Further steps towards the tuning of inertial controllers for broadband‐frequency‐varying structures

There is a large number of slender structures that are prone to vibrate due to pedestrian loads. The well‐known tuned mass damper (TMD) is increasingly used in practice, but this device may detune under several circumstances drastically reducing its expected performance. A semiactive TMD (STMD) provides more robustness under structure uncertainties than the passive one and may be recommended for use when several vibration modes are in the excitation range and/or when the mode to cancel has uncertainty or varies over time. The tuning of the semiactive device for these cases cannot be the same as the passive one. This paper presents a methodology for the tuning of an STMD under broad frequency‐band uncertainties using an ON/OFF phase control law. This control law is experimentally implemented considering interesting practical aspects, such as the estimation of the inertial mass velocity from the measured acceleration or a low‐pass filter to avoid unnecessary control actions, among others. An optimization process using a proposed performance index is carried out. Thus, it has been demonstrated that the optimal STMD is achieved for much lower damping and higher tuned frequency than the passive one. Finally, the proposed methodology is validated with experimental results.


INTRODUCTION
Lightweight and high-span pedestrian structures are usually prone to vibrate. Although current codes may be fulfilled, these structures may vibrate excessively and are not as comfortable as expected. Control devices can mitigate vibration significantly and improve comfort while fulfilling vibration serviceability requirements, simultaneously increasing the lifespan of the structures.
Vibration damping systems for civil structures are continuously being proposed. They can be classified into passive, active, semiactive, and hybrid control. The tuned mass damper (TMD), 1 referred also to as tuned vibration absorber, is the most commonly passive device used in footbridges. 2 A review of control devices implementation in landmark footbridges have been presented in the literature. 3,4 The main drawback of TMDs is that they are highly effective only over a narrow frequency band. Hence, when structures show modal properties changing over time, and/or several vibration modes 5,6 must be cancelled by the same device, TMDs may detune and experience a significant loss of efficacy. Under these circumstances, the use of semiactive devices may become a competitive alternative. [7][8][9] However, several issues have to be tackled to achieve a successful implementation for broad frequency-band vibration: (a) the design of the control law to implement, (b) the frequency tuning in passive mode, (c) the damping in passive mode, (d) the definition of a performance index (PI) able to represent the performance in a broad frequency band, and (e) practical issues such as filtering and switching ON/OFF function. Some of these issues have been partially addressed previously. [10][11][12] Passive vibration control using TMDs [13][14][15] and optimal tuning has been extensively studied. 16,17 The option to include more than one TMD has also been considered in order to increase its robustness and expand its frequency bandwidth. 18 In this sense, the semiactive TMD (STMD) may be a solution that leads to better results in terms of costs and performance. The dynamic performance of this non-linear system has been studied in the literature. 10 The application of this control technology may be used to solve a vibration problem 19,20 or as an improvement of the structural design. 21 Numerous studies have been conducted emphasizing its performance robustness through tracking the force of the controllable damper device. 11,22 Ferreira 12 studied an optimal tuning of the controller through a profound parametric study of the parameters of the STMD. Hence, they neither set a minimization problem based on a PI related to the vibration serviceability nor include the non-linear damper behaviour.
In the set of semiactive devices, magneto-rheological (MR) dampers are usually used as smart dampers. 23 The MR damper is a damper filled with an MR fluid capable of producing adaptive damping forces that are controlled by modifying the magnetic field applied in real time. In this case, the MR damper is used as the smart device in the control strategy, giving the following advantages: rapid response, large damping force with wide dynamic range, and durability. Thus, an MR damper can be used within control systems. However, MR dampers have a strongly non-linear nature that has to be modelled and included in control design. Otherwise, the simulated performance may differ considerably from the experimental performance.
The control law used in this article was originally presented by Soong 24 and studied by other authors. 12,[25][26][27] The adapted version 27 is a simple and focus-on-implementability semiactive control law that, according to the simulations carried out, 9,28 produces promising results assuming an ideal viscous damper. Although sensitivity studies of its parameters have been accomplished, as well as optimal tuning proposals, 12,29 the main purpose of the semiactive device has not considered for its tuning: be effective in a broad frequency-band spectrum. Cases such as modal properties varying with time due to external factors (temperature, level of occupancy, or aging) and structures with several vibration modes with closely spaced frequencies prone to vibrate are common in practice. 6 Under these circumstances, much greater robustness than that provided by the TMD is required and a design approach that considers both uncertainties in structure modal parameters and an MR damper is desirable. Thus, the tuning of the semiactive system should take into account its broad spectrum of operation and the significant non-linearities, such as the control law itself, as well as the operation of the MR damper. Then, a new approach to tuning STMDs is carried out in this article and compared with the passive performance. Necessary aspects to implement the control law are also described and taken into account within the optimization problem. Although the problem is stated in a general way, this is clearly geared to highly resonant structures (for instance, lightweight footbridges), and the PI to be minimized is focused on user comfort. The following issues are considered: (a) the velocity of the inertial mass should be estimated from its acceleration, (b) a continuous trigger function should be implemented in order to avoid unnecessary damper actions when structure acceleration is of low amplitude, and (c) acceleration of the structure should be carefully filtered to avoid spillover effects and unnecessary damper action when this signal is crossing over zero.
The paper continues with a description of the semiactive inertial controller and the vibration control strategy adopted in Section 2. A sensitivity analysis of the tuning parameters of the control device is carried out in Section 3. The effect of including damper force saturation and an MR damper model instead of a linear-viscous damper is also analysed. Lastly, an optimization procedure to tune the semiactive device is proposed. Section 4 shows the experimental results conducted on a steel beam and carefully explains the implementation of the semiactive control and the practical issues that have been considered. Finally, some conclusions and suggestions for future work are given in Section 5.

