Study on the performance of a novel 1DOF isolator with QZS characteristics

A typical one-degree-of-freedom (1DOF) shock isolator that has quasi-zero stiffness (QZS) characteristics consists of two transversal springs and one portrait spring. The transversal springs can provide negative stiffness and the vertical spring is used as a load-bearing component. In this article, an improved 1DOF QZS shock isolator is obtained by utilizing the combination of lateral bars and springs. The displacement transmissibility under the harmonic excitation is obtained and the principle of minimum displacement transmissibility under the condition of system stability is proposed. Finally, numerical simulation is conducted, and the simulation results reflect that the proposed 1DOF QZS isolator significantly performs better than the previous one.


Introduction
It is of great significance to control harmful vibrations.Both active and passive methods can be used to suppress vibration.Passive vibration isolation is more stable at a low cost, so it is usually the first choice for vibration control [1].One-degree-of-freedom (1DOF) vibration isolator can be composed of a linear spring and damping element, which is the simplest linear vibration isolation element.The stiffness and damping of nonlinear isolators can be changed, which makes them have better isolation performance than linear isolators.Recently, biomimetic nonlinear isolators have received widespread attention from scholars and have achieved many important results.The body structure of animals and humans has greatly inspired researchers and some nonlinear vibration isolators with new structures have emerged, such as bionic polygonal skeleton structure [2], X-shaped structure [3][4][5][6], multi-layer and series structure [7][8].The QZS isolator has been widely used in engineering practice.It can be used for supporting floating slab tracks, dealing with issues of marine equipment, and also be used for ultra-precision sensing systems, vibration sensors, and vehicle suspension.The demand for engineering applications has driven the rapid development of QZS isolators.
Structural innovation is a significant feature of the development of QZS shock isolators.The structure of a conventional QZS shock isolator consists of a portrait spring and two gradient springs [9].The former can achieve positive stiffness, while the latter can obtain negative stiffness.This kind of QZS vibration isolator has a narrow QZS range and frequency band of shock isolation.Therefore, to enlarge the frequency band of the shock isolation, a new improved QZS isolator is proposed, which has two pairs of inclined springs [10].Obviously, for this type of isolator, the original position must be within the support points of the top pair of diagonal springs and the initial static balance place.With this arrangement, the QZS isolator's initial deflection will be very small, resulting in a significant increase in the frequency band of shock isolation.Electromagnetic components, cam mechanisms, etc. have been introduced into the vibration isolator, forming a large number of new structures.Inspired by various new structures, this article designs a new 1DOF QZS shock isolator and studies its performance.

Structure design
The traditional 1DOF QZS shock isolator consists of two horizontal springs and one standing spring.The horizontal springs are recognized as a negative stiffness mechanism.The standing spring is a load-bearing component, and its model is shown in Figure 1.For optimal performance, the two side springs are horizontal at the initial balance position.According to Figure 1, restoring force F C-QZS can be obtained as: where k v is the standing spring's stiffness, k h is the horizontal spring's stiffness, z represents the isolated mass's displacement, z r represents the harmonic excitation input, l 0 is the horizontal springs' initial length, and H is the structural parameter, as shown in Figure 1.The vertical spring's damping coefficient is c v , and structural parameter H and the bar's length L can be seen in Figure 2. It is worth noting that both the vertical and side springs are in a compressed state at the initial state shown in the figure.

Displacement transmissibility
The differential equation for isolated mass m is obtained as follows: Since the structural design can make L˃˃z r -z or l 0 =H, in either case, Equation ( 3) can be linearized to: By replacing Equation (4) in Equation ( 2), the differential equation can be changed as follows: Taking the harmonic excitation z r =Zcos(ωt) as the input, Equation ( 5) is a simple ordinary differential equation and its analytical solution can be obtained by the method of undetermined coefficients as follows: Further, we can obtain the displacement transmissibility as follows: where ||•|| 2 is the spectral norm of the matrix.After further solving Equation ( 9), we can obtain: Equation (10) shows that in the very low-frequency scope, the equation can reasonably be simplified as T D ≈1.This is also consistent with the actual situation.Compared with the force transmission transmissibility, the low-frequency range of displacement transmissibility close to 1 is wider.To obtain sufficiently small displacement transmissibility, by differentiating Equation (10) and calculating the extreme value point, we can get: There are only two extreme points of displacement transmissibility T D in the whole frequency range.Further judgment shows that ∆k 1 makes T D take the minimum value and ∆k 2 makes T D take the maximum value.At the same time, the value of ∆k must make the system given in Equation ( 2) stable.We suppose that x 1 and x 2 are state variables and , and then Equation ( 5) can be written as a state equation as follows: where 12) is stable if and only if the coefficient matrix of x has eigenvalues with non-negative real parts.Thus, the following equation is obtained as: If Equation (13) holds, it must make 0  k .∆K1<0, which makes T D get the minimum value, does not meet the stability condition.Since T D is a continuous function of ∆k, the value that satisfies the stability condition and is closest to ∆k 1 is the best.Therefore, ∆k=0 and it is the optimal value.According to the equation of ∆k, ∆k=0 can be achieved when l 0 =H and k v =2k h , and Equation ( 10) can be simplified as:

Displacement analysis
The relationship curves between displacement transmissibility T D and ∆k are shown in Figures 3-6.
The curves show that when the new isolator is a stable system (∆k≥0), the displacement transmissibility gets the minimum value when ∆k=0.

Time domain analysis of the new isolator
Time domain analysis can also verify the characteristics of the proposed shock isolator.The output acceleration of the proposed 1DOF QZS shock isolator model utilized the principle of minimum displacement transmissibility under the condition of system stability is compared with the previous model, given in Figure 7.The input signal excitation's amplitude is 0.01 m, and the angular frequency is 8 rad/s.For fairness, the structural parameters of the two models are identical, and m=10 kg, c v =100 Ns/m, k v =2, 000 N/m, and k h =1, 000 N/m are chosen.The characteristics of the new proposed 1DOF QZS shock isolator have been greatly improved.

Conclusions
The combination of oblique bars and transverse springs can improve the shock attenuation of the QZS isolator.Inspired by this, this article introduces lateral bars to improve the 1DOF QZS vibration isolator.A comprehensive analysis of displacement transfer rate and time-domain vibration isolation effect was conducted.The results indicate that the improved 1DOF QZS shock isolator performs better than the previous one.Under the condition of ensuring the stability of the whole shock isolator system, the optimization principle aiming at the minimum displacement transmissibility is effective and innovative.

Figure 7 .
Figure 7. Acceleration comparison in the time domain.