# Performance of Strengthened Accelerated Oscillator Damper for Vibration Control of Bridges
## 3 Composition and Working Principle of the NSD  
### Components and Derivation of Constitutive Equation for NSD  
The NSD is a nonlinear spring device that incorporates connecting rods and linear springs arranged in a diamond pattern. It is primarily composed of four connecting rods, four hinges (Hinge A\(-\)D), a hinge-linked spring (Spring A), a spring without a hinge-linked spring (Spring B), and a cradle. These rods, functioning as truss elements, are interconnected via the hinges. Spring A is connected to Hinge A and Hinge C at its two ends, while a cradle is positioned in the middle of Spring A to hold Spring B. In the NSD's initial position, where no external forces are applied and there is no displacement at terminals A and B, the terminals of Spring B maintain a preset distance from Hinges B and D. A detailed visual representation of the NSD's structure is depicted in Figure 2(a).  
The NSD operates similarly to a spring with changing stiffness, and we view it as an integrated unit. To delineate the relationship between the equivalent external force, denoted as \(F\), and the tension (or compression) of the NSD, represented by \(z\), we introduce the following parameters: \(k_{A}\) signifies the stiffness of Spring A, while \(k_{B}\) corresponds to the stiffness of Spring B. Additionally, \(l_{1}\) designates the length of the connecting rod, \(l_{2}\) denotes half of Spring B's length, and \(l_{3}\) represents half the distance between Hinge B and Hinge D. Given the NSD's symmetry, it is advisable to consider Terminal A as the stationary terminal  
Figure 3: The NSD’s diagram of (**a**) structural form, (**b**) geometric parameters.  
Figure 2: Time-history displacement response \(x_{1}\).  
and Terminal B as the mobile one. As a result, \(F\) and \(z\) represent the force and displacement at Terminal B, respectively. For detailed parameter specifications, please refer to Figure 3b.  
In this paper, it is clearly stated that when the NSD is under tension (specifically, when Terminal A and Terminal B are moving apart), \(z\) assumes a positive value (\(z>0\)); otherwise, it becomes negative (\(z<0\)). The value of z can be categorized into two distinct stages: the first ranging from \(-2(l_{3}-l_{2})\) to \(2(l_{1}-l_{3})\) and the second from \((-2l_{3})\) to \(-2(l_{3}-l_{2})\). These two stages correspond to two distinct situations: the first represents the NSD not in contact with Spring B, while the second represents the NSD in contact with Spring B. The relationship between \(F\) and \(z\) can be determined by applying the mechanical equilibrium equation for Hinges C and D, which are successively aligned. Following is an explanation of this relationship:  
\[F=\left\{\begin{array}{cc}\left[2k_{A}\cdot z_{p^{\prime}}(l_{4}-l_{5}) \right]/l_{5}&-2(l_{3}-l_{2})\leq z<2(l_{1}-l_{3})\\ \left[2k_{A}\cdot z_{p^{\prime}}(l_{4}-l_{5})\right]/l_{5}+F_{B}&-2l_{3}\leq z< -2(l_{3}-l_{2})\end{array}\right. \tag{20}\]  
The parameters used in Equation (20) are detailed below:  
\[l_{4}=2\sqrt{{l_{1}}^{2}-{l_{3}}^{2}} \tag{21}\]  
\[z_{p}=N+0.5\cdot z \tag{22}\]  
\[l_{5}=2\sqrt{{l_{1}}^{2}-{z_{p}}^{2}} \tag{23}\]  
\[F_{B}=k_{B}\cdot\left[z+2(l_{3}-l_{2})\right] \tag{24}\]