Introduction

The propeller is the main propulsion device of surface ships. When a ship navigates in water, an uneven wake field is inevitably formed at the stern. The rotation of the propeller in the uneven wake field generates pulsating thrust, which is transmitted to the hull through the propulsion shafting, stern bearing, intermediate bearing, thrust bearing and their bases, causing the hull to vibrate and thereby generating underwater acoustic radiation1. In recent years, the longitudinal vibration of the propulsion shaft caused by the unsteady excitation force of the propeller has received extensive attention for its detrimental effect on the acoustic stealth performance of ships2–3.

Structures with periodic variations in geometry or material are generally referred to as metamaterials. A large number of studies have shown that elastic metamaterials can exhibit attenuation domain characteristics for elastic waves, known as bandgap characteristics4. The application of the bandgap characteristics of metamaterials in the vibration reduction of ship shafting has great potential, as it can avoid the shortcomings of traditional vibration absorption techniques, such as poor absorption effect in the low-frequency or ultra-low-frequency bands, as well as issues like bending strength influence and shafting misalignment. According to the formation mechanism of elastic wave or vibration bandgaps, elastic metamaterials can be classified into two types: Bragg scattering type (Bragg) and locally resonant type (Locally resonant)5. The bandgap frequency of Bragg scattering is related to the lattice constant of the metamaterial and the material parameters of the components that make up the metamaterial. Therefore, to obtain a lower-frequency Bragg bandgap, larger basic dimensions of the metamaterial or a base material with a smaller elastic modulus must be used. However, the center frequency of the locally resonant bandgap is only related to the natural frequency of the local resonator and depends only on the inertial and stiffness characteristics of the system. Therefore, the proposal of the locally resonant bandgap mechanism breaks the limitation of the lattice constant and base material parameters of the metamaterial on the bandgap frequency, providing an opportunity to open low-frequency bandgaps using small-scale metamaterials with supporting capabilities. Consequently, locally resonant metamaterials have gained widespread application in regulating wave propagation or low-frequency vibrations within host structures6, including but not limited to beams7, plates8, shafts9, and rods10.

The superior performance of locally resonant metamaterials is primarily attributed to the innovative and advanced structural design of local resonators (LRs)11,12. Recent advancements have been observed in both the design of LRs and the investigation of locally resonant band gap properties. Fang et al.13 integrated spring-mass resonators with a lumped-mass-spring-chain structure, creating a chain-like locally resonant metamaterial. They examined the characteristics and formation mechanisms of band gaps within this structure. Drawing inspiration from this work, Banerjee14 introduced an elastic metamaterial beam that incorporates periodic spring masses for local resonance. This study explored the positioning and width of locally resonant band gaps and analyzed how vibration propagates and attenuates within the metamaterial beam. Following these contributions, Wen et al.15, El-Borgi et al.16, and Vadala et al.17 further investigated the vibration suppression and dispersion properties of locally resonant metamaterial beams augmented with additional mass LRs, delving into the mechanisms behind the locally resonant band gaps. Additionally, Sangiuliano et al.18 explored how boundary conditions influence the band gap properties of finite locally resonant metamaterial beams. Their findings revealed that edge modes exist within the frequency range predicted by unit cell analysis, which can diminish the effectiveness of vibration attenuation. To lighten the load-bearing components, Miranda et al.19, Mizukami et al.20, and Rai et al.21 utilized additive manufacturing techniques to design and fabricate these components through 3D printing. They introduced a locally resonant metamaterial that exhibits effective attenuation of low-frequency vibrations alongside high stiffness characteristics. In addition, Ma et al.22 designed tunable local resonance metamaterials for low-frequency vibration isolation by using chiral buckling structures. Stein23 investigate the behavior of acoustic metamaterials where unit cells house multiple resonating elements stacked in different configurations, aimed at instigating a wide array of wave propagation profiles that are otherwise unattainable. Several researchers have begun focusing on the study of nonlinear locally resonant metamaterials24,25,26. Bae and Oh27 identified a unique band gap phenomenon, referred to as amplitude-induced band gaps, which arises due to the excitation amplitude in these nonlinear materials.