Tuned mass damper
A TMD consists of a secondary mass (also called moving or inertial mass) attached to the structure by means of springs and dampers. The TMD mass is fixed as a fraction of the modal mass of the unique targeted vibration mode; the stiffness of the springs is selected to obtain the optimum TMD frequency, and the viscous damper ensures the operation of the TMD in a range of frequencies around the tuning frequency (providing a certain robustness for a frequency bandwidth while degrading performance at the tuning frequency). However, they have a relatively poor performance for low-level vibrations, and they still exhibit a significant lack of performance due to the off-tuning issue.  Figure 1a shows the model of a classical TMD composed of an inertial mass m T attached to a primary system by means of a spring of constant k T and a viscous damper of constant c T . The primary system is the structure modelled as a single degree of freedom system, which is composed of a mass m S , a spring of constant k S , and a viscous damper of constant c S .
The TMD has been designed using the approximate solution provided by Asami and Nishihara, 30 based on H ∞ optimization for primary systems with vanishing damping. The expressions by Asami and Nishihara 30 minimize the acceleration that is related with potential applications in footbridges and also to assess the pedestrian comfort. The TMD properties when the primary system acceleration under harmonic excitation is minimized are in which = m T ∕m S is the mass ratio and = T ∕ S is the frequency ratio, and the stiffness and damping coefficient for the TMD are obtained from respectively. This solution is the initial solution for the optimization process presented in Section 2.3.

Semiactive control strategy
A phase control strategy for the TMD damping is considered here (see Figure 1b). The phase control presented in Soong and Dargush 24 and Chung et al. 26 and adapted by Moutinho 27 has been adopted because this is clearly geared to practical implementation due to the measured real-time parameters employed: the structure acceleration instead of displacement and the inertial mass velocity instead of the relative velocity. This control law will then be effective when structure velocity is negligible with respect to the inertial mass velocity, which usually happens in civil engineering structures under resonant events. The lowest structural response is achieved when the velocity of the inertial mass and structure acceleration have opposing phases, so the semiactive device objective is to bring the inertial mass motion as close as possible to this phase. This is equivalent to a phase lag of 90 • between the structure acceleration and the control force. The control law adopted is of the ON/OFF type due to its simplicity. Thus, the adopted control law when ideal viscous damping is assumed is defined as in which c max is the maximum damping achieved by the damper, c min is the optimal damping obtained from Equation (3), x S is the structure acceleration (measured by an accelerometer), and .
x T is the velocity of the inertial mass (which might be obtained from the integration of an accelerometer signal installed on the inertial mass). When an MR damper is assumed FIGURE 2 Simplified control scheme with an inertial controller (see Figure 1d), the law of Equation (5) is rewritten as follows: in which I is the current applied to the MR damper. The maximum and minimum values, I max and I min , depend on the MR damper employed. This strategy is fairly simple and implementable as it uses only two magnitudes easily measured. However, a few aspects must be taken into account before the practical implementation. These are described in Section 4.3.

