Design of a Low-Frequency Vibration Isolator With Large-Stroke and Variable-Payload Capabilities Using Gear-Spring Units Open Access

Vibration is a common phenomenon in the natural world and engineering fields. While it can occasionally be harnessed for specific purposes, in most engineering applications, vibrations are detrimental, causing undesirable consequences [1–3]. For example, vibrations in the optical payloads of remote sensing satellites reduce imaging resolution, vibrations in machine tools compromise machining precision, and vibrations in vehicle seats diminish ride comfort [4–6]. Vibration isolation is a reliable approach that places a mechanical system between the vibration source and the protected object to suppress the transmission of motion and force [7–9]. Vibration isolation enhances the precision of machines and mechanical systems while also extending their operational lifespan [10–13].

Active vibration isolation uses sensors, actuators, and controllers to counteract vibrations in real time [14,15]. Although active vibration isolation systems can provide excellent performance and adaptability to varying disturbances, high cost and system complexity are considered their main disadvantages. In contrast, passive vibration isolation systems use mechanical elements to absorb and dissipate vibrational energy [16–18]. These systems are favored for their simplicity, reliability, and low cost. A linear vibration isolator is commonly used, and it can provide a frequency band of vibration isolation of “the square root of two” times its natural frequency [7,19,20]. To expand the frequency band of vibration isolation to a lower region, nonlinear vibration isolation using quasi-zero (or zero) stiffness mechanisms was adopted [21–23]. These mechanisms can generate high static stiffness and low dynamic stiffness, enabling low-frequency vibration isolation without decreasing the load capacity of mechanical devices [24–26].

The literature in mechanical sciences has presented a variety of low-frequency vibration isolators designed with various quasi-zero stiffness mechanisms [27,28]. For instance, Ling et al. [29] proposed a low-frequency vibration isolator based on a cockroach-inspired structure with linear springs. Qi et al. [30] used a magnetically modulated sliding structure by combining a sliding beam with a pair of repulsive magnets. Zhou et al. [21] used double-arc flexible beams. Wang et al. [31] proposed a compact quasi-zero-stiffness device by adopting a negative stiffness magnetic spring, a spiral flexure spring, and coils. Yao et al. [32] used a cam-roller-spring mechanism. Yan et al. [33] presented a novel modulated tetrahedron structure with nonlinear stiffness and inertia modulation by using a pair of repulsive magnets. Yu et al. [34] developed an origami-inspired truss-spring structure combined with an inertial mechanism. Han et al. [35] developed a lightweight nonlinear vibration isolator using the Kresling origami modules. Xu et al. [36] adopted a topology optimization method to construct quasi-zero stiffness meta-structures with smooth boundaries. Xiao et al. [37] combined the snap-through behavior of inclined beams and the support of folded beams. Yan et al. [38] presented a bio-inspired toe-like structure. Yao et al. [39] used a cam-roller-spring mechanism.

Recent advances in low-frequency vibration isolation research have revealed a new challenge that focuses on extending the quasi-zero stiffness region of the mechanisms [40–42]. Increasing the quasi-zero stiffness region can enlarge the isolation stroke, enabling the vibration isolation of substantial displacement amplitudes without compromising effectiveness. For example, Chong et al. [41] used a nonlinear X-combined structure with linear springs. Sun and Jing [43] adopted a multi-layer scissor-like structure. Yan et al. [44] designed with a symmetric polygon structure. Liu et al. [45] proposed an elastic origami-inspired structure. Ling et al. [46] introduced a passive click-beetle-inspired structure. Zhao et al. [47] proposed a limb-like quasi-zero-stiffness mechanism by adopting two pairs of oblique springs. Although these mechanisms can provide a large quasi-zero stiffness region, the requirement for an extended transition region to reach the quasi-zero stiffness behavior is unavoidable and may limit the isolation stroke.

