Tunable negative stiffness spring using maxwell normal stress

Highlights

• Negative stiffness online tunable spring based on Maxwell normal stress.

• Improved stiffness tunable range and energy efficiency through a novel magnetic circuit design.

• High-static-low-dynamic isolator constructed by the parallel connection of the negative spring and a linear mass-spring-damper system.

• Tunable vibration isolator with reduced starting isolation frequency.

Abstract

A vibration isolator based on a tunable negative stiffness mechanism combines the advantages of high-static-low-dynamic stiffness (HSLDS) isolators to expand the isolation frequency band and variable stiffness isolators to suppress resonance. In this paper, a novel tunable negative stiffness spring using Maxwell normal stress (SMNS) is proposed. The stiffness tunable range and energy utilization efficiency are greatly improved due to the newly designed magnetic circuit. Moreover, the electromagnetic negative stiffness device has the advantages of no friction, no backlash, compact structure and easy control. An analytical model of the electromagnetic force is built based on magnetic circuit analysis, and the parameter analysis is performed. An HSLDS isolator is constructed by connecting the SMNS in parallel with a linear isolator. The stiffness and vibration isolation performance are measured. The experimental results show that the SMNS produces an online tunable negative stiffness, which expands the isolation bandwidth and significantly improves the vibration isolation performance.

Introduction

With the improvement of manufacturing and measurement accuracy, the demand for low-frequency vibration isolation increases rapidly. Not only the requirement of vibration attenuation has been increased, but also many additional constraints need to be met, such as excessive ambient temperature and limited volume [1], [2]. It is hard for conventional linear isolators to meet the requirements, because effective isolation requires a natural frequency that is less than $\sqrt{2}/2$ times the excitation frequency. However, the mass of an isolated object is usually unchangeable in practice, and the support stiffness has to be relatively high to avoid excessive static deflections and maintain stability [3], [4], [5].

High-static-low-dynamic stiffness (HSLDS) isolators have received considerable attention because they overcome the inherent contradiction between carrying capacity and vibration isolation performance in conventional isolators [6], [7], [8]. HSLDS refers to high static stiffness to support the load with small static deflection and low dynamic stiffness to realize low natural frequency and hence a wider isolation frequency range [9]. The HSLDS characteristic can be implemented by exploiting the nonlinear characteristics of the elastic element exhibited during the deformation process [10]. Another more common method is to connect the negative stiffness element and the positive stiffness element in parallel [4].

The Negative stiffness mechanism (NSM) provides decreasing resistance to external force with increasing displacement [11], [12], which is associated with instability and has one unstable equilibrium position where the force is zero [13]. The NSM is a key component of the HSLDS system and has been investigated by many researchers: bistable plates [14], oblique springs [15], oblique linkages connecting horizontal springs [16], [17], Euler buckled beams [18], cam-roller-spring mechanisms [19], and magnetic springs [20], [21], [22]. Negative stiffness in the rotation direction has also been realized by both mechanical [23] and magnetic approaches [24]. Moreover, the NSMs have been used in two-stage isolators [25], [26] to further improve the isolation performance, as well as multi degree of freedom systems [27], [28] and floating raft systems [2]. Nearly all theoretical and experimental studies have proven their superiority than conventional one especially in low-frequency vibration isolation.

However, most of the NSMs cited above are passive mechanisms whose stiffness cannot be adjusted. In general, passive isolators with immutable stiffness are incapable of dealing with the changing excitation frequencies. Some active control methods were introduced to the HSLDS systems to improve the stability and isolation performance [29], [30], but the high costs should be considered. Semi-active system with variable stiffness combines the tunability of active system and the reliability of passive system, so it may be a better scheme [31], [32]. Variable stiffness elements have the ability to change one or more properties to obtain the desired stiffness required to reduce a vibratory input. Variable stiffness elements that produce positive stiffness have been studied in many papers, such as magneto-rheological elastomers (MREs) [33] and helical springs with variable active coil numbers [34]. However, the positive stiffness element has the function of carrying the load weight and maintaining the static equilibrium position of the load. Adjusting the positive stiffness may change the static equilibrium position of the bearing system, making it inconsistent with the equilibrium position of the negative stiffness element, thus deteriorating the performance of the isolator. The use of a tunable negative stiffness element makes it possible to change stiffness without changing the static equilibrium position.

