Active Microvibration Control of Tool Platforms Installed on the Floors Subjected to Moving Vehicles in Industrial Factories

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Introduction
Microvibrations directly induced by internal automated material handling systems (AMHSs) [1], such as automated guided vehicles (AGVs) [2,3], rail guided vehicles (RGVs), stockers (STKs) or moving cranes [4], and overhead hoist transfer (OHT) [5], on the production foors in semiconductor or liquid-crystal-display (LCD) fabrication factories (or fabs) have rarely been considered while designing fab buildings.Consequently, majority of the AMHS-induced foor and platform vibrations surpass the allowable Bolt Beranek & Newman Vibration Criterion (BBN-VC) [6]. Figure 1 illustrates the process tools supported by tool platforms (or tool foundations) that are installed on the production foors in cleanrooms of high-tech fabs.Te tool platforms are typically composed of steel material (e.g., steel beams, columns, and plates), with some being manufactured using concrete-inflled steel tubes.Te short (or rigid) platforms, lower than 1.0 m in height, are typically employed in semiconductor fabs, whereas the tall (or fexible) ones, higher than 2.0 m in height, are generally fabricated by the LCD fabs for manufacturing larger glass panels.Te lateral stifness of the tall platforms is substantially lower than that of the short ones such that their natural vibration frequencies generally lie within the bandwidth of the BBN-VC (4-100 Hz).Accordingly, the vibration response amplifcations of the tall fexible platforms subjected to foor excitations at platform base are more remarkable than those of the short ones, and the vibration response on the platform top is larger than the allowable vibration limit within the required bandwidth.Additional horizontal vibration response on the top of the tall platforms can be attributed to the base rotation excitations induced from the uneven relative vertical foor vibrations generated by the moving AGVs.Te taller the platform, the higher the microvibration level.Hence, the monitored vibration levels of tall platforms under the AGV moving loads in LCD fabs are substantially higher than the desired VC-B level or even the VC-A level.Here, VC-B denotes the vibration criterion-(VC-) B level.Furthermore, the VC curves are a set of allowable root mean square (RMS) velocity spectra (referred to as VC-A, VC-B, VC-C, VC-D, and VC-E) corresponding to the amplitude range of 50.8-3.175μ-m/s (0.0051-0.00032 cm/s) within a frequency range of 8-100 Hz.
Te foor and platform microvibrations at precision high-tech fabs induced by external trafc such as nearby vehicles and trains have been extensively studied over the last two decades.Yang and Agrawal [7] proposed a hybrid foor isolation system with rubber bearings and actuators (electric or magnetostrictive) that were installed on the frst foor of a three-story fab; they aimed to evaluate its control performance using the modifed Kanai-Tajimi power spectral density (MKT-PSD) function to quantify the traininduced horizontal ground accelerations.Simulation results indicated that the root mean square (RMS) velocity spectrum of the hybrid foor isolation system satisfes VC-E if appropriate stifness and damping of the hybrid foor isolation system are adopted.Xu et al. [8] established an analytical model considering a three-story fab building with a hybrid-controlled platform subjected to the train-induced vertical ground motion generated using a Fourier spectrum for a given distance from the track.Te hybrid control platform consisted of active hydraulic actuators and passive mounts with springs and dashpots between the platform and building.Numerical results demonstrated that the vibration of the passive control platform may exceed VC-B, whereas that of the hybrid-controlled platform could be efectively suppressed to meet VC-E.Hong et al. [9] constructed a three-dimensional fnite element building model with a hybrid platform subjected to three-dimensional ground excitations.Te hybrid platform mounted on the second foor of the building consisted of several pairs of passive devices and active actuators.Numerical results demonstrated that the hybrid platform can efectively reduce the vibration responses in all directions to lie below VC-E.
Vibrations induced directly by the internal moving AMHSs or trucks on the production foor in foundry fabs or industrial factories have rarely been considered while designing the facilities.Pan et al. [10] investigated the container trucks traveling with varied speeds within the multistory factory buildings at the site of vibration-sensitive equipment installation.Moreover, the road roughness was considered and generated though a PSD function according to ISO 8606.Results demonstrated that the vertical foor vibrations induced by container trucks traveling at the speeds below 40 km/h generally meet the allowable vibration criterion.However, the peak vertical vibration may possibly exceed the more stringent vibration limit for certain extreme precision equipment (e.g., steppers).Hu et al. [5] proposed an analytical model including fab buildings with foundation, the OHT system, and the isolation system-controlled tool platform to investigate the OHT-induced foor microvibrations in semiconductor fabs.Results revealed that the predicted foor microvibrations are consistent with the in situ measurements beyond 10 Hz; however, they are underestimated below 10 Hz.Moreover, an 80% microvibration reduction for the isolated tool platform could be achieved.Ju et al. [4] constructed a realistic analytical model integrating a high-tech fab with wafe slabs, steel rails, and a crane to assess moving crane-induced foor microvibrations, and numerical results from fnite element analysis agreed well with the experimental measurements.Moreover, they suggested that smoothing the wheel and rail, decreasing the speed of the crane, and increasing the slab and beam depths of the fab building are efcient in reducing foor microvibrations.Lee et al. [2,3,11] explored the AGVinduced foor microvibrations under diferent AGV moving conditions (e.g., weights and speeds) in LCD fabs.Moreover, fragility analysis was conducted to develop foor microvibration fragility surfaces that can be used to evaluate the exceedance probabilities of the preferred vibration level.
