em}3\tau +b_{11}\hspace{0.17em}\sin \hspace{0.17em}\tau +b_{13}\hspace{0.17em}\sin \hspace{0.17em}3\tau \\ z_2=q_2=a_{21}\hspace{0.17em}\cos \hspace{0.17em}\tau +a_{23}\hspace{0.17em}\cos \hspace{0.17em}3\tau +b_{21}\hspace{0.17em}\sin \hspace{0.17em}\tau +b_{23}\hspace{0.17em}\sin \hspace{0.17em}3\tau \end{array}$

(13)

where $\tau =\omega t$ , $a_{ij}$ and $b_{ij}$ are the unknown harmonic coefficients, with $i=1,2$ and $j=1,3$ . Newton-Raphson iteration is terminated when the error falls below 1×10^-10, and for the periodic orbit stability analysis, $N_k$ is set to 10,000. To characterize the response amplitude of periodic motion containing higher-order harmonics, the root-mean-square value of the harmonic coefficients is used as an approximate measure:

$z_{ir}=\sqrt{a_{ij}^2+b_{ij}^2}$

(14)

The stability of the periodic solutions obtained by the IHB method is determined on the basis of Floquet theory. Stability is assessed by calculating the Floquet multipliers (i.e., the eigenvalues of the monodromy matrix) associated with the linearized perturbation equation about the periodic solution. ³¹

By selecting the parameters $\epsilon =0.005$ , ${\xi }_1=1.2\mathrm{e}-3$ , ${\xi }_2=1.2\mathrm{e}-3$ , $\beta =1.5$ , $f=0.01$ the amplitude-frequency characteristic curves obtained using the IHB method and the Runge-Kutta method for the NES response are shown in Figure 7. Figure 7 shows that when the number of harmonic terms is set to $N_c=N_s=2$ , the results of the two methods agree well over most of the frequency range. However, small discrepancies appear within the interval $\omega \in \left(1.09,1.25\right)$ . The discrepancy can be attributed mainly to two factors: first, the IHB method includes an insufficient number of harmonic terms and therefore cannot capture responses beyond the third harmonic; second, the Runge-Kutta method also introduces numerical error, although the former effect is dominant. Although increasing the number of preset harmonic terms in the IHB method can reduce the computational error, it significantly increases the computation time. Overall, the discrepancy between the two methods is small on the scale shown in the figure.

Figure 7.

Amplitude-frequency characteristics curve of the nonlinear energy sink response.

3.2. Vibration suppression effect analysis of nonlinear energy sink

To evaluate line-spectrum suppression under varying excitation conditions, two indicators are considered: resonance-peak attenuation and the vibration absorption bandwidth of the NES for the mechanical equipment. ³² Common evaluation methods mainly include the generalized transfer-rate method based on the nonlinear frequency-response function and the classical energy-analysis method. The generalized transfer-rate method is mainly suitable for NES systems under base excitation, whereas the energy-based method is more general. For the system described by equation (6), the total energy ${\mathrm{E}}_{\text{main}}$ stored in the primary system over one period $T_0=2\pi /\omega$ is used to assess the vibration suppression performance of the NES.

It should be noted that since the Incremental Harmonic Balance (IHB) method can only obtain the system’s steady-state periodic solution, when the periodic solution becomes unstable, the system may exhibit quasi-periodic or chaotic responses. In such cases, the unstable response obtained by the IHB method will differ from the true response of the system. For unstable periodic solutions, the integration duration $T_0$ should be adjusted according to the actual response frequency of the system. Therefore, to reduce analysis errors, ${\mathrm{E}}_{\text{main}}$ is taken as the root mean square value of the system’s response within one period $T_0$ after the system’s response stabilizes.

For the case without a coupled NES, the equation of motion of the primary system is given by:

${\ddot{z}}_1+{\xi }_1{\dot{z}}_1+z_1=f\hspace{0.17em}\cos \hspace{0.17em}\omega t$

(15)

The steady-state periodic solution can be assumed in the pre-set form $z_1=a_1\hspace{0.17em}\mathit{cos}\hspace{0.17em}\omega t+b_{1}sin\omega t$ by substituting into equation (15), the harmonic coefficients are obtained as:

$a_1=\frac{f\left(1‐{\omega }^2\right)}{{\left(1‐{\omega }^2\right)}^2+{\xi }_1^2{\omega }^2},b_1=\frac{f{\xi }_1\omega }{{\left(1‐{\omega }^2\right)}^2+{\xi }_1^2{\omega }^2}$