The need to accommodate variable payloads introduces an additional challenge in achieving effective low-frequency vibration isolation in dynamic systems [4,21]. Changes in payload weight shift the system's equilibrium position and alter its dynamic behavior. Although payload variation is common in practice, few low-frequency vibration isolators capable of accommodating different payloads have been reported in the literature. For instance, Ye et al. [48] developed a multi-cam mechanism with multiple levels in which the number of cams can be adjusted according to the applied payload. Zheng et al. [49] combined a pair of identical oblique beams with a pair of identical semicircular arches to construct a quasi-zero stiffness layer. In Zheng et al.'s design, the number of layers is appropriately arranged to provide multiple quasi-zero stiffness characteristics for dealing with different payloads. Although these isolators can accommodate different payloads, the use of multiple beam layers and cams makes the overall system bulky and requires substantial installation space for necessary components.

This article presents a low-frequency vibration isolator with large-stroke and variable-payload capabilities. The vibration isolator is constructed with a parallel mechanism comprising six identical RRR legs. Each leg is equipped with a gear-spring unit (GSU), which enables the realization of zero-stiffness characteristics. Compared to traditional low-frequency vibration isolators, the proposed design offers several key advantages:

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  The proposed isolator simultaneously provides a large isolation stroke and adaptability to variable payloads, enabling effective isolation of large-amplitude excitations across a broad payload range. The large-stroke capability is achieved by eliminating the extended transition region, thus maintaining zero-stiffness characteristics throughout the entire workspace of the mechanism. The adaptability to varying payloads is realized by adjusting the positions of the spring ends via lock sliders.

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  The adoption of a parallel mechanism in the vibration isolator enhances structural rigidity and compactness, thereby extending its applicability to heavy-load scenarios.

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  The required spring stiffness for achieving zero-stiffness characteristics can be predetermined, thereby simplifying the spring selection process. As a result, off-the-shelf springs can be readily integrated into the design.

The remainder of this article is organized as follows. Section 2 presents the conceptual design of the proposed vibration isolator. Section 3 details the vibration isolation analysis. Sections 4 and 5 provide the performance evaluation and simulation validation, respectively. Finally, Sections 6 and 7 discuss the results and present the conclusion.

The structure of the proposed low-frequency vibration isolator is illustrated in Fig. 1. It is constructed with a parallel mechanism consisting of six identical RRR legs. Leg i (i = 1, 2, …, 6) consists of links i1 and i2 in which link i1 is connected to the base (link 0) while link i2 is connected to the moving platform (link 3). Assume that the lengths of links O_iC_i and C_iD_i are equal to each other, denoted as l. A GSU is attached to each leg so that the center of gear i1 aligns with the rotation joint of link i1 (joint O_i). Each GSU consists of two spur gears, a crank, a lock slider, and a spring. For simplicity, two consecutive GSUs share the same spring. Gear i1 is fixed to link i1, while gear i1 is rotated about joint A_i. Spring i (i = 1, 2, 3) is attached between gears i2 and (i + 1)2 through cranks and sliders i and i + 1, respectively. Crank i is fixed to gear i2, while slider i is connected to the spring. Slider i is arranged so that its sliding axis coincides with the longitudinal axis of crank i. The translational motion of the slider enables an adjustment in the position of the spring end. For a given payload, the positions of the spring ends are fixed when the moving platform moves upward or downward as the sliders are locked. Given the motion constraints of the links and gears described above, the springs extend as the moving platform displaces.

As the displacement of the moving platform s_p is variable and the link length l is constant, Eq. (13) cannot be satisfied.

In summary, the mechanism can realize zero-stiffness characteristics throughout the range of angle θ = (0, π/2) when Eq. (14) is satisfied.

This section describes the dynamic modeling of the vibration isolator by adopting Lagrange's equation [50]. The analysis of system response and design for variable payloads are also presented.

In Eqs. (23)–(27), f₁, f₂, and f₃ represent the generalized damping forces produced by the air, rotational friction induced by the joints, and sliding friction induced by the gears, respectively; c₁, c₂, and c₃—the air damping coefficient, friction coefficient of a revolute joint, and friction coefficient of a pair of gears, respectively; n_x and n_y—the numbers of revolute joints and gear pairs, respectively.

Equation (38) indicates that the response of the moving platform z(t) can be determined when the excitation at the base x(t) is given.