Piezoelectric actuators (PEAs) [12] have been widely employed as drivers or control units in precision positioning platforms, equipment, and civil structures owing to their specifc advantages including compact size, high resolution, fast reaction time, low heat generation, low power consumption, and no electromagnetic interference (which is prohibited in high-tech fabs).Liu et al. [13] presented the recent advances in microvibration isolation in a state-ofthe-art review paper and introduced various types of passive, active, semiactive, and hybrid isolation systems or devices implemented with piezoelectric materials or actuators.Song et al. [14] reviewed the vibration control of civil structures (e.g., beams, trusses, frames, and cable-stayed bridges) using piezoelectric patch or stack actuators.Kamada et al. [15] proposed an active control strategy for the fexural-shear type frame structures using the PEAs installed underneath the column bases.Te installation locations of the PEAs can be appropriately managed to control the bending moments and axial forces of the columns.Experimental results of a four-story steel frame under the broader band random excitations indicated that the structural acceleration responses can be efectively reduced using the proposed control schemes.Hora et al. [16] proposed an active microvibration control system using PEAs to yield a reverse couple moment to suppress the vertical vibrations of the 2 Structural Control and Health Monitoring foors in precision manufacturing factories simulated as a two-dimensional steel truss frame.Te experimental results revealed that the foor acceleration response can be reduced to 1/32 of the noncontrolled case under the harmonic excitation.Etedali et al. [17] proposed a novel piezoelectric isolation system integrating an isolation system and piezoelectric friction dampers (PFDs) for seismic vibration control of isolated buildings.Using genetic algorithms, the optimal semiactive controllers can be obtained to generate the control forces of PFD.Simulation results of a benchmark fve-story building demonstrated that the proposed PFD can reduce the seismic foor acceleration responses with less displacement demand of the base isolator under the cases of near-and far-feld earthquakes.Fan et al. [18] explored the dynamic behavior of variable friction dampers using a piezoelectric ceramic tubular actuator simulated using ANSYS.A constitutive equation governing the force and displacement of the damper subjected to the harmonic load was proposed, and certain formulas for calculating the dissipated energy of the damper within one cycle were derived.Vestad and Steinert [19] developed a simple fexible passive vibration isolation table (or mass block) suspended by four steel springs and connected by four viscous dampers.Te natural frequency of the suspended table is 1.2 Hz (in the vertical direction), and it successfully mitigates vibration responses in the frequency range of 5-20 Hz, allowing vibration-sensitive experiments or high-resolution measurements to be performed on the table.Yang et al. [20] investigated the dynamic behavior and vibration isolation of a fexible Stewart platform with six supporting legs consisting of fexible joints and uniaxial piezoelectric isolators.Moreover, a model-based decoupled controller was proposed to determine the active control force based on the leg's feedback force and position information.Both the simulation and experimental results demonstrated that the system's vibration isolation capabilities can be improved under vertical excitation generated by a shaker.Jiang et al. [21] proposed a three-degree-of-freedom stabilization platform that consists of three force-position integrated electric cylinders (or supporting legs) as active motion actuators.Te platform was designed to adjust the stifness of the cylinders to have good vertical vibration isolation performance and to keep the platform horizontally based on an adaptive active compliance control law.Simulation and experiment results indicated that the platform can perform well under a singlecylinder impact, three-dimensional disturbances, and irregular random excitations.Fang and Fang et al. [22][23][24] developed a series of passively damped outrigger systems for seismic vibration control of fexible high-rise buildings.Te core of the high-rise building was simulated based on the cantilevered Timoshenko beam theory, and the viscous dampers were vertically implemented between outriggers and perimeter columns.Moreover, the authors also proposed optimization and genetic algorithms to determine the damper placement locations and corresponding parameters  to minimize structural responses, such as inter-story drifts and foor accelerations of high-rise buildings.Most of the aforementioned fexible platforms were proposed primarily to support light-weight precision mechanical systems and metrology equipment in research laboratories; however, in real-life high-tech production fabs, there are several complicated pipelines connected between foors and heavy processing tools (or platforms) to supply gas and fuid materials (some are toxic) to processing tools for the manufacturing of wafers or glass panels, as shown in Figure 2. If isolation or supporting systems are to be installed underneath a heavy platform base for vibration isolation, higher control force demands are required, and numerous fexible pipeline joints or fexible connectors should be used to prevent pipeline damage due to relative displacements, which could be a challenging task.Terefore, microvibration control devices that can be installed on the top of specifc tall production platforms, such as active tuned mass dampers (ATMDs), active mass drivers, or the proposed active piezoelectric mass damper (APMD), without requiring supplementary damping and spring elements to tune the natural frequency of the mass block, as will be introduced later, could be promising and practical alternatives.
Lin et al. [25] conducted the shaking table tests using a semiactive friction tuned mass damper (SAF-TMD) to assess its structural control performance under seismic excitation.Te SAF-TMD can adjust the friction or clamping force provided by a PEA.Experimental results indicated that the control performance of the SAF-TMD is superior to that of the typical passive friction TMD under the same peak TMD stroke.Ju et al. [26] proposed a semiactive piezoelectric friction mass damper, the desired friction force of which can be adjusted from the input voltage command of PEAs.Te performance of the damper was validated using a real-time hybrid testing method.Moreover, the other semiactive piezoelectric friction dampers developed for seismic vibration control of frame structures can be found in the literature [27][28][29][30].
Te present study aimed to assess the active microvibration control performance of the equipment platform under the internal AGV movements in precision high-tech fabs.Te motion equation of an analytical model including the three-span continuous beam (or foor) system and platform under the action of AGV moving loads (simulated using an MKT-PSD function [7,31]) and active control force provided by a proposed active piezoelectric mass damper (APMD) was derived.Moreover, the direct output feedback control algorithm [32,33] was adopted to determine the optimal feedback gain matrix for calculating the active control force.Dynamic time history analyses of the proposed model under diferent AGV weights moving at the same speed were further conducted using the state space procedure (SSP) [34].Moreover, the corresponding RMS foor and platform velocity vibration spectra were obtained via a one-third octave band spectrum analysis [35].Finally, the control performance of the APMD and its required control demands such as the control force, stroke, and applied voltage were assessed and discussed.