(16)

Further, the steady-state vibration energy of the system can be accurately expressed by the maximum potential energy equation:

${\mathrm{E}}_{\text{main}}^{\prime }=\frac{a_1^2+b_1^2}{2}=\frac{f^2}{2\left({\left(1-{\omega }^2\right)}^2+{\xi }_1^2{\omega }^2\right)}$

(17)

First, the influence of NES damping is discussed. Keeping the parameters fixed as $\epsilon =0.05$ , ${\xi }_1=1.2\mathrm{e}-3$ , $\beta =0.5$ and $f=0.01$ , the system’s vibration energy curve under different damping conditions is shown in Figure 8. As shown in the figure, when no NES is coupled to the system, the primary system exhibits a distinct resonance peak at its natural frequency $\omega =1$ . When ${\xi }_2=0.012$ , the bifurcation points of the system’s energy curve are at 0.9759 and 1.037. In the unstable section between the bifurcation points, the vibration energy is lower than that of the condition without the coupled NES, but there is also a distinct distorted ring-shaped unstable energy curve (shown in red in part (b) of the figure), which leads to the vibration energy in the range $1.037\le \omega \le 1.045$ being higher than that of the condition without the coupled NES. At this point, the NES not only fails to suppress vibration but also aggravates the vibration of the mechanical equipment. When the damping ${\xi }_2=0.02$ is slightly increased to 0.02, the distorted ring-shaped branch disappears and, near resonance, the system merges into a single unstable branch. At this point, the system’s bifurcation points are at 0.9806 and 1.025. Between these bifurcation points, the NES exhibits good broadband vibration absorption performance. However, further increasing the damping of the NES to ${\xi }_2=0.05$ and ${\xi }_2=0.12$ causes the system’s energy curve to lose the unstable periodic solution completely, and the system response approaches the characteristics of a linear system, showing a single resonance peak near the new resonance frequency. Furthermore, when ${\xi }_2=0.05$ , the system’s resonance frequency is 0.9852 with a resonance energy of 0.0374; when ${\xi }_2=0.12$ , the resonance frequency is 0.9781 with a resonance energy of 0.1376. Therefore, it can be concluded that as the damping is increased again, the system’s resonance frequency shifts to the left, and the vibration energy near the new resonance peak increases. The NES loses its broadband vibration absorption ability. Thus, the damping of the nonlinear energy sink can be categorized into four types: very low damping ( ${\xi }_2=0.012$ ), weak damping ( ${\xi }_2=0.02$ ), strong damping ( ${\xi }_2=0.05$ ), and excessive damping ( ${\xi }_2=0.12$ ). These results indicate that under very low damping, the system’s nonlinear dynamics are complex, and the NES does not provide vibration suppression within a local frequency range. Under weak damping conditions, the NES exhibits broadband vibration absorption performance. Under strong damping conditions, the NES is effective in suppressing resonance peaks but has limited low-frequency vibration absorption capability. In conditions of excessive damping, the NES can still suppress some resonance peaks but results in a degraded vibration suppression effect near the new resonance peak.

Figure 8.

Influence of nonlinear energy sink damping on vibration suppression effects. (a) Frequency range $0\le \omega \le 2$ , (b) local enlarged view $0.95\le \omega \le 1.05$ .

Next, the influence of the NES mass is discussed. Keeping the parameters fixed as ${\xi }_1=0.0012$ , ${\xi }_2=0.02$ , $\beta =0.5$ and $f=0.01$ , the system’s vibration energy curve under different mass ratio conditions is shown in Figure 9. As seen in Figure 9(a), when $\epsilon =0.167$ , a small curved unstable branch still exists near the resonance frequency. When the mass ratio is slightly reduced to $\epsilon =0.083$ , the curved branch degenerates into a smooth unstable branch. As the mass ratio continues to decrease, the unstable solution region near the resonance frequency gradually widens, and the system’s vibration energy continuously decreases, thus enabling the NES to provide stronger broadband vibration suppression effects. As shown in Figure 9(b), when the mass ratio is further reduced from 0.04, the system’s solution branches split, forming multiple unstable branches, and the system’s vibration energy increases sharply. If the mass ratio is reduced further, the system begins to exhibit linear characteristics and generates a new single resonance frequency. The frequency value gradually approaches $\omega =1$ , and the resonance peak energy also increases, indicating that the NES no longer provides broadband vibration absorption effects.