Assume that an extra load ΔM is added to the vibration isolator, resulting in a new payload (M + ΔM). In this case, the dynamic equation (Eq. (38)) is not maintained, and the equilibrium position varies as the term g(M, r₃) is no longer zero. From Eq. (36), to make Eq. (37) satisfied with the payload change, either the spring stiffness k or crank length r₃ must be adjusted. Adjusting the spring stiffness is possible (e.g., changing the number of active coils), but it is relatively complex. In contrast, the adjustment of the crank length is more practical and is used in the current design. As shown in Fig. 1, a lock slider is added to each crank so that its position can be adjusted along the longitudinal axis of the crank.

From Eq. (56), one can see that when the payload is varied, an appropriate crank length can be chosen to attain the equilibrium condition. Moreover, as the crank length can be adjusted, it is possible to predefine the spring stiffness k, simplifying the spring selection procedure.

The prescribed parameters of the vibration isolator are listed in Table 1. The link lengths and payload are selected as l = 0.1 m and M = 6 kg. A linear spring (MISUMI AWS18-90) with a stiffness coefficient k = 2650 N/m is chosen. Figure 4 illustrates the allowable excitation amplitude of the vibration isolator. Here, x+ and x– represent the positive (moving up) and negative (moving down) excitation amplitudes, respectively. The positive excitation amplitude can achieve up to 2l when the equilibrium position (where the excitation starts) is chosen at θ = 0. The positive excitation amplitude is decreased to zero at θ = π/2. In contrast, the negative excitation amplitude increases from 0 to 2l when the joint angle θ increases from 0 to π/2. The intersection of the positive and negative excitation regions forms a region that represents the allowable excitation amplitude of the vibration isolator. The allowable excitation amplitude is zero at θ = 0 and π/2, and it can achieve up to l at θ = π/3. In other words, the isolation stroke can achieve up to 2l.

Figure 5 shows the transmissibility of the vibration isolator. Here, the equilibrium position θ = π/4 is set, and the base excitation is defined as x = 0.02cos(2πft) (m) with time t from 0 s to 20 s. The obtained results show that the vibration isolator works effectively within the frequency range f = [0, 10] Hz, as the platform acceleration (response) is significantly smaller than the base acceleration (excitation). The starting isolation frequency is f = 0 Hz. The increase of the frequency leads to a reduction in the transmissibility T_d (e.g., T_d = –58 dB with f = 1 Hz and T_d = –103.7 dB with f = 10 Hz).

The transmissibility of the vibration isolator at different equilibrium positions is shown in Fig. 6. It can be seen that the vibration isolation is effective at all the considered positions (θ = π/6, π/5, π/4, π/3). With the same frequency, the increase of the joint angle θ leads to a decrease in the transmissibility. For example, with the frequency f = 2 Hz, the transmissibility at the positions θ = π/6 is –64.8 dB, and it is reduced to –75.9 dB at θ = π/3. It can be concluded that the isolation performance is increased with the increase of the joint angle θ.

Figure 7 illustrates the parameter selection for the vibration isolator with different payloads. For a given payload, there are infinite choices for the spring stiffness k, link length l, and crank r₃. With the same stiffness and link length, the larger the payload is used, the longer the crank is required. For example, with k = 3000 N/m and l = 0.1 m, the required crank for M = 6 kg is r₃ = 0.036 m, and that for M = 24 kg is r₃ = 0.072 m. Figure 8 shows the transmissibility of the vibration isolator with different payloads (M = 6, 12, 18, 24 kg). It is shown that, with the same frequency, the increase of the payload leads to a reduction in the transmissibility. For example, with f = 2 Hz, the transmissibility is –71.8 dB with M = 6 kg and decreased to –99.5 dB with M = 24 kg. More results with different payloads are presented in Table 2. It can be concluded that the vibration isolation performance is effective with variable payloads and increases with the increase in payload.

