Research on a Novel CRSM for a Type of QZS Vibration Isolator

Quasi-zero stiffness is usually abbreviated as QZS. This kind of QZS isolator has a negative stiffness mechanism, which is usually a spring mechanism (NSSM), thus possessing excellent isolation performance. However, it is prone to instability under low-frequency and large amplitude excitation. In response to this situation, a novel type of cam and roller spring mechanism (CRSM) is designed. This mechanism is composed of an arc-shaped groove, a rolling element, a spring, and a sliding pair. Use the combination of CRSM and NSSM to improve the stability of QZS isolators and prevent instability. Under two typical excitations, simulation tests in SIMULINK are conducted to analyze the vibration attenuation performance of the improved isolator and the role of CRSM in improving stability. The conclusion is that CRSM can greatly improve stability without reducing vibration damping performance.


Introduction
QZS means quasi-zero stiffness.Positive and negative stiffness mechanisms are the core of QZS vibration isolators.Springs are usually used to provide positive stiffness, and in many cases, springs are also used to form negative stiffness mechanisms (NSSM).Therefore it has good isolation performance.In recent years, many research achievements have been made in this field [1][2][3][4].Structural innovation is a hot research topic.Many kinds of QZS vibration isolators with innovative structures have emerged in large numbers.Each new structure is designed to solve specific problems.For cutting down the load static deflection, two sets of diagonal mechanical springs are introduced to the vibration isolator [5].In order to upgrade the attenuation performance when the amplitude of excitation is large, a shear-thinning viscous damper is adopted in QZS vibration isolator [6].Static bearing capacity and vibration isolation performance are often difficult to achieve simultaneously, but both are equally important.Regarding this, an absorber for vibration that utilizes springs to provide negative stiffness is designed and installed between the isolated mass and the base [7].To reduce the size of the QZS isolator, a novel configuration consisting of a coupled pair of magnetic rings arranged in parallel with a wave-spring is introduced [8].For improving the isolation effect of ultra-low frequency excitation, a proposed vibration isolator with an enhanced double QZS structure is presented, comprising two secondary QZS structures arranged in parallel and connected by vertical springs [9].
Cam and roller spring mechanism (CRSM) is very famous for vibration attenuation.In recent years, the research and application of CRSM in QZS isolators have received great attention.For the vibration isolator, if a couple of titled springs or a spring and rod structure is employed, an irregular jumping phenomenon will emerge when damping is minor and amplitude of excitation is substantial.In order to cope with this problem, an improved CRSM with rods is introduced into the vibration isolator [10].
Since CRSM is capable of yielding a specified restoring force irrespective of the displacement and the X-shape structure has strong load support ability, a novel QZS vibration isolator combining an X-shape configuration and CRSM structure is proposed [11].A new parabolic cam-roller is designed and used in QZS isolator, and some important theoretical equations are given and the conditions for QZS characteristics are obtained [12].In addition, CRSM also can be used as negative stiffness mechanism [13][14][15].
The new CRSM proposed in this paper is different from the aforementioned CRSM.The aforementioned CRSM mainly utilizes the tangent relationship between the rolling element and the outer contour of the cam to achieve the effect of variable stiffness.The CRSM proposed in this paper utilizes the relative motion relationship between the rolling element and the inner contour (arc groove) of the cam to limit travel and avoid instability.The mix of CRSM and NSSM is an innovation of this paper, which not only ensures the excellent isolation performance of QZS isolator, but also greatly improves its stability under low-frequency and large amplitude excitation.
The organization is as follows.Firstly, a novel CRSM is designed for isolator.Then, the performance is analyzed in theory.Finally, simulations are carried out and conclusions are presented.

Design of the CRSM in Isolator
Here, a novel CRSM for a type of QZS isolator is designed and its model is given in figure 1.From this figure, we can see that CRSM and NSSM are arranged in parallel on each side of the isolator.It should be noted that after placing the load, the compression amount of the main spring must be adjusted first to make the connecting rod in a horizontal state.At this point, by designing the structural dimensions, the roller in the CRSM is positioned in the middle of the arc.In fact, the negative stiffness mechanism of the isolator is a composite structure.The coupling relationship is one of the key research focuses, which achieves the optimal combination through parameter matching to achieve the best quasi-zero stiffness characteristics of the whole.x is upward displacement of mass relative to the initial position, y is the displacement of input excitation.kv and cv, represent the stiffness and damping of the spring, respectively, with different subscripts representing springs at different positions.The meanings represented by other letters can be seen from figure 1.Assuming that the excitation input is applied to the base in the initial state, and the motion between the various components of the system is not synchronized, the base will first move upwards, causing the connecting rod to tilt, thereby triggering a nonlinear relationship between the base and the load.The nonlinear dynamic equation is as follows: (1) where , , , , , and is the initial maximum deformation.Therefore, based on the mathematical relationship between force and stiffness, the system stiffness equation can be obtained as: (2) Equation ( 2) exhibits a high degree of nonlinearity and is very complex.In order to achieve good performance, we can simplify it, for example, making kv equal to 2kh and L equal to ∆max, as follows: (3) Under the parameter equivalence relationship given in equation (3), the system has a certain range of quasi-zero stiffness characteristics and excellent performance.Equation ( 2) can then be transformed into another form: (4) To reflect more general relationships, dimensionless processing is a particularly common and necessary means.The specific method is to divide both sides of equation ( 4) by kr simultaneously.And z/R-r itself is a ratio of length units, which is already dimensionless.Equation ( 4) after dimensionless processing is as follows: (5) To make equation ( 5) physically meaningful, the absolute value of dimensionless z (z/R-r) should be less than 1.The stiffness of the dimensionless system exhibits a nonlinear dependence on the dimensionless relative displacement.The relationship between them can be seen clearly in figure 2. The size of the excitation energy has a very significant impact on the performance of the system.This is easy to understand.If the amplitude of the excitation is too large, any isolation system will deteriorate and will not have a good isolation effect.Therefore, the same applies to the vibration isolation system designed in this paper, which must ensure that the relative displacement z is within a certain range, where z<L.And due to z<R-r, the simplest way to meet the requirements is to establish the following equation: (6)

Numerical Simulations
Based on the aforementioned theoretical analysis, the simulations can be carried out to test the performance of the isolator and the functions of the SRSM.Variable values are given in table 1.The excitations given in equations ( 7) and ( 8) are chosen to input the isolator for simulation, case a) and b) respectively.
Under case a) and b), the simulations are successfully implemented.Results are obtained and given in figure 3, figure 4 and figure 5.

Conclusions
An innovative CRSM used for a type of QZS isolator is presented.Theoretical analysis and simulation are conducted on the entire new vibration isolation system.The simulations show that the combination of CRSM and NSSM can greatly improve the performance of QZS vibration isolators.CRSM can improve system stability and avoid instability in extreme situations.Meanwhile, CRSM did not significantly weaken the isolation performance of NSSM.

Figure 2 .
Figure 2. Stiffness-displacement curve of the system after dimensionless processing.

Figure 3 .
Figure 3. Vertical acceleration curves under case a). r

Figure 4 .
Figure 4. Acceleration curves of isolator with CRSM and no CRSM under case b).

Figure 5 .
Figure 5. Displacement curves of isolator with CRSM and no CRSM under case b).