Developing new localized resonators for low-frequency band gaps generation remains a hotspot in metamaterial research. In order to achieve ultra-low frequency vibration reduction with a large mass in a confined space and resolve the contradiction between load-bearing capacity and small dynamic stiffness, the QZS system presents an opportunity for innovative design of localized resonators. The QZS system enables the vibration system to possess high static stiffness and low dynamic stiffness characteristics, primarily achieved through parallel connection of positive and negative stiffness structures or by employing structures with nonlinear mechanical properties. Currently, QZS structures are primarily utilized in vibration isolators. In addition to the geometric parallel model of three groups of springs proposed by Carrella28, there are also collapsible cylindrical structure29, cam-roller type with triangular structure30, double parallel quadrilateral swing type31, double-chamber solid-liquid mixture type32, electromagnetic type33, and bistable hybrid laminated plate type34 etc. These QZS structures, despite their ability to achieve significant load-bearing capacity and low-frequency vibration isolation effects, are plagued by issues such as inadequate load-bearing capacity, complex structure, poor stability, and limited effective working stroke. Recently proposed biomimetic limb-shaped structures not only possess simpler designs that are easier to adjust but also exhibit higher load-bearing capacities and lower natural frequency characteristics35. This includes X-shaped structures36,37,38,39,40,41,42,43, limb-shaped structures44,45, and polygonal framework structures46,47. Deng et al.48 introduced a biomimetic vibration isolation structure inspired by the multi-layered neck of birds which expands the effective range of dynamic displacement. Yan et al., on the other hand, designed a large-stroke QZS vibration isolator based on simulations of cats’ movements49 and toe mechanics45.

In addition, many scholars have conducted research on the mechanical modeling and solution methods of locally resonant oscillators with nonlinear characteristics. Lazarov and Jensen50 et al. unveiled the band structure of the chain configuration featuring local resonators with cubic stiffness nonlinearity. Fang51 et al. assembled a nonlinear acoustic metamaterial by interconnecting cubic stiffness resonators into a chain and studied its scattering characteristics. Chakraborty and Mallik52 employed perturbation theory to analyze the wave propagation properties of the nonlinear periodic chain structure. Manktelow53 examined the interaction of feeble nonlinear waves in the metamaterial using multiscale methodology, thereby proposing potential applications for nonlinear wave interaction. Rothos and Vakakis54 investigated dynamic interactions among traveling waves within linear chain structures incorporating local nonlinear resonators. Khajehtourian and Hussein55 explored the initiation frequency of band gaps in one-dimensional spring metamaterials with mass-spring resonators while analyzing the influence of nonlinearity on these band gaps.

This paper adopts a bionic QZS as the local oscillator to construct the metamaterial RUC of the ship shafting, which can effectively solve the contradiction between the bearing capacity and the smaller dynamic stiffness, and achieve low-frequency longitudinal vibration reduction in the narrow space of the shafting. Moreover, the proposed bionic limb-like structure not only has a simple structure and is easy to adjust, but also has higher bearing capacity and lower natural frequency characteristics. This paper is organized as follows: Sect. "Analysis of mechanical characteristics of bionic limb-shaped quasi-zero stiffness structures" focuses on the parameter design of the RUC for metamaterials using a bionic limb-shaped QZS structure. In Sect. "Analysis of band gap characteristics of ship shafting with bionic limb-shaped quasi-zero stiffness local resonance oscillator", we model the QZS metamaterials as a lumped-mass-spring chain and derive the dispersion relationship through the Harmonic Balance method, then discuss the impact of damping, stiffness ratio, mass ratio, and excitation amplitude on this band gap. In Sect. 4, the resulting band gap is then confirmed through numerical simulations. Finally, in Sect. "Conclusion", we present our conclusions based on these findings.

Analysis of mechanical characteristics of bionic limb-shaped quasi-zero stiffness structures

Mechanical model

The model of the bionic limb-shaped QZS structure is shown in Fig. 1, which is composed of a bionic negative stiffness structure and a vertical positive stiffness spring in parallel, where\({k_v}\)and \({c_1}\)are the stiffness and damping of the vertical positive stiffness spring respectively. Bionic negative stiffness structure is an artificial structure inspired by the way in which animals absorb ground vibrations and impacts during landing by bending their legs, deforming their muscles. It consists of linkages a, b, c, and d that simulate the skeleton, 7 rotary joints that simulate the joints, horizontal stiffness\({k_h}\)and horizontal damping\({c_3}\)that simulate the stiffness and damping coefficients of muscle deformation. The specific structure and parameters are shown in Fig. 2(a) and (b). At the same time, in order to facilitate the design and adjustment of structural parameters, the bionic limb-shaped structure studied in this paper is a centre symmetric structure with respect to the point O in Fig. 2(b), that is, the length relationship of the rod can be expressed as \({L_a} = L{}_d = {L_1}\), \({L_b} = L{}_c = {L_2}\), \(\phi\) and \(\theta\)are the angles between the connecting rod a and c as for the horizontal direction, respectively. The parameters of bionic limb-shaped QZS structure are shown in Table 1.

Fig. 1
figure 1

Bionic limb-shaped quasi-zero stiffness structure.

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Fig. 2
figure 2

Bionic limb-shaped negative stiffness structures.