Optimal control design
The purpose of this section is to provide a procedure to find an optimal tuning for STMD control devices in structures with uncertainty in their modal parameters and/or with several critical vibration modes. The term closed-loop control implies the use of a feedback controller to bring the output of the system (movement of structure) to a desired value. The controller changes the dynamics of the system. The general feedback control scheme is shown in Figure 2, in which the inertial controller changes in real time following the semiactive control law. The optimization will minimize a PI over a broad frequency band that is geared to structures that show resonant behaviour. The PI is defined according to the following expression: in which F 0 is the force amplitude of F(t), S i ( ) is the spectrum of the acceleration response of the structure (ẍ S (t)), S i ( ) is denoted as the normalized spectrum, and N may be the number of vibration modes to be cancelled or a set of models representative of a particular uncertain vibration mode. This set may be derived considering a normal distribution of the modal parameters. Note that theŜ( ) is used instead of the frequency response function due to control system non-linearities. The optimization problem consists of minimizing the PI defined in Equation (7) subjected to the following constraints: in which f T and T are the frequency and damping ratio of the controller device. The variable |x T | is the displacement of the inertial mass that must be limited by the maximum stroke and is considered fixed. Note that in Equation (8), the damping ratio T , equivalent to c min in Equation (5), is substituted by I min (see Equation (6)) when the MR damper model is included in the design. The perturbation force F(t) used to evaluate the structural response is a chirp waveform whose frequency increases at a linear range with time, as follows: where the instantaneous excitation frequency is defined as a function of time (t) : k being the rate of frequency change, f f the final frequency, f i the starting frequency, and T f the required time to sweep from f i to f f . The frequency range is chosen to excite a broad frequency band, and T f is chosen to ensure that sweeping at each frequency is sufficiently slow.

SIMULATION RESULTS
Two kinds of analysis, described in Sections 3.1 and 3.2, are carried out for the four configurations model shown in Figure 1. The first analysis is a sensitivity analysis of the tuning parameters of the control device. The second one consists of running the optimization procedure defined in Equation (8).

Sensitivity analysis
The following parameters have been adopted to undergo the sensitivity analysis: • The excitation has been applied using a chirp signal (see Equation (9) Table 1 shows the parameters used in the sensitivity analysis. The PI is studied for the tuning frequency ratio and the damping ratio T . A total of 2,601 simulations have been carried out. The inclusion of the MR damper model aims to study the effect of degradation in the operation of the control device, passive, and semiactive.

Control design parameters
Having completed the sensitivity study, the optimization problem is run. The stroke of the device has been limited to 0.3 m, and the value of the mass ratio has been set again at 2%. Note that both variables are related, the higher the mass ratio is, the smaller the stroke achieved.
The PI minimization has been computed using the patternsearch function from MATLAB software. 31 The pattern-search method seeks a minimum based on an adaptive mesh that, in the absence of linear constraints, is aligned with the coordinate directions. 32 Several optimization algorithms have been compared, and pattern-search has always been the most efficient in achieving the best solution.
The optimization is carried out assuming that the parameters involved are variables following a normal distribution randomly generated. Each varying parameter has the following characteristics: indicates the normal or Gaussian distribution with i and 2 i being the mean and the standard deviation, respectively. Each randomly generated value of each parameter is grouped with those that occupy the same position in the generation sequence, obtaining a set of randomly generated models for the same structure. The PI has been computed from 100 models, that is, N = 100 in Equation (7).