Figure 9.

Influence of nonlinear energy sink mass on vibration suppression effects. (a) $0.05\le \epsilon \le 0.167$ , (b) $0.0125\le \epsilon \le 0.04$ .

Finally, the influence of the NES stiffness is discussed. Keeping the parameters fixed as $\epsilon =0.05$ , ${\xi }_1=0.0012$ , ${\xi }_2=0.02$ and $f=0.01$ , the system’s vibration energy curve with changes in nonlinear energy sink stiffness is shown in Figure 10. From the figure, it can be seen that when the stiffness is 0.0625 and 0.125, the system exhibits linear characteristics, and the resonance peak energy decreases gradually as stiffness increases. When the stiffness is further increased to 0.25, 0.5, and 1, the system begins to show unstable solutions. The unstable solution interval widens gradually as stiffness increases, and the energy amplitude decreases, indicating that the broadband vibration absorption performance of the NES near resonance is gradually enhanced with increasing stiffness. When the nonlinear stiffness is increased further, the curve begins to split again, generating multiple unstable solutions. The system’s characteristics become more complex, and high-energy resonance peaks are generated. At this point, the NES essentially loses its vibration absorption capability.

Figure 10.

Influence of nonlinear energy sink stiffness on vibration absorption effect. (a) $0.0625\le \beta \le 1$ , (b) $2\le \beta \le 4$ .

4. Local parameter optimization of the flexible hinge-type nonlinear energy sink system

4.1. Nonlinear energy sink parameter optimization

To determine the optimal damping and stiffness of the NES, the optimization variables and their ranges are defined as follows:

$0\le {\xi }_2\le 0.06,0\le \beta \le 2$

(18)

The optimization objective is to minimize the maximum response amplitude and vibration energy of the mechanical equipment near resonance for each parameter set $({\xi }_2,\beta )$ . The optimization functions are:

${\mathrm{H}}_1=\min \left(\max \hspace{0.5em}{\mathrm{z}}_{1\mathrm{r}}\right)$

(19)

${\mathrm{H}}_2=\min \left(\max \hspace{0.5em}{\mathrm{E}}_{\text{main}}\right)$

(20)

Because the above objective functions do not admit explicit analytical expressions, direct local optimization is carried out using the Runge-Kutta method combined with an enumeration procedure. The initial conditions are set to 0, the excitation frequency range is $0.8\le \omega \le 1.2$ the frequency sweep interval is 0.01, the interval for ${\xi }_2$ is set to 0.0025, and the interval for $\beta$ is set to 0.025.

When the mass ratio is taken as $\epsilon =0.04$ , the 3D plot of the maximum amplitude and maximum vibration energy of the mechanical equipment is shown in Figure 11, and the 2D contour plot in the $\left({\xi }_2,\beta \right)$ plane is shown in Figure 12. As seen in Figure 11 without the coupling of the NES, the maximum amplitude is 5.80, and the maximum vibration energy is 33.68. After coupling the NES, the maximum response amplitude and vibration energy of the mechanical equipment are reduced over most of the parameter range. Figure 12 shows that under both optimization criteria, the optimized regions corresponding to amplitude and energy are within 0.1 and 0.01, respectively. If the NES is designed with parameters within this optimized region, compared to the uncoupled case, the resonance peak amplitude of the mechanical equipment is reduced by 98.28%, and the maximum vibration energy is attenuated by 99.97%. Moreover, the optimized parameters are mainly concentrated in the weak damping region of $0.01\le {\xi }_2\le 0.02$ . In Figure 12(a) and (b), there exist contour lines with maximum amplitudes of 0.5 and maximum vibration energy of 0.1, respectively. This indicates that when the stiffness parameter increases beyond the optimized region, the vibration suppression performance of the NES deteriorates sharply. Therefore, the vibration suppression performance is more sensitive to changes in stiffness than to changes in damping.

Figure 11.

3D plot of the maximum amplitude or maximum vibration energy of the mechanical equipment versus damping and stiffness when the mass ratio is ε = 0.04.

Figure 12.

Contour plot in the 2D plane of the two optimization results when the mass ratio is ε = 0.04.