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Table 1 The calculation parameters of bionic limb-shaped QZS structure.
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Static analysis of bionic limb-shaped QZS structures

Figure 3(a) and (b) respectively show the relationship before and after the deformation of the bionic limb-shaped structure under the action of force F, from which the expressions of the horizontal relative displacement x and the vertical relative displacement z can be obtained as follows:

$$x={L_1}(\cos \varphi - \cos {\varphi _0})$$
(1)
$$z=2{L_1}(\sin {\varphi _0} - \sin \varphi )+2{L_2}(\sin {\theta _0} - \sin \theta )$$
(2)

Then the force relationship is obtained as follows:

$$\frac{F}{{2{\text{tan}}\varphi }}+\frac{F}{{2{\text{tan}}\theta }}={k_h}x$$
(3)

From the geometric relations \({L_1}cos{\phi _0} = {L_2}\cos {\theta _0}\), \({L_1}\cos \phi = {L_2}\cos \theta \), we can obtain

$$\left\{ \begin{gathered} \sin {\theta _0}{\text{=}}\sqrt {1 - L_{r}^{2}{{\cos }^2}{\varphi _0}} \hfill \\ \sin \theta {\text{=}}\sqrt {1 - L_{r}^{2}{{\cos }^2}\varphi } \hfill \\ \end{gathered} \right.$$
(4)

where Lr=L1/L2.

The Eqs. (1) and (4) are brought into the Eq. (3), and the Eq. (4)is brought into the Eq. (2) to obtain the static equilibrium equation of the bionic limb-like structure

$$F=\frac{{2{k_h}{L_1}(\cos \varphi - \cos {\varphi _0})}}{{\frac{{\cos \varphi }}{{\sqrt {1 - {{\cos }^2}\varphi } }}+\frac{{{L_r}\cos \varphi }}{{\sqrt {1 - L_{r}^{2}{{\cos }^2}\varphi } }}}}$$
(5)
$$z=2{L_1}(\sqrt {1 - {{\cos }^2}{\varphi _0}} - \sqrt {1 - {{\cos }^2}\varphi } )+\;2{L_1}/{L_r}(\sqrt {1 - L_{r}^{2}{{\cos }^2}{\varphi _0}} - \sqrt {1 - L_{r}^{2}{{\cos }^2}\varphi } )$$
(6)

The dimensionless form of the static equilibrium equation of the bionic limb-like structure can be obtained from Eqs. (5) and (6)

$${F_{\text{b}}}=\frac{{2{k_{\text{b}}}{L_b}(\varPhi - {\varPhi _0})}}{{\frac{\varPhi }{{\sqrt {1 - {\varPhi ^2}} }}+\frac{{{L_r}\varPhi }}{{\sqrt {1 - L_{r}^{2}{\varPhi ^2}} }}}}$$
(7)
$${Z_{\text{b}}}=2{L_{\text{b}}}(\sqrt {1 - {\varPhi _0}^{2}} - \sqrt {1 - {\varPhi ^2}} )+2{L_b}/{L_r}(\sqrt {1 - L_{r}^{2}{\varPhi _0}^{2}} - \sqrt {1 - L_{r}^{2}{\varPhi ^2}} )$$
(8)

Here,

$$\begin{gathered} {L_{\text{b}}}{\text{=}}{L_1}/{L_0}\;\;\;{k_b}={k_h}/{k_0}\;\;\;{F_b}=F/{k_0}{L_0} \hfill \\ {Z_{\text{b}}}=z/{L_0}\;\;\;{\varPhi _0}^{{}}=\cos {\varphi _0}\;\;\;\varPhi =\cos \varphi \hfill \\ \end{gathered} $$

where L0 and k0 are the reference rod length and reference stiffness of the bionic limb-like structure.

The dimensionless form of the stiffness equation of the bionic limb-like structure is further obtained from Eqs. (7) and (8)

$$\frac{{d{F_{\text{b}}}}}{{d{Z_{\text{b}}}}}=\frac{{d{F_{\text{b}}}}}{{d\varPhi }} \times \frac{{d\varPhi }}{{d{Z_{\text{b}}}}}$$
(9)

where \(\frac{{d{F_{\text{b}}}}}{{d\varPhi }}{\text{=}}\frac{{2{k_{\text{b}}}{L_1}}}{{\frac{\varPhi }{{\sqrt {1 - {\varPhi ^2}} }}+\frac{{{L_r}\varPhi }}{{\sqrt {1 - {L_r}^{2}{\varPhi ^2}} }}}} - \frac{{2{k_{\text{b}}}{L_1}\left( {\varPhi - {\varPhi _0}} \right)}}{{{{\left( {\frac{\Phi }{{\sqrt {1 - {\Phi ^2}} }}+\frac{{Lr\Phi }}{{\sqrt {1 - L{r^2}{\Phi ^2}} }}} \right)}^2}}} \times \left( {\frac{{{\varPhi ^2}}}{{{{\left( {1 - {\varPhi ^2}} \right)}^{3/2}}}}+\frac{{{L_r}^{3}{\varPhi ^2}}}{{{{\left( {1 - {L_r}^{2}{\varPhi ^2}} \right)}^{3/2}}}}+\frac{1}{{\sqrt {1 - {\varPhi ^2}} }}+\frac{{{L_r}}}{{\sqrt {1 - {L_r}^{2}{\varPhi ^2}} }}} \right)\)

