## 2 Explicit CDM for STST  
### CDM for Dynamic RTHS  
The discretized equation of motion of the NS at the \(i\)th integration time step for the dynamic RTHS can be expressed as follows:  
\[\mathbf{M}_{\text{N}}\mathbf{a}_{\text{N},i}+\mathbf{C}_{\text{N}}\mathbf{v}_{ \text{N},i}+\mathbf{K}_{\text{N}}\mathbf{d}_{\text{N},i}+\mathbf{F}_{\text{E}, i}=\mathbf{F}_{i}, \tag{1}\]  
where \(\mathbf{M}_{\text{N}}\), \(\mathbf{C}_{\text{N}}\), and \(\mathbf{K}_{\text{N}}\) denote the mass, damping, and stiffness matrices of the NS, respectively; subscript \(i\) means the integration time step; \(\mathbf{F}_{\text{E}}\) and \(\mathbf{F}\) are the reaction force vector of the ES and the external excitation force vector, respectively; and \(\mathbf{a}_{\text{N}}\), \(\mathbf{v}_{\text{N}}\), and \(\mathbf{d}_{\text{N}}\) are the acceleration, velocity, and displacement response vectors of the NS, respectively. Several integration algorithms are available for solving Equation (1), but this study prefers the CDM for its concision and widespread use. The conventional velocity and acceleration approximations of the CDM at the \(i\)th step are as follows:  
\[\mathbf{v}_{\text{N},i}=\frac{\mathbf{d}_{\text{N},i+1}-\mathbf{d}_{\text{N}, i-1}}{2\Delta t}, \tag{2}\]  
\[\mathbf{a}_{\text{N},i}=\frac{\mathbf{d}_{\text{N},i+1}-2\mathbf{d}_{\text{N},i}+\mathbf{d}_{\text{N},i-1}}{\Delta t^{2}}, \tag{3}\]  
where \(\Delta t\) is the predetermined integration time interval. Substituting Equations (2) and (3) into Equation (1), one obtains the following:  
\[\mathbf{d}_{\text{N},i+1}=\left(\frac{\mathbf{M}_{\text{N}}}{\Delta t^{2}}+ \frac{\mathbf{C}_{\text{N}}}{2\Delta t}\right)^{-1}\left[\mathbf{F}_{i}-\left( \mathbf{K}_{\text{N}}-\frac{2\mathbf{M}_{\text{N}}}{\Delta t^{2}}\right) \mathbf{d}_{\text{N},i}-\left(\frac{\mathbf{M}_{\text{N}}}{\Delta t^{2}}- \frac{\mathbf{C}_{\text{N}}}{2\Delta t}\right)\mathbf{d}_{\text{N},i-1}- \mathbf{F}_{\text{E},i}\right]. \tag{4}\]  
According to Equation (4), the target displacement response \(\mathbf{d}_{\text{N},i+1}\) at the \((i+1)\)th step can be calculated directly using the measured reaction force \(\mathbf{F}_{\text{E},i}\) and the known \(\mathbf{F}_{i}\), \(\mathbf{d}_{\text{N},i}\), and \(\mathbf{d}_{\text{N},i-1}\). However, based on Equations (1)-(4), it is impossible to explicitly express the target velocity \(\mathbf{v}_{\text{N},i+1}\) and acceleration \(\mathbf{a}_{\text{N},i+1}\) at the \((i+1)\)th step. The interface displacement in \(\mathbf{d}_{\text{N},i+1}\) is conventionally regarded as the loading target on the ES. This means that the sole loading control target is displacement regardless of the velocity and acceleration. However, the specimens in STST are known to be displacement, velocity, and/or acceleration-dependent devices. Not only the displacement target but also the target velocity and/or acceleration should be predetermined by integration algorithms and realized by the shaking table. To obtain explicit targets of displacement and acceleration, [PERSON] et al. [45] proposed the CDM for dynamic real-time substructure testing (RTST). For an actuator controlled in a traditional displacement mode, the achievement of the explicit acceleration target is dependent on how the displacement command is issued with respect to time. To this end, a constant acceleration in the time interval \([t_{i}\,t_{i+1}]\) was assumed, resulting in a displacement command profile as a quadratic function in time \(t\), namely:  
\[d_{\text{EC},i+1}(t)=d_{\text{EC},i}+v_{\text{EC},i}(t-t_{i})+0.5d_{\text{EC},i+1}(t-t_{i})^{2}, \tag{5}\]  
where \(d_{\text{EC}}\), \(v_{\text{EC}}\), and \(a_{\text{EC}}\) are the displacement, velocity, and acceleration targets for the specimen, respectively. The effectiveness of this CDM for dynamic RTST was verified on a mass-spring specimen loaded by a 2500 kN servo-hydraulic actuator [45]. Even though several instances of STST based on displacement control have been conducted in recent years, there are two issues in such STST.