## 2 Development of an Analytical Model for LGTS Disk Springs  
The slotted disk spring with a number of evenly distributed slots is a modification of the conical disk spring. A linearly gradient thickness slotted (LGTS) disk spring is presented in Figure 1. As the model of slotted disk springs is derived from that of conical disk springs, the assumptions proposed previously [20] are still employed: (a) the axial compressive force is evenly distributed over the edges; (b) the cross-section will not change its shape during the process of loading-unloading and rotation around a neutral point; (c) angular deflections are sufficiently small; (d) different contact edges have the same friction coefficient; and (e) the radial stresses are negligible. The slotted disk spring consists of two parts: the non-slotted segment and the slotted segment [17; 32]. The non-slotted segment is known as the closed-ring section, and the slotted segment is referred to as tongues, as shown in Figure 2. Consequently, the total deflection of the slotted disk spring can be expressed as  
\[f=f_{2}+f_{3}=\frac{a-d}{a-b}f_{1}+f_{3} \tag{1}\]  
where \(a\) is the outer radius; \(b\) is the inner radius; \(d\) is the minimum inner radius; \(f_{1}\) is the deflection due to the rotation of the non-slotted segment; \(f_{2}\) is the deflection caused by the rotation of the slotted segment; \(f_{3}\) is the deflection caused by the bending of the slotted segment; \(w_{b}\) is the width of the slotted part at radius \(b\); \(w_{d}\) is the width of the slotted part at radius \(d\); and \(L\) is the length of the slotted part. [PERSON] et al. [26] introduced the dimensionless thickness variation parameter \(\tau\) to describe the form of the disk spring with a linearly gradient thickness, which can be expressed as  
\[\tau=\frac{t_{b}-t_{a}}{t_{a}+t_{b}} \tag{2}\]  
where \(t_{a}\) is the thickness at radius \(a\) and \(t_{b}\) is the thickness at radius \(b\). Simultaneously, the thickness is given as  
\[t(x)=T_{0}+T_{1}x \tag{3}\]  
Figure 1: Three-dimensional model of the LGTS disk spring. Figure 2: Schematic of the LGTS slotted disk spring. where \(T_{0}\) and \(T_{1}\) are denoted as  
\[T_{0}=t_{m}\bigg{(}1+\tau\frac{a+b}{a-b}-2\tau\frac{c}{a-b}\bigg{)} \tag{4a}\] \[T_{1}=2\tau\frac{t_{m}}{a-b} \tag{4b}\]  
where \(t_{m}\) is the thickness located at radius \(m\) and \(m=(a+b)/2\). For a non-slotted disk spring, the parameter \(\tau\) has an invariable region of \(-1\) to \(1\) that is independent of the geometric parameters. However, the region of the dimensionless parameter \(\tau\) for a slotted disk spring is subjected to the geometric parameters. Then, a new dimensionless parameter \(\tau^{\prime}\), which has a stationary region (\(-1\), \(1\)) and is independent of geometrical parameters, is defined to characterize the thickness characteristics of the LGTS disk spring and shown as  
\[\tau^{\prime}=\frac{t_{d}-t_{a}}{t_{d}+t_{a}} \tag{5}\]  
where \(t_{d}\) is the thickness at the radius \(d\). Combining Equations (2) and (5), the relationship between \(\tau\) and \(\tau^{\prime}\) can be derived as  
Figure 2.0  
where \(C\) is the ratio of \(a\) to \(b\) and \(C_{1}\) is the ratio of \(a\) to \(d\). The schematic of an LGTS disk spring is illustrated in Figure 2.1 In addition, the thickness is defined as  
Figure 2.2  
where \(T_{0}^{\prime}\) and \(T_{1}^{\prime}\) are denoted as  
Figure 2.3  
Figure 2.4  
where \(t_{m}^{\prime}\) is the thickness at radius \(m^{\prime}\) and \(m^{\prime}=(a+d)/2\) and \(c^{\prime}\) is the radius of the neutral point, which can be given by  
Figure 2.5  
The radius \(c_{0}\) of the neutral point for the uniform slotted disk spring is calculated by taking \(\tau^{\prime}=0\) into Equation (9). Figure 2.6  
On the basis of the assumption that all contacts are frictionless, the load-deflection relationships of uniform disk springs were derived from the equilibrium equation of the external applied moment and internal moment [20; 26; 28].