and\(\frac{{d\varPhi }}{{d{Z_{\text{b}}}}}={(\frac{{2{L_{\text{b}}}\varPhi }}{{\sqrt {1 - {\varPhi ^2}} }} - \frac{{2{L_{\text{b}}}{L_{\text{r}}}\varPhi }}{{\sqrt {1 - {L_{\text{r}}}^{2}{\varPhi ^2}} }})^{ - 1}}\).

Fig. 3
figure 3

The relationship diagram of the bionic limb-shaped structure before and after deformation.

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The QZS nonlinear restoring force can be expressed as:

$$f(z) = {f_1}(z) + {f_v}(z)$$
(10)

where\({f_1}(z)=4{k_h}x\frac{{dx}}{{dz}}\frac{{dz}}{{dy}}\) and \({f_v}(z) = {k_v}z\).

Then, by using Taylor series, we can expand\({f_1}(z)\)in Eq. (10) as follows:

$${f_1}(z) \approx {\bar {f}_1}(z)={n_1}z+{n_2}{z^2}+{n_3}{z^3}+{n_4}{z^4}$$
(11)

Therefore Eq. (10) can be rewritten as:

$$f(z) = ({n_1} + {k_v})z + {n_2}{z^2} + {n_3}{z^3} + {n_4}{z^4}$$
(12)

It can be further written as:

$$f(z) = {k_1}g(z) = {k_1}(z + {r_2}{z^2} + {r_3}{z^3} + {r_4}{z^4})$$
(13)

where\({k_1}={n_1}+{k_v},{r_2}=\frac{{{n_2}}}{{{n_1}+{k_v}}},{r_3}=\frac{{{n_3}}}{{{n_1}+{k_v}}},{r_4}=\frac{{{n_4}}}{{{n_1}+{k_v}}}\).

By substituting the calculated parameters from Table 1 into Eqs. (10 ~ 13), the relationship between force, stiffness, and displacement of the biomimetic limb-like QZS structure can be obtained and is shown in Fig. 4.

Fig. 4
figure 4

The static mechanical characteristics of bionic limb-like quasi-zero stiffness structure.

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From Fig. 4(a), it can be seen that the force-displacement curve of the structure is relatively flat in a wide displacement range near the static equilibrium position, which means that the stiffness is close to zero. However, when the displacement exceeds the QZS range, the force will suddenly increase, indicating that the stiffness will gradually increase beyond the QZS displacement range. As mentioned above, the low resonant frequency of the local oscillator is an important condition for opening the low-frequency band gap, therefore, the ultra-low stiffness of the local oscillator at the static equilibrium position is crucial for achieving the low-frequency band gap.

Analysis of band gap characteristics of ship shafting with bionic limb-shaped quasi-zero stiffness local resonance oscillator

Damped nonlinear lumped mass-spring chains system

Figure 5 shows the structural diagram of the RUC of the metamaterials with QZS structure in ship shafting, where 7 rotary hinges simulate the joints of the biomimetic limb-like structure. The negative stiffness structure is connected in parallel with a horizontal positive stiffness spring, which is then connected to the oscillator with a mass of m. The horizontal spring bears the inertial force of the oscillator mass, \({k_v}\)and\({c_1}\) represent the stiffness and damping of the positive stiffness spring in the horizontal direction. The vertical stiffness\({k_h}\)and damping\({c_3}\)simulate the stiffness and damping coefficients of the muscle deformation process. When the biomimetic limb-like negative stiffness spring reaches its static equilibrium position, the local resonant oscillator has QZS characteristics, thereby achieving a low-frequency longitudinal vibration band gap for the propulsion shaft. Figure 6 shows the installation diagram of RUC with the biomimetic QZS in the inner hole of the ship’s shafting. The installation offers space-saving benefits, enhances structural compactness, and ensures greater stability, thereby mitigating the adverse effects caused by the rotation of the propulsion shaft system on the oscillator.

Fig. 5
figure 5

Structure diagram of RUC of metamaterials with bionic limb-shaped quasi-zero-stiffness.

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Fig. 6
figure 6

The arrangement of RUC in the inner hole of ship shafting.

