# Enhancing Vibration Isolation Performance by Exploiting Novel Spring-Bar Mechanism
## 2 Mathematical Modelling  
### Model Description  
As shown in Figure 1a, the current vibration isolation system is developed from a nonlinear isolator model from our previous study, shown in Figure 1b[14]. In both systems, a vibrating mass \(m\) is subjected to a harmonic excitation with amplitude \(f_{\text{e}}\) and frequency \(\omega\). To isolate its vibration from transmitting to base point \(D\), a usual approach is to insert a linear isolator, which comprises a linear vertical spring of stiffness \(k_{1}\) and a viscous damper with a damping coefficient \(c\). In order to improve vibration isolation performance at low frequencies, a spring-bar mechanism (SBM) can be configured with the linear isolator. Here, the new isolator includes two rigid bars of a fixed length \(l\), with one end pinned together at point \(E\) with the mass. Compared to our previous design with a pair of compressed bars [14], there are now horizontal springs with stiffness \(k_{2}\) connected to the bars so as to enlarge the design space for the stiffness characteristics. The joints \(A\) and \(B\) can move freely in fixed frictionless horizontal channels. The motion of the system is constrained to the plane of the paper. The static equilibrium position of the mass, at which the bars are horizontal, is taken as the reference position of \(x=0\). Due to the symmetry of the system, the mass \(m\) can only have a vertical displacement. In real-world applications, the mass of a force-excited machine is usually much heavier than that of the spring-bar mechanism; therefore, the latter is considered negligible in the current study. It is straightforward to construct the vibration isolator shown in Figure 1a by joining the side walls with the ground wall. In other words, the spring-bar mechanism as well as the vertical mass-spring-damper are configured in a frame. In this way, the experimental prototype of the proposed isolator can be made, and the proposed isolator can be inserted between an applied force and the mounted point, applicable to most engineering scenarios in which vibration isolators are used. Compared to the previous vibration isolator design shown in Figure 1b, which requires the generation of constant compressive force \(P\), the current design in Figure 1a comprising the SBM only requires widely available physical springs and bars and can be realized conveniently in practice. Here, we investigate the behaviour of the current SBM alone before studying the dynamics of the integrated system. As shown in Figure 2a, compared to the original length of the horizontal springs, there is an initial deflection of \(\delta_{0}\) when the bars are horizontal. Here, \(\delta_{0}\) is positive when the springs are under tension and negative when under compression. When the bar joint \(E\) is subject to a vertical displacement of \(x\), as shown in Figure 2a, the length of each spring increases by \(\delta=l(1-\cos\theta)\), where \(\theta\) denotes  
Figure 1: Schematics of the vibration isolation systems with (**a**) the SBM and (**b**) an NSM [14]. the angle between the bar \(A^{\prime}E^{\prime}\) and the horizontal line \(A^{\prime}B^{\prime}\). Correspondingly, the spring force becomes  
\[f_{p}=k_{2}(\delta_{0}+\delta). \tag{1}\]  
Based on the geometrical relations, we have  
\[\sin\theta=\frac{x}{l},\ \cos\theta=\frac{\sqrt{l^{2}-x^{2}}}{l},\ \tan\theta=\frac{x}{ \sqrt{l^{2}-x^{2}}}\, \tag{2}\]  
where \(x\
eq l\). As the springs and bars are considered to be rigid with negligible mass, the total resultant force applied on the joint \(E\) in the vertical direction is  
\[f_{r}=2f_{n}=2f_{p}\tan\theta, \tag{3}\]  
which is also the restoring force of the SBM, as shown in Figure 2b.