Analytical Model of the Floor-Platform System under Moving Loads and Control Forces
Figure 3 illustrates the analytical model of the proposed foor-platform system under the AGV moving loads in LCD factories.Te three-span continuous foor (or beam) system was simplifed from a real LCD fab [11], and the platform with a width b and height h installed on the foor and implemented with the APMD (Figure 4) was modeled as a single-degree-of-freedom (SDOF) system subjected to the base rotation excitation and active control force.Notably, the rotation excitation at the platform base could be attributed to the uneven vertical foor vibrations under the AGV movements.Te displacement response in the vertical direction, u(x, t), of a multispan continuous beam can be calculated from its modal displacements and the mode shape functions (i.e., the classical mode superposition method), as follows [36]: where c i (t) is the i-th modal coordinate, ψ i (x) is the i-th mode shape function, and n is the total number of vibrational modes to compute the vibration response.Moreover, the linear equation of motion of the analytical model subjected to N moving concentrated axle loads (Figure 3) can be derived employing the Bernoulli-Euler beam theory without considering the shear deformation efect as follows [2,3]: where M, C, and K represent the n × n mass, damping, and stifness matrices, respectively; location matrix of the moving axle loads on the beam; and Figure 5: Analytical model of the platform implemented with passive base isolation system (two spring-dashpot models).