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Table 2 Parameter of spring-mass chains model.
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To derive the dispersion relationship of longitudinal waves, the ship shafting with the biomimetic limb-shaped QZS local resonance oscillator shown in Fig. 6 is simplified into a one-dimensional chain-like QZS metamaterial model with a concentrated mass-spring system as shown in Fig. 7. The more chain units there are, the closer it approaches a continuous structure. In the chain, we simplify the supporting frame and central mass as lumped masses M and m, respectively. The elastic frame between each RUC is represented by a spring \({k_p}\), while the positive stiffness and negative stiffness elements connecting the central mass and frame are modeled as nonlinear springs \({k_q}\left( { = {k_1}f'\left( {{u_j} - {v_j}} \right)} \right)\)with QZS characteristics. The parameters of the chain are listed in Table 2 and can be determined by using parameters in Table 1 and the stiffness shown in Fig. 4 (b). When the number of RUC is significantly large or even approaching infinity, a state of periodicity emerges within the chain, causing every RUC in the chain to remain in a consistent situation. The investigation process for any individual RUC is identical and has the ability to unveil all dynamic characteristics pertaining to the entire model. In relation to the jth RUC, one can derive equations of motion as follows.

Fig. 7
figure 7

Simplified spring-mass chains model.

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$$M{\ddot u_j} + 2{k_p}{u_j} - {k_p}{u_{j - 1}} - {k_p}{u_{j + 1}} + {k_1}g({u_j} - {v_j}) + {c_q}({\dot u_j} - {\dot v_j}) = 0$$
(14)
$$m{\ddot v_j} + {c_q}({\dot v_j} - {\dot u_j}) + {k_1}g({v_j} - {u_j}) = 0$$
(15)

where\({c_q}\)is the damping coefficient, \({k_1}g\left( {{v_j} - {u_j}} \right)\) is the nonlinear restoring force, which is shown in Fig. 4(a). To derive the dispersion relationship analytically, the restoring force in Eq.(13)can be written as follows:

$${k_1}g({v_j} - {u_j}) = {k_1}({v_j} - {u_j}) + {k_1}{r_2}{({v_j} - {u_j})^2} + {k_1}{r_3}{({v_j} - {u_j})^3} + {k_1}{r_4}{({v_j} - {u_j})^4}$$
(16)

By introducing the relative displacement \({q_j} = {v_j} - {u_j},{\omega _0} = \sqrt {{{{k_p}} \mathord{\left/{\vphantom {{{k_p}} M}} \right.\kern-\nulldelimiterspace} M}} \)and non-dimensional parameters

$$\tau = {\omega _0}t,\hat u = \frac{u}{{{u_s}}},\zeta = \frac{{{c_q}}}{{2\sqrt {{k_1}m} }},\hat q = \frac{q}{{{u_s}}},\alpha = \frac{{{k_1}}}{{{k_p}}},\beta = \frac{m}{M}$$
(17)

where \({u_s} = 2mm\) is displacement of mass M when the stiffness of RUC starts begins to increase sharply. Subsequently, Eqs. (14–16) can be rephrased as follows:

$${\hat u^{\prime\prime}_j} + 2{\hat u_j} - {\hat u_{j - 1}} - {\hat u_{j + 1}} - 2\zeta \beta \kappa {\hat q^{\prime}_j} - \alpha f\left( {{{\hat q}_j}} \right) = 0$$
(18)
$${\hat q^{\prime\prime}_j} + 2\zeta \kappa {\hat q^{\prime}_j} + {\kappa ^2}f\left( {{{\hat q}_j}} \right) + {\hat u^{\prime\prime}_j} = 0$$
(19)
$$g({\hat q_j}) = {\hat q_j} + {\chi _2}\hat q_j^2 + {\chi _3}\hat q_j^3 + {\chi _4}\hat q_j^4$$
(20)

where \(\kappa = \frac{{\sqrt {{{{k_1}} \mathord{\left/{\vphantom {{{k_1}} m}} \right.\kern-\nulldelimiterspace} m}} }}{{{\omega _0}}} = \sqrt {\frac{\alpha }{\beta }} \) represents the non-dimensional natural frequency of the linearized local resonator, \(()^\prime\) or ()” denote the first- and second-order derivatives with respect to non-dimensional time τ, respectively, \({\chi _2} = {r_2}{u_s}\), \({\chi _3} = {r_3}u_s^2\) and \({\chi _4} = {r_4}u_s^3\).

