Analytical and experimental investigation of a high-static-low-dynamic stiffness isolator with cam-roller-spring mechanism
Graphical abstract
Introduction
Vibration is a common physical phenomenon in engineering systems, which can cause unstable or damage to the equipment, so that vibration isolator is widely used in industrial applications [1], [2], [3]. Vibration isolators can be divided into three types: active isolator, semi-active isolator and passive isolator. Compared with the active and semi-active isolators, passive isolator is the common choice, because it is simple and energy efficient [4]. Linear spring isolator is a usual passive isolator, but its isolation performance in low-frequency is not so well. As we all know, the linear spring isolator starts having isolation effect when the excitation frequency is above [5], where ω0 is the first-order natural frequency of the linear spring isolator. The weakness of the linear spring isolator is that when the isolated mass is small, the stiffness of the spring should also be extremely small to achieve vibration isolation at low-frequency. Hence, the most urgent problem is to find a kind of isolator with large loading capacity and low dynamic stiffness around the equilibrium position.
The high-static-low-dynamic stiffness (HSLDS) isolator is a kind of emerging isolator which contains high static stiffness to bear the isolated object and low dynamic stiffness (approximates to zero) in the working phase around the equilibrium position to achieve low-frequency vibration isolation. The structure of the HSLDS isolator is a positive stiffness mechanism and a negative stiffness mechanism set in parallel, in which the positive stiffness mechanism provides the positive stiffness force in the whole phase and the negative stiffness mechanism provides the negative stiffness force to counteract the positive stiffness force in the working phase around the equilibrium position, so that the dynamic stiffness of the working phase is very low. There are many HSLDS isolator models have been proposed in research works, such as: three-spring-combination (TSC) model, Euler buckled beams (EBB) model, and cam-roller-spring mechanism (CRSM) model, and so on.
TSC model is first put forward by Carrella et al. [5] and becomes the most popular and widely studied HSLDS isolator in research field. They used two oblique springs as a negative stiffness mechanism to design a TSC model isolator [5]; discussed the chosen of optimum parameters to obtain the HSLDS characteristic [6]; analyzed the force transmissibility of the TSC model isolator [7]; and compared the difference between the force transmissibility and displacement transmissibility of the isolator [8]. Xu et al. [9] designed the TSC model isolator using five springs, in which four inclined springs were used as the negative mechanism, and the experiment results showed that the proposed isolator had a better performance than the linear isolator in low-frequency but they were equivalent in high-frequency. Le and Ahn [10,11] using the rod and vertical spring as the inclined spring is another form of the TSC model. It was applied as the driver seat to isolate the low frequency vibration, and the experiment results demonstrated that it contains wide isolation frequency range and low resonance amplitude. Further, Le and Nguyen [12] proposed an adjustable mechanism to change the structure parameters of the isolator. Based on the rod-TSC model, Dong et al. [13] added the geometric nonlinear damping (GND) to the HSLDS isolator, and the results demonstrated that this kind of nonlinear damping performs better than the cubic damping under base excitation condition. Wang et al. [14] discussed the influence of two parameters on the dynamic response and isolation effectiveness, and the result was beneficial to the design, analysis and application of the isolator.
Huang et al. [15] replaced the inclined springs into Euler buckled beams, at that time, the TSC model was transformed into the EBB model, and the parameters choosing principles were proposed [16]. The analysis showed that the EBB model isolator had a superior performance to the linear isolator under small duration shock [17] and stiffness and load imperfections [18].