Ideal viscous damper
The sensitivity analysis presented in Section 3.1 is carried out for TMD and STMD. For the latter, only the c min is tuned because c max is assumed to be enough for blocking (see Equation (5)). The PI is computed for each case. Figure 3 shows the contour plots including isolines. It is clear that the STMD is more robust than the TMD. How the PI is obtained is explained here. First, the set of randomly generated models is obtained (Figure 4a). Then, Figure 4b shows the result of Equation (8), in which the maximum value of the sum of the spectra is the PI. Figure 5 shows the envelope curve of the PI with the initial solution together with the optimized one obtained for passive ( Figure 5a) and semiactive (Figure 5b). Although the maximum value of the curve (PI) is the value to be optimized, it must be appreciated that all the points of the curve in the semiactive case are below that of the passive case. Table 2 shows a summary of the results associated to Figure 5a,b. On the one hand, the optimum design of the TMD improves the PI obtained from the classical approach (Equations (1) and (2)). To achieve this, the TMD damping should be much greater in order to deal with the system uncertainty. On the other hand, for the considered control law, the optimal design parameters achieved significantly improves the performance reduction, and interestingly, the unblocking damping of the STMD should be much slower than the one obtained from the classical approach. Obtaining the PI

Effect of damper force saturation
Now, a maximum damper force is added to the model: the force saturation. All the cases shown in Figure 3 are repeated with different force saturation values in order to evaluate the performance degradation of the device because its capacity to block is limited, that is The running root mean square (RMS) acceleration computed using a -second time interval is defined as The maximum value of the running RMS acceleration with = 1 s is usually known as the maximum transient vibration value (MTVV): This value is used to asses the vibration serviceability of structures under human loading. 33 The cumulative distribution function curves for the MTVV are plotted in Figure 6a for several values of F D max (without control and with the TMD are also included for comparison). For the case of the TMD, F D max has no effects because demanded forces are always lower than that value. For the STMD, increasing the values of F D max improves the performance, because the blocking capacity is enhanced. In order to have a single value to compare the results with different F D max , the total area between the cumulative distribution function curve and the y-axis is obtained, resulting in Figure 6b. Thus, this figure shows the performance of each case referred to the case of 500 N (which is considered as perfect blocking). Initially, small F D max increments produce important performance improvements; however, from a certain F D max value, the increments do not result in significant performance improvements. With only F D max = 75 N, 95% of the maximum performance is achieved.

Effect of considering an MR damper model
In order to consider the MR damper model in the controller design, a model identification of the MR damper used for the experimental test was carried out. A Bingham model was identified. The objective is to study how the TMD and STMD performances degrade when an MR damper model is considered (MR-TMD, Figure 1c and MR-STMD, Figure 1d) instead  of an ideal viscous damper. This model assumes that a body behaves as a solid until minimum yield stress is exceeded and that it exhibits a linear relationship between the stress and the rate of shear deformation. 34 From the equilibrium of the mechanical element configuration, the force generated by the MR damper can be expressed as where c 0 is the viscous damping parameter, f c is the frictional force, and f 0 is the force offset. Several experimental tests were carried out for several combinations of displacements, frequencies, and damper currents in order to calibrate the    Figure 7 shows the experimental setup together with the MR damper to be identified: sponge-type MR damper from the Lord Corporation company. A steel frame combined with an electromagnetic actuator (APS Dynamics Model 400), an accelerometer, a linear variable differential transformer, and a load cell has been used in order to measure the damper acceleration, the displacement, and the force, respectively. Each test was done with a sinusoidal excitation with fixed frequency, fixed shaker gain (amplitude displacement measure by an linear variable differential transformer), and fixed damper current. This process has been repeated for several combinations. Therein, the damping ratio is changed by the input current of the MR damper. The selected range of frequencies, shaker gains, and current supplies involved in the experimental procedure are specified in Table 3. A total of 252 tests were completed. Figure 8a shows, in a single plot, all the parameters involved in the identification process. The amplitude of the force, displacement, and acceleration in the steady-state response of each test are shown. Figure 8b shows that the dependency of the acceleration with the displacement and the frequency is much lower than that with the input current, in such a way that the Bingham model parameters are considered to be dependent only on the current I. Thus, the parameters of the model of Equation (14) were calibrated using lsqcurvefit MATLAB routine (least-square method) and these are Hence, the control law adopted is the one of Equation (6), and the initial condition for the optimization process will be assumed for I = 0 A. For the semiactive case, it is necessary to optimize the minimum damper current, because the current to block the inertial mass is the maximum value for the ON/OFF control law. Figure 9 shows the degradation obtained when the Bingham MR damper model is used for the sensitivity analysis with the MR-TMD and the MR-STMD. The following aspects are observed from Figure 9 as compared with Figure 3: (a) In both cases, the performance has degraded clearly when the MR model is considered, and (b) the MR-STMD behaves much better than the MR-TMD, showing a greater robustness, that is, it is more effective for a broader frequency band.
The optimization analysis for designing the control parameters is carried out for MR-TMD and MR-STMD. Figure 10 shows the PI obtained for both cases. The PI of the MR-TMD is almost the same as that of the TMD (see Figure 5a). However, the PI of MR-STMD is worse than the STMD (see Figure 5b), so the inclusion of the MR model significantly degrades the effectiveness of the semiactive control. Table 4 shows the initial tuning and that obtained from the optimization process. Interestingly, the tuning frequency for the MR-STMD is over the structure frequency when a broad frequency band is considered. Besides, the optimal minimum current is I min = 0 A, which indicates that the frictional force f c of the device is excessive at 0 A and the results would improve significantly for lesser values of f c .  Note. The improvement is computed from the initial solution of each device. Abbreviations: MR, magneto-rheological; PI, performance index; TMD, tuned mass damper; STMD, semiactive tuned mass damper.