The Harmonic balance method is employed to solve the equations of motion for the damped nonlinear mass-spring chain, and it is assumed that the solution for the jth local resonator can be expressed as

$${\hat q_j} = {B_1}{e^{i\left( {\Omega \tau - j\mu } \right)}} + {\bar B_1}{e^{ - i\left( {\Omega \tau - j\mu } \right)}}$$
(21)

where \({\bar B_1}\)represents the conjugate complex number of B1, \(\Omega = \omega /{\omega _0}\)represents the ratio of the external excitation frequency to the natural frequency of the main oscillator in the cell, and \(\mu \) represents the wave propagation constant of longitudinal wave.

By inserting Eq. (21) into Eq. (19) and performing two integrations with respect to time τ, the displacement response of mass M at position j can be derived

$${\hat u_j} = {B_1}\left[ { - 1 + \frac{{2i\zeta \kappa }}{\Omega } + \frac{{{\kappa ^2}}}{{{\Omega ^2}}}\left( {1 + 3{\chi _3}B_1^2\bar B_1^2} \right)} \right]{e^{i\varphi }} + c.c. + O\left( {B_1^3{e^{i3\varphi }}} \right)$$
(22)

where \(\varphi = \Omega \tau - j\mu \) , c.c. represents the conjugated term related to the first term mentioned in Eq. (22), and \(O\left( {B_1^3{e^{i3\varphi }}} \right)\) denotes a negligible contribution to the response. Consequently, solely the terms concerning \({e^{i\varphi }}\) and \({e^{ - i\varphi }}\)are retained in the expression provided above.

By inserting Eqs. (21) and (22) into Eq. (18) and equating the coefficients of \({e^{i\varphi }}\) to zero, one can derive the equation that relates amplitude to frequency

$$\left[ { - 1 + \frac{{2i\zeta \kappa }}{\Omega } + \frac{{{\kappa ^2}}}{{{\Omega ^2}}}\left( {1 + 3{\chi _3}{B_1}{{\bar B}_1}} \right)} \right]\left( { - {\Omega ^2} + 2 - {e^{i\mu }} - {e^{ - i\mu }}} \right) - \left[ {2i\zeta \beta \kappa \Omega + \alpha \left( {1 + 3{\chi _3}{B_1}{{\bar B}_1}} \right)} \right] = 0$$
(23)

Subsequently, the dispersion relationship can be obtained by utilizing Eq. (23).

$$\cos \left( \mu \right) = 1 - \frac{{{\Omega ^2}}}{2} - \frac{1}{2}\frac{{{\Omega ^2}\left[ {2i\zeta \beta \kappa \Omega + \alpha \left( {1 + 3{\chi _3}{B_1}{{\bar B}_1}} \right)} \right]}}{{\left[ { - {\Omega ^2} + 2i\zeta \kappa \Omega + {\kappa ^2}\left( {1 + 3{\chi _3}{B_1}{{\bar B}_1}} \right)} \right]}}$$
(24)

The dispersion relation of the chain is illustrated in Fig. 8, with the right and left planes displaying the real and imaginary components of the solution, respectively. Consequently, the relationship between the wave amplitudes of adjacent masses \({M_j}\) and \({M_{j + 1}}\)can be demonstrated as

$${\hat u_{j + 1}} = {e^{{\mathop{\rm Im}\nolimits} \left( \mu \right)}}{\hat u_j}{e^{ - i{\mathop{\rm Re}\nolimits} \left( \mu \right)}}$$
(25)
Fig. 8
figure 8

The dispersion relation between frequency and µ.

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The chains exhibit wave attenuation within the sky blue region (band gap) highlighted, where the rate of attenuation \({e^{{\mathop{\rm Im}\nolimits} \left( \mu \right)}}\) is below 1. The propagation of waves at frequencies beyond the band gap in the chain is characterized by minimal energy loss, resulting in a transmission rate \({e^{{\mathop{\rm Im}\nolimits} \left( \mu \right)}}\) that is close to 1.

In addition, the phase difference \(Re(\mu )\) between wave propagation through two adjacent masses \({M_j}\) and \({M_{j + 1}}\) is subject to variation with \(\mu \). Within the band gap, the attenuation rate \({e^{{\mathop{\rm Im}\nolimits} \left( \mu \right)}}\) reaches its minimum when \({e^{{\mathop{\rm Im}\nolimits} \left( \mu \right)}}\). In simpler terms, this implies that the frequency matches the resonator’s natural frequency \(\left( {\omega = \sqrt {{{{k_1}} \mathord{\left/{\vphantom {{{k_1}} m}} \right.\kern-\nulldelimiterspace} m}} } \right)\).

Figure 9 shows the effect of damping on the band gap. The imaginary part of the coefficient \(\mu \) decreases as the damping increases, which means that the local resonant shaft’s attenuation effect on the longitudinal wave in the band gap range becomes worse. However, the increase in damping leads to a significant broadening of the band gap, which helps the local resonant shaft to control the longitudinal wave over a wider frequency range.