Figure 9 illustrates the correlation between the isolation stroke and payload capability of the vibration isolator. Here, the spring stiffness k = 2650 N/m is chosen. As shown in Fig. 9(a), when the crank r₃ is defined, the isolation stroke is reduced with the increase of the applied payload. For example, with r₃ = 0.1 m, the isolation stroke is 0.1 m with M = 80 kg and is increased to 0.3 m with M = 26.7 kg. In other words, there is a compromise between the isolation stroke and payload capability. The transmissibility of the vibration isolator with different isolation strokes is presented in Fig. 9(b). It shows that, with a prescribed frequency, the increase of the isolation stroke leads to a reduction in transmissibility. For example, when f = 1 Hz is given, the isolation stroke is 0.2 m with T_d = –59 dB and is 0.9 m with T_d = –88.4 dB. It can be concluded that increasing the isolation stroke leads to a reduction in the payload capability and an increase in the vibration isolation performance.

Damping is a critical factor for low-frequency vibration isolation, as it governs energy dissipation during mechanical transmission. This section investigates the effect of air damping and friction damping on the performance of the proposed vibration isolator.

Figure 10 shows the effect of air damping on the transmissibility of the vibration isolator with different applied payloads. Here, the air damping coefficient c₁ ranges from 0 Ns/m to 0.5 Ns/m, while the joint and gear friction damping coefficients c₂ and c₃ are fixed to 0.01 Ns/m. The spring stiffness k = 2650 N/m and the equilibrium position θ = π/4 are chosen. The base excitation is defined as x = 0.02cos(2πft) (m) with time t from 0 s to 20 s. As can be seen, with a prescribed frequency and payload, the increase of the air damping leads to reducing the transmissibility. For example, with f = 10 Hz and M = 6 kg, the transmissibility is –105.2 dB with c₁ = 0.053 Ns/m and is –116.8 dB with c₁ = 0.342 Ns/m. In other words, the vibration isolation performance increases with the increase in air damping. It also shows that the low-transmissibility region is enlarged when the applied payload increases. This result indicates that the vibration isolation performance can be improved when the applied payload is increased.

Figure 11 shows the effect of joint friction damping on the transmissibility of the vibration isolator. Here, the joint friction damping coefficient c₂ ranges from 0 Ns/m to 0.5 Ns/m, the air damping coefficient c₁ is fixed to 0.005 Ns/m, and the gear friction damping coefficient c₃ is fixed to 0.01 Ns/m. The spring stiffness k = 2650 N/m and the equilibrium position θ = π/4 are selected. The figure shows that, with a prescribed frequency and payload, the increase of the joint friction damping leads to an increase in transmissibility. For example, with f = 10 Hz and M = 6 kg, the transmissibility is –77.4 dB with c₂ = 0.053 Ns/m and is –39.6 dB with c₂ = 0.316 Ns/m. These results imply that the vibration isolation performance is lowered with increased joint friction damping. Moreover, the increase in the payload reduces the high-transmissibility regions, enhancing the vibration isolation performance. For instance, with c₂ = 0.053 Ns/m, the transmissibility is –77.4 dB with M = 6 kg and –99.4 dB with M = 18 kg.

The effect of gear friction damping on the transmissibility of the vibration isolator is shown in Fig. 12. Here, the gear friction damping coefficient c₃ ranges from 0 Ns/m to 0.5 Ns/m, the air damping coefficient c₁ is fixed to 0.005 Ns/m, and the joint friction damping coefficient c₂ is fixed to 0.01 Ns/m. The spring stiffness k = 2650 N/m and the equilibrium position θ = π/4 are used. With a prescribed frequency and payload, the increase of the gear friction damping leads to an increase in transmissibility. For example, with f = 10 Hz and M = 6 kg, the transmissibility is –80 dB with c₃ = 0.158 Ns/m and is –64.9 dB with c₃ = 0.342 Ns/m. These results mean that the vibration isolation performance is lowered when the gear friction damping increases. The figure also shows that the high-transmissibility regions are diminished when the applied payload increases. For instance, with c₃ = 0.158 Ns/m, the transmissibility is –80 dB with M = 6 kg and is –102 dB with M = 18 kg. Thus, it can be concluded that the vibration isolation performance is enhanced with the increase of the payload.