If the mass ratio is taken as $\epsilon =0.05$ , the calculation results are shown in Figures 13 and 14. At this point, the corresponding amplitude and energy in the optimization region are within 0.08 and 0.008, respectively, indicating that the resonance peak amplitude of the mechanical equipment is reduced by 98.62%, while the maximum vibration energy attenuation efficiency reaches 99.98%. Therefore, compared to $\epsilon =0.04$ , the vibration suppression effect of the NES is slightly improved, and the range of the optimization region has significantly increased, greatly enhancing the robustness of the NES parameter design.

Figure 13.

3D plot of the maximum amplitude or maximum vibration energy of the mechanical equipment versus damping and stiffness when the mass ratio is 0.05. (a) Maximum amplitude of the mechanical equipment, (b) maximum vibration energy of the mechanical equipment.

Figure 14.

Contour plot in the 2D plane of the two optimization results when the mass ratio is 0.05.

If the parameters ${\xi }_1=0.0012$ , $f=0.01$ , and $\omega =1$ are kept constant, and the parameter sets $\epsilon =0.04$ , ${\xi }_2=0.0125$ , $\beta =0.5$ and $\epsilon =0.05$ , ${\xi }_2=0.02$ , $\beta =1.15$ are selected from the optimization region, the measured vibration responses of the mechanical equipment are shown in Figures 15 and 16. The results indicate that the mechanical equipment exhibits a quasi-periodic motion state under these parameter conditions.

Figure 15.

Vibration response of the mechanical equipment with $\epsilon =0.04$ , ${\xi }_2=0.0125$ , $\beta =0.5$ . (a) Time history diagram, (b) phase diagram, (c) poincaré section diagram (d) power spectral density diagram.

Figure 16.

Vibration response of the mechanical equipment with $\epsilon =0.05$ , ${\xi }_2=0.02$ , $\beta =1.15$ . (a) Time history diagram, (b) phase diagram, (c) poincaré section diagram (d) power spectral density diagram.

4.2. Sensitivity analysis of nonlinear energy sink parameters

Using the optimized parameters $\epsilon =0.05$ , ${\xi }_2=0.02$ , and $\beta =1.15$ , the sensitivity of NES vibration suppression performance to changes in excitation frequency and amplitude is further analyzed. The excitation frequency is varied within $\pm$ 20% of the natural frequency, whereas the excitation amplitude is varied within $\pm$ 50% of 0.01. The 2D contour plots of the mechanical equipment response amplitude and vibration energy are shown in Figures 17 and 18, respectively. As shown in the figures, when the excitation frequency varies, the NES consistently maintains strong resonance-peak suppression and vibration-energy attenuation compared with the uncoupled system. When the excitation amplitude changes, there is a noticeable increase in the vibration amplitude and vibration energy of the mechanical equipment within the green contour regions in Figures 17 and 18. However, by comparing Figures 17 and 18, it is evident that this increase is primarily due to the elevated input energy from the external excitation. Even in the least favorable parameter region, the NES still achieves approximately 80% suppression of resonance peaks and 98% attenuation of vibration energy. Therefore, with the NES structural parameters properly designed, the system retains excellent vibration suppression performance even when the external excitation frequency and amplitude vary within a certain range. These results demonstrate the broadband vibration absorption capability and robustness of the NES.

Figure 17.

2D contour plots of the mechanical equipment’s response amplitude versus excitation frequency and amplitude. (a) Coupled with nonlinear energy sink, (b) without nonlinear energy sink.

Figure 18.

2D contour plots of the mechanical equipment’s vibration energy versus excitation frequency and amplitude. (a) Coupled with nonlinear energy sink, (b) without nonlinear energy sink.

4.3. Comparison of vibration suppression effects between nonlinear energy sink and equivalent linear vibration absorber

For Eq. (6), the equation of motion of the equivalent linear vibration absorber system can be written as follows:

$\{\begin{array}{l}{\ddot{z}}_1+{\xi }_1{\dot{z}}_1+z_1+{\xi }_l\left({\dot{z}}_1-{\dot{z}}_2\right)+{\beta }_l\left(z_1-z_2\right)=f\hspace{0.17em}\cos \hspace{0.17em}\omega t\\ \epsilon {\ddot{z}}_2+{\xi }_l\left({\dot{z}}_2-{\dot{z}}_1\right)+{\beta }_l\left(z_2-z_1\right)=0\end{array}$

(33)

where ${\xi }_l$ and ${\beta }_l$ correspond to the damping and stiffness parameters of the linear vibration absorber, and $\epsilon$ represents the mass ratio. Applying the Laplace transform to Eq. (33) and performing straightforward manipulations yields the expression for the amplitude amplification factor of the mechanical system response:

