# Performance of Strengthened Accelerated Oscillator Damper for Vibration Control of Bridges
## 2 Topological Configuration and Motion Equations of the SAOD-LS
### Derivation and Analytical Solution of the Motion Equation for SAOD-LS  
Under the assumption that the system is not impacted by friction and the transmission is idealized with no energy dissipation [39], we can establish a generalized coordinate system by setting the equilibrium positions of \(m_{1}\) and \(m_{2}\) as the reference points. We  
Figure 1: Schematic diagram of SAOD-LS. mathematically formulate the total kinetic energy \(T\) and elastic potential energy \(V\) of the SAOD-LS, respectively. \[T =\frac{1}{2}m_{1}\dot{x}_{1}{}^{2}+\frac{1}{2}m_{2}\dot{x}_{2}{}^{2} \tag{1}\] \[V =\frac{1}{2}k_{1}{x_{1}}^{2}+\frac{1}{2}k_{2}{x_{k}}^{2} \tag{2}\]  
In Equations (1) and (2), the symbols \(x_{1}\) and \(\dot{x}_{1}\) indicate the displacement and velocity of \(m_{1}\), respectively. The symbol \(\dot{x}_{2}\) represents the velocity of \(m_{2}\), while \(x_{k2}\) represents the relative displacement between the two terminals of \(k_{2}\). Please note that in this article, the symbols \(x_{1}\) and \(x_{2}\) are used to indicate the primary and secondary mass displacement, respectively. This convention is consistently followed throughout the entire paper. Given a linear damping force, we can express the Rayleigh dissipation function (\(D\)) as follows:  
\[D=\frac{1}{2}c_{1}\dot{x}_{1}{}^{2}+\frac{1}{2}c_{2}\dot{x}_{c2}{}^{2} \tag{3}\]  
In Equation (3), the symbol \(\dot{x}_{c2}\) represents the relative velocity between the two terminals of \(c_{2}\). Because the displacement of the link-secondary-structure end is \(r\) times that of the link-primary-structure end, the idealized relationship meets the following equation:  
\[ \begin{cases}x_{2}=-r\cdot x_{1}\\ \dot{x}_{2}=-r\cdot\dot{x}_{1}\\ \ddot{x}_{2}=-r\cdot\ddot{x}_{1}\end{cases}  \tag{4}\]  
Due to the extremely small mass ratio (i.e., \(m_{2}/m_{1}\)) in actual bridges, as mentioned in [39], this study assumes that the forcing function acts only on the primary structure. By combining Equations (1)-(4) with the Lagrange Dynamic [39], we can formulate the motion equation of SAOD-LS as follows:  
\[\big{(}m_{1}+m_{2}r^{2}\big{)}\ddot{x}_{1}+\big{[}c_{1}+c_{2}\big{(}r+1)^{2} \big{]}\dot{x}_{1}+\big{[}k_{1}+k_{2}\big{(}r+1)^{2}\big{]}x_{1}=-m_{1}\ddot{ x}_{g} \tag{5}\]  
where \(\ddot{x}_{g}\) is the horizontal ground acceleration [41]. The analytical solution to Equation (5) can be derived from its equivalent equation form, Equation (6). \[\ddot{x}_{1}+\frac{c_{1}+c_{2}\big{(}r+1)^{2}}{m_{1}+m_{2}r^{2}}\dot{x}_{1}+ \frac{k_{1}+k_{2}\big{(}r+1)^{2}}{m_{1}+m_{2}r^{2}}x_{1}=\frac{-m_{1}\ddot{x} _{g}}{m_{1}+m_{2}r^{2}} \tag{6}\]  
The structure as a whole can be regarded as a SDOF system, as the displacement and velocity of \(m_{1}\) and \(m_{2}\) are connected by the transmission device. Equation (6) represents a standard differential equation for a SDOF system. To simplify the expression, the constant term in Equation (6) is substituted as below:  
\[ \begin{cases}M=m_{1}+m_{2}r^{2}\\ C=c_{1}+c_{2}\big{(}r+1)^{2}\\ K=k_{1}+k_{2}\big{(}r+1)^{2}\end{cases}  \tag{7}\]  
where \(M\), \(K\), and \(C\) represent the generalized mass, generalized spring coefficient, and generalized damping coefficient, respectively [39]. Assuming the external excitation function is harmonic oscillation, the \(\ddot{x}_{g}\) acts upon the following proposed dynamic system:  
\[\ddot{x}_{g}=a\cdot sin(\theta t) \tag{8}\]  
where \(a\) and \(\theta\) represent the amplitude and frequency of the input acceleration, respectively. The analytical solution of Equation (6) can be expressed as follows [42]:  
\[x_{1}=x_{t}+x_{s} \tag{9}\]  
where \(x_{t}\) and \(x_{s}\) denote transient response and steady-state response, respectively.