msc adams software was used to verify the performance of the proposed vibration isolator, as illustrated in Fig. 13. In this model, the link length l = 0.1 m and spring stiffness k = 2650 N/m were chosen. The base excitation was given as x = 0.02cos(2πft) (m) with time t from 0 s to 20 s. The simulation of the vibration isolator was performed through three scenarios: varying equilibrium positions, varying frequencies, and varying payloads. Figures 14–16 show the simulated accelerations of the vibration isolator. In scenario I, the maximum platform acceleration (response) is generally smaller than the maximum base acceleration (excitation). This result implies that vibration isolation is achieved at all the equilibrium positions. For example, at the equilibrium position θ = π/6, the maximum base and platform accelerations are 0.79 m/s² and 0.072 m/s², showing an acceleration reduction of 90.9%. The accelerations at the equilibrium positions θ = π/5, π/4, and π/3 are reduced by up to 91%, 91.3%, and 91.6%, respectively. In scenario II, the increase of the frequency leads to an increase in the maximum base acceleration (e.g., from 0.79 m/s² with f = 1 Hz to 54.9 m/s² with f = 8 Hz). The increase of the frequency also increases the maximum platform acceleration (e.g., from 0.069 m/s² with f = 1 Hz to 0.94 m/s² with f = 8 Hz). These results indicate that the acceleration can be reduced by 91.3% with f = 1 Hz and 98.3% with f = 8 Hz. In scenario III, the vibration isolation performance is attained when the payload varies. With M = 6 kg, the maximum base and platform accelerations are 0.79 and 0.069 m/s², showing a reduction of 91.3%. Increasing the applied payload reduces the platform acceleration, leading to an increase in the vibration isolation performance. For instance, with M = 12 kg, the maximum platform acceleration is 0.051 m/s², corresponding to a reduction of 93.5%.

Recent literature indicates a growing number of design approaches for low-frequency vibration isolators, with some efforts aimed at increasing the isolation stroke [43–47], while others focused on accommodating variable payloads [48,49]. However, there is no low-frequency vibration isolator that can achieve both large-stroke and variable-payload capabilities. In other words, the proposed low-frequency vibration isolator, incorporating the GSUs, represents an innovative design that makes a significant contribution to the advancement of low-frequency vibration isolation technologies. From the analytical results presented in Sec. 4, one can observe that the increase of joint and gear friction damping lowers the vibration isolation performance. This result underlines the importance and meaning of reducing friction in the design of vibration isolators. In contrast, air damping is proportional to the vibration isolation performance. Thus, incorporating more damping between the base and the platform proves to be a more effective strategy for enhancing vibration isolation performance.

In this article, the HBM method is used to derive the analytical solution for the dynamic response of the vibration isolator. A comparison between the analytical results obtained using matlab and the simulation results from msc adams is presented in Fig. 17. Here, the payload M = 6 kg, the equilibrium position θ = π/6, and the spring stiffness k = 2650 N/m are used. With the frequencies f = 1 and 2 Hz, the analytical and simulated results are quite similar. To measure the similarity of the analytical and simulated responses, their cross-correlation amplitude is calculated, as illustrated in Figs. 17(c) and 17(d). The details of the calculation of the cross-correlation amplitude of two responses can be found in Refs. [51,52]. With f = 1 Hz, the peak cross-correlation amplitude is 0.086 at lag 0, indicating that the simulated response matches the analytical response the most. The similarity is reduced with the increase and decrease of the lag. Similar results are obtained with f = 2 Hz, where the peak cross-correlation amplitude is 0.814 at lag 0. The results above imply that the HBM method is relatively accurate in analyzing the response of the low-frequency vibration isolator.

This article presented a novel low-frequency vibration isolator capable of achieving both large-stroke motion and adaptability to variable payloads. The isolator was based on a 6-RRR parallel mechanism, with a GSU integrated into each leg to realize a quasi-zero-stiffness characteristic. The necessary link and spring parameters were analytically determined based on the zero-stiffness condition. The HBM was utilized to analyze the dynamic response of the vibration isolator. Numerical examples demonstrated that the response acceleration was significantly lower than the excitation acceleration, with a reduction exceeding 97%. Effective vibration isolation was achieved across a wide range of excitation frequencies—from 0 to ∞—as well as under varying equilibrium positions and payload conditions. The results also indicated that vibration isolation performance improved with larger payloads, reduced friction damping, and increased air damping.

The APC for this publication was supported by VinUniversity.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.