## 2 NSC Formed by CRMs with Quadratic Polynomial Trajectory  
In nonlinear vibration isolators, the NSC and the positive supporting structure are usually connected in parallel along the vibration isolation direction. Figure 1 illustrates the principle of the NSC in an isolation system for vertical vibration. When the vibration isolator just bears the rated mass, it moves down from the position shown in Figure 1a to the static equilibrium position in Figure 1b,c. At this time, the NSC only generates restoring force in the horizontal direction, so the load-bearing capacity of the vibration isolator, here provided by a vertical linear spring, will not be affected. When isolating vibrations, the NSC begins to deviate from the static equilibrium position, as shown in Figure 1d, and produces a vertical component force in the same direction as the movement of the isolated object. The new component force is related to displacement and results in a reduction in the dynamic stiffness of the system. Actually, the NSC, as a part of the stiffness unit, together with the supporting spring determines the stiffness characteristics of the vibration isolator. The NSC makes the stiffness of the vibration isolator nonlinear. It does not interfere with the static stiffness provided by the supporting spring, but changes the dynamic stiffness during vibration by the superposition of a variable vertical restoring force. Correspondingly, the force-displacement principles of the linear supporting spring, the NSC and the entire vibration isolation system are given in Figure 2. At the working range pointed out on the curves, the negative stiffness offsets part of the positive stiffness, and makes the system possess ultra-low dynamic stiffness while maintaining the static supporting capacity unchanged. The NSCs formed by QCRMs are indicated by dash-line boxes in Figure 1c,d. The rollers in the NSC, which have the same displacement as the isolated object, are installed on the upper loading platform by horizontal springs. As the rollers move along the quadratic polynomial trajectory of the cam mounted on the baseplate, the compression length of the horizontal springs is affected, enabling the NSC to provide a varying force to the vibration isolation system. Figure 1: Negative stiffness corrector (NSC) in a nonlinear vibration isolator. (a) Unloaded linear spring; (b) loaded linear spring at static equilibrium position; (c) isolator with NSC at static equilibrium position; (d) isolator with NSC in operation. Figure 2: Force-displacement curves of linear supporting spring, NSC, and the vibration isolator they build. The main function of the cam is to convert the vertical displacement along the vibration direction into the horizontal displacement of rollers, and then the direction of the elastic reaction force of the rollers back into the vertical direction. In CRM-NSCs, since the cam contours are easy to design and have regularity, their trajectory functions can be derived inversely according to the desired stiffness characteristics of the vibration isolator. In fact, both the negative stiffness structures with inclined-springs [10] and with link-springs [25] can be transformed into the unified form constructed by CRMs. To further understand the QCRM-NSC in our research, the forces between a set of cam and roller are illustrated in Figure 3. Define \(y\) as the absolute displacement of the roller with the upward positive direction, and only consider the vibration in the vertical direction. The roller center at the static equilibrium position in Figure 3a is set as the origin of \(y\), which is on the horizontal centerline of the cam. When the base of the vibration isolation is subjected to an external excitation \(z\), there is a vertical relative displacement \(u=y-z\) between the roller and the cam, as shown in Figure 3b.