Structural Control and Health Monitoring
Figure 3), its equation of motion in the horizontal direction under the rotational base excitation (θ(t)) and a single active control force (F c (t), Figure 3) can be represented as follows [36]: where m a , c a , and k a denote the system parameters (i.e., mass, damping, and stifness, respectively) of the APMD; m p , c p , and k p correspondingly represent the system parameters for the platform simulated as a SDOF system in this study; x a represents the APMD stroke (displacement) of the mass block relative to the platform; x p represents the displacement of the platform relative to the foor; and F c (t) represents the active control force provided by the piezoelectric stacks or actuators [12] (Figure 4) that can achieve the precision displacement control.Te angular base acceleration can be further derived from the relative vertical foor accelerations at locations x 1 and x 2 as follows: where represent the row vectors that collect the mode shape values of each mode at locations x 1 and x 2 , respectively.Moreover, the base coupling force (F b (t)) underneath the platform at x 1 and x 2 induced by the initial and active control forces can be derived using (4) as follows: Te modal coupling force (F * b (t)) exerted on the foor system at x 1 and x 2 can be represented as follows: Te motion equation of the foor-platform system under the AGV moving loads and active control force can be represented using equations ( 2)-( 4) and ( 6) as follows: 6 Structural Control and Health Monitoring x p (t) x p (t) where  M,  C, and  K represent the (n + 2) × (n + 2) mass, damping, and stifness matrices of the foor-platform system, respectively; E c represents the (n + 2) × N location matrix of moving axle loads; and B c denotes the (n + 2) × 1 infuence vector of the active control force, provided that only an APMD is considered in this study.
Te dynamic vibration response or state T including the displacement and velocity of the analytical model at each time step k for (7) can be recursively obtained using the SSP, as follows [33]: where in which s denotes an integer value.
To further compare the control performance of tall fexible platforms using diferent control schemes, a passive base isolation system (PBIS) and typical ATMD were also investigated in this study.Te tall platform controlled using the PBIS consisting of two spring-dashpot devices is illustrated in Figure 5. Te equation of motion of the base-isolated platform supported by the foor under AGV moving loads can be derived as follows: Structural Control and Health Monitoring x b (t) where m b and I b denote the mass and moment of inertia of the rigid platform base, respectively; c b and k b denote the damping and stifness coefcients of the base isolation system (two spring-dashpot model), respectively; and x b and θ b denote the vertical displacement (measured from the position of static equilibrium) and angular rotation of the rigid platform base, respectively.Moreover, the spring and damping forces can be represented as ), respectively.Te stifness coefcient of the base isolation system was defned and determined by a given frequency, f b , as k b � (m p + m b ) × (2πf b ) 2 .Similarly, the damping coefcient of the base isolation system was determined by a given damping ratio, ξ b , as