$\frac{z_1}{z_{st}}\left(\omega \right)=H\left(\omega \right)=\sqrt{\frac{{\left(-C_l{\omega }^2+1\right)}^2+F_l^2{\left({\xi }_l\omega \right)}^2}{{\left(A_l{\omega }^4-B_l{\omega }^2+1\right)}^2+{\left(-D_l{\omega }^2+E_l\right)}^2{\left({\xi }_l\omega \right)}^2}}$

(34)

where $z_{st}$ represents the initial displacement produced by the excitation force $f$ on the mechanical system in the absence of the vibration absorber. For the case where the mechanical primary system includes damping, the coefficients $A_l$ , $B_l$ , $C_l$ , $D_l$ , $E_l$ , $F_l$ are defined as:

$A_l=\frac{1}{{\omega }_n^2{\mathrm{\Omega }}_n^2},B_l=\left[\frac{1+\epsilon }{{\mathrm{\Omega }}_n^2}+\left(\frac{1}{{\omega }_n^2}\right)\left(1+4{\xi }_l{\xi }_1\right)\right],C_l=\frac{1}{{\omega }_n^2}{D}_l=\left(\frac{2}{{\omega }_n^2{\mathrm{\Omega }}_n}\right)\left(1+\epsilon +\frac{{\xi }_1}{{\xi }_l}\right),$

$E_l=2\left(\frac{{\mathrm{\Omega }}_n}{{\omega }_n^2}+\frac{1}{{\mathrm{\Omega }}_n}\frac{{\xi }_1}{{\xi }_l}\right),F_l=2\frac{{\mathrm{\Omega }}_n}{{\omega }_n^2}$

where ${\omega }_n$ is the natural frequency of the linear vibration absorber, and ${\mathrm{\Omega }}_n$ is the natural frequency of the mechanical primary system. Since both the mass and stiffness of the mechanical primary system are 1, the following relationships hold: $z_{st}=f$ , ${\omega }_n=\sqrt{{\beta }_l/\epsilon }$ , and ${\mathrm{\Omega }}_n=1$ . The vibration suppression effect of the linear vibration absorber is closely related to the variables ${\xi }_l$ , ${\beta }_l$ . Based on the optimal tuning theory for displacement response, the analytical expression for the optimal frequency ratio of the linear vibration absorber is given by:

${\gamma }_{\text{opt}}={\frac{{\omega }_n}{{\mathrm{\Omega }}_n}|}_{\text{opt}}=\frac{1}{1+\epsilon }-\left(0.241+1.74\epsilon -2.6{\epsilon }^2\right){\xi }_1-\left(1-1.9\epsilon +{\epsilon }^2\right){\xi }_1^2$

(35)

With the primary system parameters ${\xi }_1=1.2\mathrm{e}-3$ and $f=0.01$ fixed, the optimization parameters for the NES are chosen as $\epsilon =0.05$ , ${\xi }_2=0.02$ , $\beta =1.15$ , and the optimization parameters for the linear vibration absorber are $\epsilon =0.05$ , ${\beta }_l=\epsilon {\gamma }_{\text{opt}}^2$ , ${\xi }_l={\xi }_2$ . The amplitude-frequency characteristics of the mechanical system’s vibration response are shown in Figure 19. The results show that both the linear vibration absorber and the NES are capable of suppressing resonance peaks, with the performance of the linear vibration absorber even surpassing that of the NES. However, when the excitation frequency moves far from the natural frequency of the primary system, the linear vibration absorber introduces two additional fixed resonance frequency points to the system. In contrast, this phenomenon does not occur after the NES is coupled to the system. Therefore, over a wide frequency range, the NES exhibits superior vibration suppression performance.

Figure 19.

Comparison of amplitude-frequency characteristics between nonlinear energy sink and equivalent linear vibration absorber.