Fig. 9
figure 9

The effect of damping on the band gap.

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Figure 10 shows the effect of nonlinear parameters on the band gap. The result indicates that the influence of nonlinear coefficient \({B_1}{\bar B_1}\) on the band gap structure is more obvious than that of damping. With the increase of nonlinear constant, the position and width of the band gap undergo obvious changes, i.e., the band gap moves to the high-frequency region and the band gap width increases. The reason is that with the increase of excitation amplitude, the stiffness of local oscillator increases significantly, leading to the movement of the band gap to the high-frequency region and the increase of the band gap width.

Fig. 10
figure 10

The effect of nonlinear parameters on the band gap.

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Figure 11 shows the effect of stiffness ratio on the band gap. It shows that, as the stiffness ratio increases, the band gap shifts to the high-frequency region and the band gap width increases. The reason is that the stiffness of the local oscillator increases obviously.

Fig. 11
figure 11

The effect of stiffness ratio on the band gap.

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Figure 12 shows the effect of mass ratio on the band gap. It can be seen that the increase in the quality factor leads to a shift of the band gap towards lower frequencies and a reduction in its width.

Fig. 12
figure 12

The effect of mass ratio on the band gap.

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Nonlinear finite chain mode

In this section, we consider a chain of finite length consisting of n RUCs, as depicted in Fig. 13. The primary mass M of the first RUC is connected to a fixed base via a spring, while the tail RUC has a free boundary condition. To analyze the wave attenuation performance, we establish and numerically solve the equations of motion for the entire chain using the Runge-Kutta method. We evaluate transmittance as a measure of wave attenuation, defined as the difference in horizontal displacement between the first and last RUCs.

Applying Newton’s second law of motion leads to the derivation of the equation of motion in a matrix format.

$${\bf{M\ddot x}} + {\bf{C\dot x}} + {{\bf{F}}_{\bf{k}}}\left( {\bf{x}} \right) = {\bf{F}}$$
(26)

where \({\bf{x}} = {\left[ {{u_1}{\rm{ }}{q_1}{\rm{ }}{u_2}{\rm{ }}{q_2}{\rm{ }} \cdots {\rm{ }}{u_n}{\rm{ }}{q_n}} \right]^T}\) is the displacement vector, which contains the displacement \({u_i}\left( {i = 1\sim n} \right)\) of primary masses and the relative displacements \({q_i} = {u_i} - {v_i}\left( {i = 1\sim n} \right)\)between the primary mass and the attached oscillator. M and C are given by

$${\bf{M}} = \left[ {\begin{array}{*{20}{c}}M&0&0&0& \cdots &0&0\\m&m&0&0& \cdots &0&0\\0&0&M&0& \cdots &0&0\\0&0&m&m& \cdots &0&0\\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\0&0&0&0& \cdots &M&0\\0&0&0&0& \cdots &m&m\end{array}} \right]$$
(27)
$${\bf{C}} = \left[ {\begin{array}{*{20}{c}}0&{ - {c_q}}&0&0& \ldots &0&0\\0&{{c_q}}&0&0& \ldots &0&0\\0&0&0&{ - {c_q}}& \ldots &0&0\\0&0&0&{{c_q}}& \ldots &0&0\\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\0&0&0&0& \ldots &0&{ - {c_q}}\\0&0&0&0& \ldots &0&{{c_q}}\end{array}} \right]$$
(28)

The force vector \({\bf{F}} = {\left[ {f\sin \left( {\omega t} \right){\rm{ }}0{\rm{ }} \cdots {\rm{ }}0{\rm{ }}} \right]^T}\) represents the only harmonic excitation force acting on the first primary mass M. And Fk(x) is given by

$${{\bf{F}}_{\bf{k}}}\left( {\bf{x}} \right) = \left[ {\begin{array}{*{20}{c}}{2{k_p}}&{ - {k_q}}&{ - {k_p}}&0&0& \cdots &0&0&0&0&0&0\\0&{{k_q}}&0&0&0& \cdots &0&0&0&0&0&0\\{ - {k_p}}&0&{2{k_p}}&{ - {k_q}}&{ - {k_p}}& \cdots &0&0&0&0&0&0\\0&0&0&{{k_q}}&0& \cdots &0&0&0&0&0&0\\\vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\0&0&0&0&0& \cdots &{ - {k_p}}&0&{2{k_p}}&{ - {k_q}}&{ - {k_p}}&0\\0&0&0&0&0& \cdots &0&0&0&{{k_q}}&0&0\\0&0&0&0&0& \cdots &0&0&{ - {k_p}}&0&{{k_p}}&{ - {k_q}}\\0&0&0&0&0& \cdots &0&0&0&0&0&{{k_q}}\end{array}} \right]{\bf{x}}$$
(29)

where \({k_p}\) is the stiffness of primary, and \({k_q}\left( { = {k_1}f'\left( {{u_j} - {v_j}} \right)} \right)\) is the nonlinear springs with QZS characteristics can be derived from Eq. (13) and written as

$${k_q} = {k_1} + 2{k_1}{r_2}{q_j} + 3{k_1}{r_3}{q_j}^2 + 4{k_1}{r_4}{q_j}^3 = 270 + 1.51 \times {10^5}{q_j} + 2.56 \times {10^8}{q_j}^2 + 2.7 \times {10^9}{q_j}^3$$
(30)