Direct Output Feedback Control Algorithm
Te present study adopted the direct output feedback control algorithm [32,33] to determine the optimal feedback gain matrix and compute the control force using only the partial output state responses (y[k]), which can be represented as follows: where y[k] represents the r × 1 output state vector (if the number of the output state response is r, r ≤ 2(n + 2)), G represents the r × r feedback gain matrix, and D represents the r × 2(n + 2) location matrix of the output state.Herein, we only considered the velocity response of the platform (i.e., r � 1 and D � 0 0 0 1×n 0 1 0 1×n  ) as the output state.Te performance index, J[k], for each time instant k can be defned as follows: where an optimal solution exists only if the weighting matrices (or scalars) Q and R are semipositive and positive defnites, respectively.Herein, the weighting matrix was adopted to reserve the frst term on the R.H.S. of (11) as the total strain and kinetic energy of the platform.R is a scalar quantity, as only a single APMD was considered.Hence, the optimal feedback gain matrix, G (or gain factor G for single output response), can be obtained using the equation as follows [33]: where 8 Structural Control and Health Monitoring Te optimal gain matrix (G) shown in ( 12) is represented implicitly and can be determined using an iterative method as follows [32,33]: (1) Consider i � 1 and G (i) � 0.

Assessment of AGV-Induced Floor and Platform Microvibrations
4.1.Target LCD Fab Model. Figure 3 illustrates a three-span typical LCD fab [11] with a long span (36 m) mega truss foor system under AGV movements.Te mass per unit length and fexural rigidity of the beam model are 9.1505 × 10 3 kg/m and 2.4608 × 10 11 N-m 2 , respectively; accordingly, the eigenanalysis yielded the frst fve vibration frequencies of the continuous beam model as follows: 6.285, 8.055, 11.762, 25.142, and 28.653 Hz, respectively.Notably, the total number of vibrational modes, n � 12, was considered to compute the vibration response of the foor system.Moreover, an equivalent damping ratio of 5% was assigned to each vibration mode.

Simulation of AGV Moving Forces.
Herein, a single AGV with two equal axle loads (N � 2) was assumed to traverse the foor system.Te AGV moving force (or dynamic engine force) can be obtained using an MKT-PSD function or spectrum [7,31] given by the following expression: where f 1 , f 2 , ξ 1 , and ξ 2 denote the constant parameters; f 1 and f 2 represent the maximum and minimum frequencies of the predominant bandwidth of the moving force, respectively; ξ 1 and ξ 2 control the confguration of the PSD functions; and S 0 denotes the magnitude of the force.To investigate the efects of the AGV weight on the foor and platform microvibrations, fve diferent AGV weights with w 5 � 21582 N (i.e., the maximum AGV weight equivalent to the mass of 2200 kg), w 4 � 0.8w 5 , w 3 � 0.6w 5 , w 2 � 0.4w 5 , and w 1 � 0.2w 5 were considered for the time history analysis.Notably, the AGVs herein were moving at v � 2.0 m/s. Figure 6 demonstrates the dynamic engine force MKT-PSD spectra of the fve AGV weights, and their corresponding governing parameters with f 1 � 50 Hz are listed in Table 1.Te intensity and frequency bandwidth of the MKT-PSD functions vary and increase with increasing AGV weights.Tese calibrated moving force PSD spectra can be obtained using the previously reported regression formulas [31].Moreover, the PSD spectra were further transformed into dynamic engine force time histories [37], as shown in Figure 6, for conducting the dynamic time history analyses.Te maximum force of each axle of the AGV for w 5 � 21582 N and v � 2.0 m/s was scaled to be 2453 N (i.e., approximately 11.4% of the AGV weight or equivalent to a mass of 250 kg) to better predict the in situ microvibration level as suggested by Lee et al. [11].

Parameters of the Platform and Piezoelectric Actuator.
Te horizontal natural frequency and equivalent damping ratio of the 3000 kg platform installed at central location of midspan (x � 54 m) were 25 Hz and 2%, respectively.Moreover, the width (b) and height (h) of the platform were 2.0 and 2.6 m, respectively.Te cross-sectional area (A d ) of the piezoelectric stack actuator was 5.07 cm 2 (equivalent to the diameter of 2.54 cm), total number of layers (n d ) of piezoelectric material was 200, thickness (t d ) of each layer was 0.051 cm, and piezoelectric strain constant (d 33 ) and the modulus of elasticity (E d ) of the piezoelectric material were 3.71 × 10 − 10 (m/V) and 4.62 × 10 7 kN/m 2 , respectively [30].Moreover, the mass (m a ) of the APMD mass block was 50 kg, and the damping (c a ) and stifness (k a ) of the APMD were assumed to be 0.   the simulated time histories of the active control force, applied voltage, stroke of mass block, and power demand of the APMD, respectively.Te relationship between the stroke (x a (t)) of the mass block (or the deformation of the piezoelectric stacks) and active control force (F c (t)) of the piezoelectric stacks can be represented as follows [30]:

Microvibration Simulation Results.
where ) is the stifness of the piezoelectric stacks in the polarization direction (axial direction) and V(t) represents the applied voltage.Using the control force and stroke of the mass block, the corresponding applied voltage command of the APMD can be determined using (15) as follows:    Structural Control and Health Monitoring Moreover, the power demand of the APMD can be calculated using the control force and velocity of the mass block as follows: To explore the efect of the weighting factor (R) of the active control force on the performance of the APMD, the weighting factors between R � 10 − 5 and R � 10 − 8.5 were considered in this study.As presented in Table 2, the smaller the R value, the larger the feedback gain factor and the corresponding active control force.Notably, the feedback gain factors were obtained using the step factor λ � 0.01 and tolerance of error ε � 10 − 7 .Moreover, the maximum control force, applied voltage, stroke, and power demands of the APMD with m a � 50 kg and diferent R values simulated from the time history analyses under diferent AGV weights are summarized in Table 2. Te maximum control demands of APMD increase with decreasing R values.Moreover, the APMD demonstrates the control performance only if the R  value meets the condition of R ≤ 10 − 8 , within which the APMD can provide considerable active control force to suppress the AGV-induced platform microvibrations.Te satisfactory control performance, as shown in Figures 8 and  9 (m a � 50 kg), can be achieved at R � 10 − 8.5 (corresponding to G � 318915.02).For example, for the maximum AGV weight (w 5 ), the platform microvibration can be efectively controlled to be below the allowable VC-B level at the maximum active control force of only 38.74 N, applied voltage of 242.99 V, stroke of 18.16 μm, and power of 0.069 Watt (W), as illustrated in Figures 10-13, respectively.Compared to the application of the PEAs in structural vibration control of civil structures under earthquake or wind excitations, the application of the proposed APMD with relatively smaller control demands is feasible for microvibration control of tall platforms in LCD fabs, as suggested by the simulated results in this study.
For better understanding the efect of the mass block on control demands of the APMD, diferent masses of the mass block are further considered for dynamic time history analyses.Figures 14-16 demonstrate the maximum stroke, applied voltage, and power demands with the varying mass of the APMD (10-100 kg) and the active control force weighting factor (10 − 8.5 -10 − 5 ), provided that the optimal active control force time history has been obtained as shown in Figure 10.Te results indicate that the peak control demands decrease with increasing APMD mass and decreasing active control force weight factor.If the appropriate mass block, say m a ≥ 50 kg (equivalent to approximately 1.7% of the total weight of the platform), and the weighting factor of R � 10 − 8.5 (or R ≤ 10 − 8 ) are adopted, the APMD can also achieve a satisfactory vibration control performance requiring less and more reasonable control demands than those summarized in Table 2 for m a � 50 kg.Regarding the mechanisms of the APMD, the reaction force from the mass block of the APMD is considered as the active control force exerted to the tall platform.Unlike typical ATMD systems, the APMD did not require tuning its natural frequency in resonance with the horizontal frequency (25 Hz) of the tall platform such that the APMD could require less stroke of the mass block to achieve satisfactory vibration control performance.In addition, under the same active control force or energy demands, the larger the mass of the mass block, the smaller the corresponding stroke of the mass block (or the piezoelectric stack actuator), applied voltage, and power demand.Terefore, the mass of the mass block can be easily selected based on the available installation space and the desired capacities of the stroke and applied voltage of the piezoelectric stack actuator from ( 15) and ( 16).Furthermore, unlike typical seismic active control systems that are operational for only dozens of seconds with larger control force or external energy for suppressing earthquake-induced structural vibrations, the APMD used for mitigating platform microvibrations induced by AGVs in real-life hightech production fabs is operational all the time and requires less power consumption, as depicted in Figure 13 and Table 2. Te proposed APMD driven by piezoelectric stack actuators has a high precision control capability.It provides a preliminary economical solution and demonstrates the potential for facilitating persistent microvibration control of special taller platforms subjected to such AGV-induced platform base rotation excitations with larger intensity and broader bandwidth in LCD fabs or similar industrial factories.