5. Experimental investigation of the flexible hinge-type nonlinear energy sink system

5.1. Prototype of the flexible hinge-type nonlinear energy sink

Figure 20 shows the three-dimensional structure and sectional view of the prototype flexible hinge-type NES. The main components in the figure are shown in Table 1. As shown in the figure, the device consists of four flexible hinges (two pairs). To reduce the overall design size, the flexible hinges are symmetrically folded and mounted through the frame. The The preload is adjusted uniformly by the threaded screw mechanism (3). The vertical helical spring is arranged on the mounting seat, and the height of the mounting seat is adjusted by the worm gear mechanism to meet the different mass-carrying requirements of the NES. The design range of the additional mass is 8-12 kg. When the static equilibrium position and the cubic stiffness state indicator pin are located at the center of their respective indicator holes, this indicates that the NES is in its normal initial operating state. In addition, the main structural parameters of the flexible hinge and the vertical helical spring are listed in Tables 2 and 3, respectively. When the preload adjustment spring is removed, the flexible hinge is no longer preloaded, and the prototype can be regarded as an equivalent linear vibration absorber.

Figure 20.

Three-dimensional structure and sectional view of the prototype nonlinear energy sink.

Table 1.

Main components of the nonlinear energy sink principal prototype.

Label	Component	Label	Component
1	Additional-mass tray	9	Flexible hinge mounting frame
2	Cubic stiffness state indicator hole	10	Preload-adjustment screw
3	Threaded preload-adjustment mechanism	11	Preload-adjustment spring
4	Vertical push rod	12	Static equilibrium position indicator line
5	Vertical helical-spring mounting seat	13	Vertical helical spring
6	Worm gear spring height adjustment mechanism	14	Height adjustment gear
7	Mounting base	15	Housing
8	Flexible hinge	​	​

Table 2.

Main structural parameters of the flexible hinge.

Structural parameter	Value	Structural parameter	Value
Material	Al-6061	Flexible hinge width	50mm
Elastic modulus	72GPa	Elastic beam length	12.7mm
Number of flexible hinges	2	Elastic beam thickness	0.4mm
Shape factor	2.49	Rigid beam length	31.6mm
Total length of flexible hinge	100mm	Rigid beam thickness	2mm

Table 3.

parameters of the vertical helical spring.

Structural parameter	Value	Structural parameter	Value
Material	60Si2Mn	Wire diameter	3mm
Outer diameter	43mm	Free height	75mm
Inner diameter	37mm	Stiffness value	3.16N/mm

5.2. Static mechanical characteristics test of the flexible hinge-type nonlinear energy sink

The static stiffness of the prototype was measured using an electronic universal testing machine is shown in Figure 21. Figure 22 presents the static stiffness test results of the NES prototype. In the figure, the green curve represents the measured data, whereas the red curve represents the cubic polynomial fit. The figure, shows that the prototype exhibits clear nonlinear characteristics near the static equilibrium position (x=3 mm). Additionally, the static stiffness curve exhibits hysteresis during both loading and unloading. This hysteresis may be attributed partly to the measurement precision of the testing machine and partly to the elastic potential energy stored in the preloaded flexible hinge, which makes its stiffness characteristics sensitive to external loading. Although some deviation exists between the fitted and theoretical values at large displacements away from the static equilibrium position, the overall trends of the curves agree well.

Figure 21.

Static mechanical characteristics test of the nonlinear energy sink prototype.

Figure 22.

Static stiffness test results of the nonlinear energy sink prototype. (a) Measured values, (b) theoretical values.

5.3. Dynamic characteristics test of the flexible hinge-type nonlinear energy sink

Photographs of the dynamic test setup are shown in Figure 23. In this setup, the primary system is represented by a single-layer vibration-isolation platform, with a uniformly distributed mass of mass of $m_1$ =200kg used to simulate mechanical equipment. The mass is symmetrically placed on four metal vibration isolators. Rigid blocks are used to support both the metal isolators and the rigid base. The exciter is fixed on the base, with heat dissipation provided by a fan. An impedance sensor is installed at the output end of the exciter, which is connected to the center of the lower surface of the reduced vibration mass via a guide rod. The impedance sensor simultaneously measures the input force and acceleration signals of the exciter. The nonlinear energy sink device is mounted at the center of the upper surface of the reduced vibration mass, with an additional mass $m_2$ =10kg. Accelerometers are installed on the upper surfaces of both mass blocks $m_1$ and $m_2$ . Additionally, a magnetorheological damper is installed between the two mass blocks to adjust the damping of the NES device. When the magnetorheological damper is present, the system operates under high damping conditions, and when it is absent, the system operates under low damping conditions.

Figure 23.

Photos of the test site.

