## 3 Design Methodology Based on the Common Framework
### Identifications and FE Model Calibration  
The experimental results were conducted on a full-scale laboratory structure, as depicted in Figure 2. This structure comprises a slender steel-concrete composite slab with dimensions of 5.8 m \(\times\) 1.8 m \(\times\) 0.14 m, resulting in a thickness-to-span ratio of only 1/40. The control of this structure was achieved using two APS electrodynamic actuators, driven by two power amplifiers (APS 144 and APS 145, APS Dynamics, Inc., Sand Diego, CA, USA). Acceleration measurements were performed utilizing a piezoelectric accelerometer (PCB Piezotronics model 393B31, PCB Piezotronics GmbH, Huckelhoven, Germany). To induce excitations and obtain the experimental frequency response functions (FRFs), a large-sledge impulse hammer (PCB model 086D50, PCB Piezotronics GmbH, Huckelhoven, Germany) was employed. This experimental setup enables comprehensive investigation and analysis of the structural response under controlled conditions, providing valuable insights into the efficacy of the proposed active vibration control strategies. To commence, a finite-element (FE) model was constructed using ANSYS Multiphysics 2022R2 [42], employing shell elements and encompassing 45 nodes (refer to Figure 3). This FE model serves as the basis for selecting appropriate actuator locations capable of mitigating the most dominant vibration modes. Consequently, the actuators and accelerometers were strategically positioned at nodes N16 and N22 to control and measure the first vibration modes (see Figures 3 and 4). Figure 4 illustrates the mode shapes of the first three vibration modes, consisting of two bending modes and one torsional mode. These mode shapes aid in identifying suitable actuator locations for effectively canceling the most significant vibration modes. Notably, the hammer is utilized at node N12, identified as an optimal location for exciting all vibration modes uniformly. This meticulous approach ensures that the experimental setup is finely tuned to capture and address the dominant dynamics of the structure, facilitating accurate assessment and validation of the active vibration control strategies proposed in the study. Figure 2: Photograph of the laboratory-based floor structure employed in this work for open-loop characterization, model development, and closed-loop experimental validation of the proposed control scheme. The next step is to identify the structure and actuators. The actuators used in this word were inertial mass actuators (also known as proof-mass actuators), which generate forces through the acceleration of an inertial mass on the structure on which they are placed. The actuator consists of an inertial (or moving) mass (denoted as \(m_{A}\)) attached to a current-carrying coil moving in a magnetic field created by an array of permanent magnets. The inertial mass is connected to the frame by a suspension system, modeled by a spring stiffness \(k_{A}\) and a viscous damping \(c_{A}\)[38] (see Figure 5). Figure 4: ANSYS results showing the first three resonant modes of the floor structure. Figure 5: (a) Inertial actuator: electrodynamic shaker (APS Electro-Seis Dynamic Shaker 400). (b) Lumped mass model for the actuator. Figure 3: Node numbers and coordinates initialized on the laboratory-based floor structure shown in Figure 2.  
The actuator parameters identified in Equation (5)) are obtained by configuring the actuator and its amplifier to operate in voltage mode. The parameters within Equation (5)) have been meticulously tuned to align the 3 rd-order transfer function with the empirical data presented in Figure 6:  
* N16: \(g_{A,1}=145\), \(\xi_{A,1}=0.55\), \(\omega_{A,1}=11.3\) rad/s and \(\epsilon_{1}=62.8\);
* N22: \(g_{A,2}=180\), \(\xi_{A,2}=0.45\), \(\omega_{A,2}=11.3\) rad/s and \(\epsilon_{2}=62.8\). The next step is to identify the structure with the actuators.