Moreover, the control performance of tall fexible platforms using diferent control schemes, a passive base isolation system (PBIS), and typical ATMD was also investigated in this study.Te diferent frequency parameters (f b �10, 30, and 50 Hz) and the damping ratio parameter (ξ b �10%), along with m b � 300 kg and I b � 100 kg • m 2 , were considered in this study to evaluate the vibration isolation performance of the PBIS for the tall fexible platform.In addition, the typical ATMD considered in this study consists of an actuator, a mass block (m a �150 kg), a damping element with an equivalent damping ratio (ξ atmd ) of 20%, and a spring element with an equivalent frequency (f atmd ) of 25 Hz (i.e., equal to the horizontal natural frequency of the tall platform).Moreover, the optimal feedback gain factor of the ATMD was found to be G � 318804.88,using the same direct output feedback gain algorithm with R � 10 − 8.5 and the same feedback output state being the velocity response of the tall platform as considered by the APMD.
Figure 17 shows a comparison of horizontal RMS vibration spectra of the tall platform implemented with APMD, ATMD, or PBIS under the maximum AGV weight.Te simulated results show that all three control or isolation approaches considered in this study can suppress the horizontal microvibrations of the tall   Structural Control and Health Monitoring x a � 12.18 μm.Moreover, the PBIS can suppress the horizontal microvibrations of the tall platform, especially higher frequency responses, as shown in Figure 18, whereas it can amplify the vertical microvibrations of the tall platform at around the vertical natural frequencies of the foor system even though this study does not focus on the assessment of vertical vibration control performance of the tall platform.Moreover, as mentioned earlier, in real-life high-tech production fabs, there are several complicated gas and fuid pipelines connected between foors and processing tools or platforms.Terefore, the PBIS may not apply to such specifc tall platforms, whereas the APMD installed on the top of the tall platform without supplementary damping and spring elements to tune the natural frequency as required by the ATMD could be a feasible and promising approach in precision microvibration control.16 Structural Control and Health Monitoring third octave band spectrum analyses of the proposed beamplatform model under the action of diferent AGV moving forces generated by the MKT-PSD functions were performed.Te numerical simulation results revealed that the foor microvibrations exceed the desired VC-B level and that the tool platform microvibrations induced by the base rotation excitation without APMD generally exceed the VC-A level regardless of the AGV weight.Signifcant reductions of over 90% on the tool platform microvibrations under the AGV movements could be achieved if the platform is implemented with the APMD considering the appropriate mass (m a � 50 kg) of the mass block and weighting factor (R � 10 − 8.5 ) of the active control force.Moreover, the APMD could efectively suppress the platform microvibrations under the maximum AGV weight (w 5 � 21582 N) to satisfy the allowable VC-B level by demanding less active control force (38.74 N), stroke (18.16 μm), and applied voltage (242.99V), which can now be easily realized using the piezoelectric stack actuators.In addition, comparisons of the microvibration control performance of the tall platform implemented with APMD, ATMD, or PBIS under the maximum AGV weight have been performed.Simulation results showed that all three control or isolation schemes can suppress the horizontal microvibrations of the tall platform below the allowable VC-B level.Te APMD consisting of piezoelectric stack actuator and mass block without auxiliary spring and damping devices exhibits comparable control performance to the ATMD.Conversely, although the PBIS could further suppress horizontal microvibrations of the tall platform at a higher frequency range, it could amplify vertical microvibrations around the vertical natural frequencies of the foor system.Moreover, due to complex gas and fuid pipelines connected between foors and processing tools or platforms in real-life high-tech production fabs, it would be a challenge to realize vibration isolation by installing the PBIS underneath the platform base.Terefore, the simulated results obtained herein suggest the feasibility of using the proposed APMD to realize the horizontal microvibration control of such specifc tall platforms subjected to the AGV-induced foor vibrations with larger intensity and broader bandwidth in LCD fabs.Consequently, future studies may focus on the development of innovative active microvibration control systems and base isolation devices (hybrid control schemes) to suppress both the vertical and horizontal microvibrations of tall platforms under AGV movements.