Further dynamic-characterization tests of the nonlinear energy sink system were conducted using the VibMobile vibration testing platform developed by M+P, Germany. The test system mainly includes an exciter, acceleration sensors, an impedance sensor, an impact hammer, magnetic dampers, a controller, a data-acquisition system, a fan, the nonlinear energy sink device, and a single-layer isolation platform. The specific models of the hardware and software used in the experiments are listed in Table 4. The frequency-sweep test uses a logarithmic sweep over 1-100 Hz, with a sweep rate of 1 oct/min, a sampling frequency of 2048 Hz, and a total acquisition time of 400 s.

Table 4.

Main experimental instruments.

Instrument name	Type	Quantity	Manufacturer
Data acquisition system	M+P VibMobile	1	M+P, Germany
Vibration exciter	NTS MS50	1	Zhuoyan, Suzhou, China
Power amplifier	NTS MS50A	1	Zhuoyan, Suzhou, China
Fan	RT-H214AA	1	Beileke, Suzhou, China
Accelerometer	3097A2	2	PCB Piezotronics, USA
Impedance type sensor	288D01	1	PCB Piezotronics, USA
Impact hammer	086C03	1	PCB Piezotronics, USA

First, the gain of the power amplifier was set to 5%, and the corresponding frequency-sweep results are shown in Figure 24. Figure 24(a) shows the output force measured by the impedance sensor under different operating conditions, whereas Figure 24(b) shows the normalized average acceleration-response amplitude of the primary system under the same conditions. As shown in the figure, the first- and second-order natural frequencies of the primary system are approximately 5.05 Hz and 10.85 Hz, with average response amplitudes of 71.85 dB and 59.60 dB, respectively. When an equivalent linear absorber is added, the first-order resonance peak decreases by approximately 12.04 dB, and increasing the damping slightly improves the average vibration-absorption performance over the 4-6 Hz range. However, two new resonance peaks appear at 4.81 Hz and 5.50 Hz, especially near 5.50 Hz, and the vibration response of the primary system deteriorates accordingly. When the nonlinear energy sink is added, the overall response curve of the system shifts to the right, and the device does not exhibit a pronounced vibration-absorption effect under either damping condition. This behavior is mainly attributed to the potential energy stored in the flexible hinge under preload, which makes the nonlinear energy sink difficult to activate under small-amplitude conditions. As a result, the cubic-stiffness characteristics are not fully manifested, and the device cannot effectively absorb the vibration energy of the primary system.

Figure 24.

Frequency sweep test results of the nonlinear energy sink system under relatively lower excitation amplitude. (a) Input force, (b) main system vibration response.

Further increase the power amplifier gain to 10%, the specific frequency sweep test results are shown in Figure 25. From the figure, it can be seen that the first and second natural frequencies of the main system are located at approximately 5.00 Hz and 10.44 Hz, respectively, with average response amplitudes of 73.60 dB and 58.48 dB. When the equivalent linear vibration absorber is attached, the vibration attenuation effect is more pronounced over 4.75-6.82 Hz. Under low- and high-damping conditions, the first-order resonance peak is reduced by approximately 2.93 dB and 3.74 dB, respectively. However, at the same time, the system response exhibits two significant new resonance peaks near 7.63 Hz and 7.19 Hz. When the NES is added, under low damping conditions, the system response curve slightly shifts to the right, and the system remains difficult to activate, and effective vibration attenuation is not yet observed. However, as the damping increases, the vibration absorption performance of the NES changes significantly. On the one hand, the first-order resonance peak near 5.0 Hz decreases by 8.17 dB. On the other hand, over 4.43–5.82 Hz, the system response curve becomes smoother, in the frequency range of 4.43 Hz to 5.82 Hz, the system response curve becomes smoother, with the average line spectrum intensity decreasing by approximately 3.74 dB. The original first-order resonance peak disappears, and no new resonance peaks are generated. Therefore, compared to the equivalent linear vibration absorber, the NES exhibits good broadband vibration absorption performance under high-damping conditions. Taking 5.00 Hz as the reference frequency point, the vibration absorption bandwidth near the first-order resonance peak widens by approximately 27.8%.

Figure 25.

Frequency sweep test results of the nonlinear energy sink system under relatively higher excitation amplitude. (a) Input force, (b) main system vibration response.

Furthermore, after adding the NES, the second-order resonance peak frequency of the system slightly shifts to the right. The system response amplitude near 10.44 Hz is somewhat reduced, but the response amplitude near 9.75 Hz increases. Thus, the broadband vibration absorption advantage of the NES is mainly manifested around the first-order resonance peak, with the vibration absorption effect on the second-order resonance peak being less significant.