CRSM model provided by Zhou et al. [19] adopted the interaction force between the cam and roller as the negative force. The relationship between the cam and roller contained contact state and disengagement state, so that the dynamic model was described as a piecewise function. Wang et al. [20] designed a two-stage isolator using the CRSM model, and the analysis results showed that the two-stage isolator could bear large excitation amplitude. Zhou et al. [21] brought the CRSM model into a design of 6-DOF vibration isolation platform. The magnetic force also could provide negative force for fabrication of the HSLDS isolator [22], [23], [24], [25], [26], and the theoretical and experimental results illustrated that the isolator could lower the resonance frequency and amplitude compared with the linear isolator. Zheng et al. [27] introduced the magnetic force into the Stewart platform and the Stewart isolator could achieve six-direction low-frequency vibration isolation. There are still many other models could be employed as the HSLDS isolator, for example: the scissors-like model [28], [29], [30], [31], [32], [33], the pneumatic linear actuator model [34], disc spring model [35], origami model [36,37], and shape memory alloy model [38], [39], [40].
A pattern of constant force mechanism using cam and roller was proposed in Refs. [41] and [42]. To the author's knowledge, this kind of model has not been used as a vibration isolator. In this paper, referred to Refs. [41] and [42], we present a HSLDS isolator using cam-roller-spring mechanism, where the interaction force between the cam and roller provides the negative force and the vertical linear spring provides the positive force. Different from the CRSM model in Ref. [19], the dynamic characteristic of the proposed isolator is a continuous function and it can be designed specially to meet individual working requirement. A cam with piecewise linear function force-displacement variation is set up, and the dynamic response of the isolator under base excitation with constant displacement is obtained, meanwhile, the effects of damping ratio and base excitation amplitude are discussed. To verify the actual isolation performance, an experimental test system is constructed and the test is accomplished.
The rest of the paper is organized as follows. Section 2 describes the cam design theory and configuration of the isolator. The dynamic equation, frequency response, stability analysis and parameters studies are discussed in Section 3. The experimental test system is conducted in Section 4 to demonstrate the isolation performance of the proposed isolator. Finally, some conclusions are drawn in Section 5.
Section snippets
Structure of the HSLDS isolator
Fig. 1(a) and (b) show the structural schematic diagram and physical model of the proposed HSLDS isolator. In Fig. 1(a), the isolator could be regarded as a positive stiffness mechanism (the vertical spring) and a negative stiffness mechanism (the cam-roller mechanism) set in parallel. The behavior of the HSLDS characteristic can be described as: the vertical spring provides a positive stiffness force to support the isolated object, which is the high-static stiffness characteristic; while the
Dynamic equation
The dynamic model diagram of the HSLDS isolator at the equilibrium position under base excitation is illustrated in Fig. 7, in which c is the damping coefficient, m is the equivalent mass and y = Ycos (ωt) is the base excitation with the amplitude Y. The dynamic equation of the HSLDS isolator iswhere z = x - y, f (z) is the restoring force defined by Eq. (14).
Transform Eq. (16) into a non-dimensional form aswhere τ = ω0t, Ω = ω/ω0, ξ
Experimental setup
For propose of validating the vibration isolation performance of the proposed HSLDS isolator, an experimental test system is built up as shown in Fig. 12. The aim of the experiment is to obtain the displacement transmissibility of the HSLDS isolator (as shown in Fig. 13) and the corresponding linear isolator (the rollers and horizontal springs are removed, as shown in Fig. 14) under constant displacement base excitation.
Static force-displacement relationship verification
Firstly, the static force-displacement relationship of the HSLDS isolator
Conclusions
This paper presents a high-static-low-dynamic stiffness (HSLDS) isolator using cam-roller-spring mechanism where the cam profile of the isolator can be designed and manufactured specially to meet different working requirements. The dynamic response and isolation performance are obtained through analytical and experimental method.
Firstly, the cam design theory is described. The structure of the HSLDS isolator can be considered as the cam-roller mechanism (negative stiffness mechanism) and the
CRediT authorship contribution statement
Yuhui Yao: Conceptualization, Methodology, Validation, Formal analysis, Investigation, Writing - original draft. Hongguang Li: Writing - review & editing, Supervision, Project administration, Funding acquisition. Yun Li: Data curation, Visualization. Xiaojian Wang: Software, Resources.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
Funding: This work was supported by the National Natural Science Foundation of China [grant number 11972222].
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