Figure 1 :
Figure 1: Process tools supported by the platforms installed on the production foors.

Figure 2 :
Figure 2: Gas or fuid supply pipelines connected between foors and processing tools.

T
denotes the N × 1 moving load vector.If the tall platform is mounted on the foor at locations between x 1 and x 2 (i.e., b � x 2 − x 1 ,

Figure 3 :Figure 4 :
Figure 3: Analytical model of the foor-platform system under AGV moving loads in LCD factories.

Figure 7
presents the simulated vertical RMS foor velocity spectra at central location of the midspan (x � 54 m) under diferent AGV weights.Te foor vibrations increased with increasing AGV Structural Control and Health Monitoring weight.For example, the foor vibration could exceed VC-A for the maximum AGV weight (w 5 ), whereas they could reach VC-B for the minimum AGV weight (w 1 ).Moreover, the acceleration time histories and their corresponding RMS velocity spectra of the platform without control, shown in Figures8 and 9, respectively, substantially exceeded VC-A at approximately 25 Hz (i.e., the fundamental frequency of the platform) regardless of the weight of AGV.Tese results indicate that the AGV-induced foor and platform vibrations generally exceed the allowable VC-B level required by the LCD factory.Figures8 and 9demonstrating the comparisons of the platform vibrations without and with APMD (m a � 50 kg and R � 10 − 8.5 ) indicate the achievement of signifcant reductions (over 90%) on the platform responses in both the time and frequency domains.Tey further indicate that the platform vibrations were efectively suppressed to satisfy the desired VC-B level after implemented with the APMD.

Figures 10- 13
demonstrate the simulated time histories of the active control force, applied voltage, stroke of mass block, and power demand of the APMD, respectively.Te relationship between the stroke (x a (t)) of the mass block (or the deformation of the piezoelectric stacks) and active control force (F c (t)) of the piezoelectric stacks can be represented as follows[30]: demonstrate the simulated time histories of the active control force, applied voltage, stroke of mass block, and power demand of the APMD, respectively.Te relationship between the stroke (x a (t)) of the mass block (or the deformation of the piezoelectric stacks) and active control force (F c (t)) of the piezoelectric stacks can be represented as follows[30]:

Figure 6 :
Figure 6: Dynamic engine force PSDs and time histories of each axle under diferent AGV weights.

Figure 7 :Figure 8 :Figure 9 :
Figure 7: RMS foor vibration spectra at central location of the midspan under diferent AGV weights.

w 5 Figure 10 :Figure 11 :
Figure 10: Control force time history of the APMD under the maximum AGV weight (mass of the mass block m a � 50 kg; active control force weighting factor R � 10 − 8.5 ).

Figure 12 :
Figure 12: Stroke time history of the APMD under the maximum AGV weight (mass of the mass block m a � 50 kg; active control force weighting factor R � 10 − 8.5 ).

Figure 13 :
Figure 13: Power demand time history of the APMD under the maximum AGV weight (mass of the mass block m a � 50 kg; active control force weighting factor R � 10 − 8.5 ).

5 WFigure 14 :
Figure 14: Peak stroke demands with diferent masses of mass block and active control force weighting factor under the maximum AGV weight.

Figure 15 :
Figure 15: Peak applied voltage demands with diferent masses of mass block and active control force weighting factor under the maximum AGV weight.

Figure 16 :
Figure 16: Peak power demands with diferent masses of mass block and active control force weighting factor under the maximum AGV weight.

Table 1 :
Parameters of the simulated MKT-PSD functions for diferent AGV weights.

Table 2 :
Summary of the feedback gain factors and corresponding maximum control demands of APMD for diferent active control force weighting factors R (m a � 50 kg).