The tunable negative stiffness mechanisms that have been proposed are relatively few. One approach adds actuators to a conventional mechanical negative stiffness mechanism to change the preload [35], [36]. Palomares et al. [37] used pneumatic linear actuators (PLAs) to replace the mechanical springs, and the negative stiffness could be tuned by the air pressure. However, the complex structure and backlash may be a limitation of the practical application of mechanical methods. Adjusting the relative position between the magnets can also change the negative stiffness [38], [39]. Using electromagnets to tune the stiffness online by controlling the current may be a more effective way, especially when the tunable spring is integrated with a control system. Zhou and Liu [40], [41] proposed a tunable negative stiffness spring, which used two electromagnets with iron cores to attract permanent magnet blocks in the middle to produce negative stiffness. A semi-active control algorithm was proposed to tune the stiffness online, and a good isolation effect was obtained in a large frequency range. Ledezma-Ramirez et al. [42] used a similar mechanism for shock isolation. Su et al. [43] wound a coil around a permanent magnet to adjust the magnetic force, thus achieving tunable negative stiffness. In the study of electromagnetic negative stiffness mentioned above, the electromagnetic force was obtained by measurement and curve fitting. Sun et al. [44] used the repulsive force between ring coils and magnets to produce negative stiffness, and the configuration of the coils and magnets were studied to enlarge the stiffness tunable range [45]. The electromagnetic force were calculated using filament method based on magnetic field distribution. Wang et al. [46] used a similar mechanism to regulate the stiffness of an HSLDS resonator, and a semi-active metamaterial beam was realized by attach these resonators onto a beam periodically. In general, permanent magnets used in these studies can generate much stronger magnetic fields than electromagnets. However, the demagnetization of permanent magnets at high temperatures and their fragility may limit the application of these devices in specific environments. The heat generated by the current in the coil may also demagnetize the magnet, resulting in performance degradation. To ensure that the negative stiffness is large enough while eliminating the permanent magnet, it is necessary to improve the utilization efficiency of the magnetic field energy, which requires a reasonable magnetic circuit design to concentrate the magnetic flux in a small area. Improving efficiency can reduce the energy required to maintain the negative stiffness. Moreover, a reasonable magnetic circuit design is also conducive to improving force density. In fact, many electromagnetic actuators[47], [48] and electromagnetic bearings[49] are designed with magnetic circuit analysis. Some studies on permanent magnet negative stiffness mechanisms have also proved that the magnetic circuit design helps to improve the negative stiffness performance. Zhang et al. [50] studied the negative stiffness mechanism in which three permanent magnets attract each other. By wrapping iron material around the three magnets to form a magnetic circuit, the negative stiffness was increased by 2.4 times. Zhang et al. [51] placed a radially magnetized ring permanent magnet into a magnetic circuit, and achieved a large negative stiffness within a compact volume and light mass. But their permanent magnet circuits produce invariable stiffness, and the performance will still deteriorate due to the magnet demagnetization.

To the best of our knowledge, electromagnetic negative stiffness mechanisms considering magnetic circuit design have been rarely studied. Han et al. [52] generated a tunable negative stiffness through the asymmetric configuration of an electromagnetic tooth structure. In addition to utilizing the magnetic force between the staggered teeth, it may be more efficient to utilize the Maxwell normal force to generate a negative stiffness. Li et al. [53] proposed a tunable negative stiffness spring considering the magnetic circuit design, which used two U-shaped electromagnets at both ends to attract an iron piece in the middle. Liu et al. [54] sealed two ring coils with iron shells to form a magnetic circuit, and the tunable negative stiffness was generated by attracting an iron plate from both ends. Moreover, a part of air gap was filled with magnetorheological elastomers to improve the electromagnetic efficiency.

In this study, a novel tunable negative stiffness spring using Maxwell normal stress (SMNS) is proposed. Because of the newly designed magnetic circuit, the efficiency is significantly improved; that is, the same energy input can produce greater negative stiffness. The negative stiffness can be tuned online by controlling the current in the DC coils. There is no permanent magnet in the magnetic circuit of the spring, so it can be used under high temperature and strong impact conditions without performance degradation. In addition, similar to ordinary electromagnetic devices, the SMNS has the advantages of compact structure, no wear due to non-contact force, no backlash.

The remainder of this paper is organized as follows. Section 2 introduces the concept of the tunable negative stiffness spring using Maxwell normal stress. The electromagnetic force and negative stiffness are modelled using the magnetic circuit analysis method in Section 3 and then the parametric study is performed in Section 4. In Section 5, the negative stiffness spring is applied to a single degree of freedom (SDOF) system, which significantly improves the vibration isolation performance. Finally, the conclusions and future works are summarized in Section 6.