With the magnetorheological damper installed, the time-domain acceleration response signal of the primary system at an excitation frequency of 5.00 Hz is shown in Figure 26. The figure shows that when the NES is not added, the average acceleration response amplitude of the primary system is 0.488 m/s², and it exhibits clear periodic motion. After the NES is added, the acceleration response amplitude of the NES becomes larger than that of the primary system. The average acceleration response amplitude of the primary system is approximately 0.195 m/s², which corresponds to a 60.04% reduction in the average response amplitude relative to the system without the NES. Additionally, after adding the NES, the system no longer exhibits simple periodic motion; both oscillators exhibit clear periodic modulation in their response amplitudes.

Figure 26.

Time-domain vibration response of the main system. (a) Without nonlinear energy sink, (b) with nonlinear energy sink.

Figure 27 compares the simulated and experimental frequency-domain acceleration responses of the primary system under different excitation levels. In the figure, the blue curve represents the harmonic-balance simulation results, whereas the green symbols represent the experimental data obtained under the corresponding operating conditions. As shown in the figure, the simulated and experimental curves agree well in terms of resonance-peak position, bandwidth, and response amplitude, thereby verifying the reliability of the theoretical model. The overall response amplitude measured in the experiments is slightly lower than that predicted by the simulations, which may be mainly attributed to additional damping, nonlinear connection stiffness, and parameter-identification errors in the actual system.

Figure 27.

Simulated and experimental results for the frequency-domain acceleration response of the primary system under different excitation levels. (a) Power-amplifier gain = 5%, (b) power-amplifier gain = 10%.

The time-frequency characteristics of the acceleration time series were analyzed using a wavelet transform, as shown in Figure 28. From the figure, it can be seen that when the excitation frequency is 5.00 Hz, the dominant response frequencies of both the primary system and the NES are centered around 5.00 Hz. This indicates that the NES can spontaneously shift the resonance frequency of the coupled system, causing it to enter a 1:1:1 internal resonance with the external excitation frequency. Through this strong modulation response, it periodically absorbs vibrational energy from the primary system over a broad frequency range. Therefore, the 1:1:1 internal resonance is the primary mechanism by which the NES reduces the characteristic line-spectrum level over a broad frequency range.

Figure 28.

Wavelet transform time-frequency plot of the system’s acceleration response at 5.00 Hz. (a) Primary system, (b) nonlinear energy sink.

6. Conclusion

This paper addresses the suppression of low-frequency characteristic line spectra in ship machinery. Through theoretical analysis, numerical simulation, and experimental validation, the vibration suppression performance, parameter optimization, structural design, and testing of the NES system are investigated systematically. This study is of practical significance for improving the acoustic stealth performance of ships. The main findings are summarized as follows.

(1) Based on the principle of positive and negative stiffness in parallel, a preloaded dual-slot flexible hinge-type NES is designed. An analytical expression for the static stiffness is derived, and the critical preload of the adjustment spring is determined. The influence of structural parameters on cubic stiffness characteristics is analyzed, and the static mechanical test results agree well with the theoretical analysis.

(2) The incremental harmonic balance method is used to establish a procedure for computing steady-state periodic solutions of the NES system. The effects of three key design parameters—mass ratio, stiffness, and damping—on vibration suppression under harmonic excitation are analyzed. Using the maximum response amplitude of the mechanical equipment and the minimum peak vibration energy as objective functions, a local optimization algorithm is employed to identify the optimal parameter region of the NES.

(3) The frequency-sweep test results indicate that the NES not only reduces the first-order resonance peak near 5.00 Hz by approximately 8.17 dB, but also provides effective broadband vibration absorption over 4.43–5.82 Hz. The average line-spectrum level is reduced by about 3.74 dB, with no new resonance peaks generated in other frequency bands. The vibration absorption bandwidth is widened by approximately 27.8%. In contrast, the equivalent linear dynamic vibration absorber still generates two significant resonance peaks around 7.63 Hz and 7.19 Hz under harmonic excitation.

(4) The fixed-frequency test results show that the NES can shift the resonance frequency of the coupled system and can cause the system to undergo 1:1:1 internal resonance at the excitation frequency. At this point, both the primary system and the NES exhibit strong modulation responses, which underlie the broadband vibration absorption performance of the NES.