Then, the Eq. (26) is solved using the Runge-Kutta method to obtain the displacement transmittance for a chain consisting of nine primary masses. The resulting transmittance curve is depicted in Fig. 14. It can be observed that wave attenuation occurs within the frequency range of 7.5 Hz to 12 Hz, which is highlighted by the sky blue area and can be considered as a low-frequency band gap. It is also observed that the finite chain exhibits a numerical band gap that closely aligns with the theoretical prediction depicted in Fig. 8 for the infinite chain.

Fig. 13
figure 13

Schematic diagram of the finite chain model.

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Fig. 14
figure 14

Displacement transmittance rate of the finite chain model.

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The effect of the damping ratio on the band gap is illustrated in Fig. 15. Due to the local resonance mechanism, most of the energy of low-frequency longitudinal waves propagating on the one-dimensional chain-like QZS local resonance metamaterial is concentrated on the QZS local resonators, and is dissipated by damping during the local resonator resonance process. As the damping increases, the band gap expands to encompass a wider frequency range; however, the depth of the band gap diminishes, indicating a degradation in vibration absorption performance. Meanwhile, the center frequency of the band gap is kept unchanged.

Fig. 15
figure 15

The relationship of transmittance and frequency for different damping (ζ = 0.05, 0.08, 0.1, 0.2; α = 0.035; β = 0.286).

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The effect of the stiffness ratio on the band gap is shown in Fig. 16. As the stiffness ratio increases, the band gap shifts to the high-frequency region and the band gap width increases. It is consistent with the conclusions in Fig. 11.

Fig. 16
figure 16

Influence of stiffness ratio on the transmittance and band gap( α = 0.035, 0.07, 0.1, 0.2; β = 0.29; ζ = 0.1).

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The effect of the mass ratio on the band gap is shown in Fig. 17. It can be seen that as the quality factor increases, the band gap moves toward lower frequencies and the bandwidth narrows. The finding is in agreement with Fig. 12.

Fig. 17
figure 17

Influence of mass ratio on the transmittance and band gap ( β = 0.29, 0.5, 0.8, 1.0; α = 0.035; ζ = 0.1).

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The effect of the excitation amplitude on the band gap is illustrated in Fig. 18. Evidently, as the load increases, the band gap widens. This can be attributed to the heightened nonlinearity in response to displacement. The QZS metamaterial exhibits a nonlinear stiffness with a tendency for stiffening, and this nonlinearity becomes more pronounced with increasing displacement. Consequently, complex responses such as sub-harmonic, quasi-periodic, and even chaotic motions are induced by this strong nonlinearity. Additionally, it can be observed that the initial frequency of the band gap slightly shifts towards higher frequencies and experiences a jump-down phenomenon due to amplified stiffness and nonlinearity resulting from increased excitation.

Fig. 18
figure 18

Influence of excitation amplitude on the transmittance and band gap ( β = 0.286; α = 0.035; ζ = 0.1; f = 1.34, 1.6, 1.8, 4).

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Conclusions

This paper focuses on the design and formation mechanism of low frequency band gaps to control the low-frequency longitudinal vibration of propulsion shaft. A bionic limb-shaped QZS structure is used to construct the representative unit cell (RUC) of the metamaterial to achieve the vibration absorption. A lumped-mass chain model is employed to theoretically investigate the dispersion characteristics of the one-dimensional metamaterial and uncover its band gaps. Furthermore, this study investigated the influence of parameters on the band gaps.

The findings indicate that the bionic limb-shaped QZS structure not only effectively achieves the desired QZS characteristics but also possesses a simple design, ease of adjustment, and superior load-bearing capacity. All these features facilitate the occurrence of locally resonant band gaps at low frequencies, enhancing vibration absorption for ship shafting. The introduction of damping widens the band gap but reduces wave attenuation within it. Additionally, increasing the mass ratio and decreasing the stiffness ratio further lowers the frequency of the band gap. When subjected to larger excitations, the stiffness nonlinearity is intensified, resulting in more complex dynamic behavior for QZS metamaterials.