Structure description and experimental setup
The structure is a simply supported steel beam of 5.10-m span with UPN-200 cross-section. The structure is excited at midspan, where the first vibration mode has its maximum sag, by the same actuator used for the MR damper identification setup (see Figure 7). Two piezoelectric accelerometers mounted on a magnet plate are attached to the beam and the inertial mass, respectively. The controller hardware is a CompactRIO from National Instruments using the LabVIEW real-time module. To carry out the control law (see Equation (6)), the inertial mass velocity has to be estimated from its acceleration. Therefore, the control law programming comprises the velocity estimation, the proper control law, and other important elements needed for the practical implementation that will be developed from here. The nominal case consists of the structure without changes (Case 2). Two other cases are studied to evaluate the MR-STMD performance. Case 1 consists of adding four masses of 10 kg close to the midspan, as shown in Figure 11a. Case 3 consists of reducing the span by moving the supports (reducing 20 cm at each side, a total reduction of 40 cm of the span) and removing the masses previously added (see Figure 11b). Table 5 shows the modal parameters of the beam for the three cases. Note that the MR-TMD and MR-STMD are tuned to Case 2, and their ability to cancel vibration modes with lower and higher frequencies is studied.

Mechanical design of TMD
The Asami and Nishihara 30 solution serves as the initial estimate of the stiffness for the springs. Therefore, a physical model has been designed using two springs with a given stiffness of 2.5 kN/m each one and the MR damper identified in Section 3.5. The inertial mass is set at 3 kg (mass ratio of 4% and tuned frequency of 6.49 Hz). A final tuning was carried out from the normalized spectrum using an excitation chirp from 3 and 8 Hz. This device is studied experimentally, functioning as passive and semiactive control.

Implementation of the control law
The practice implementation of the control law is described in this section. Figure 12 shows the components of the inertial controller feedback (see Figure 2). The control force is F C = F S + F D , F S being the force of the springs and F D the force of the damper. This control scheme has been implemented using a sampling frequency of 1000 Hz and assigned first-in first-out conditions.

Switching ON/OFF function
A switching off function has been included to avoid continuous unnecessary operation of the control device. When the running RMS acceleration computed each second (see Equation (12)) is under a certain value, the control law is OFF. 35 The switching off function adopted here is a RMS,1 < 0.05 m/s 2 .

Low-pass filter
In order to avoid multiple unnecessary blocking/unblocking actions due to the noise of the acceleration of the structure at zero crossing, this signal is filtered by a low-pass Butterworth filter of fourth order with a cut-off frequency of 100 Hz. This frequency should be high enough to prevent phase shifts in the structure acceleration that may spoil the operation of the semiactive control law. Figure 13 shows the filtered acceleration for two cut-off frequencies of the low-pass filter. It is shown that the phase shift introduced by the 100 Hz cut-off frequency filter is negligible, whereas the high frequency noise is removed. Figure 14 shows the inertial mass acceleration together with the damper current as a result of applying the semiactive control law. In this figure, it can be seen that the blocking control action is applied uninterruptedly for a short time interval, that is, the damper brakes the inertial mass until tuning phase conditions are achieved.

Estimation of the inertial mass velocity
There are different ways to compute the estimated velocity of the inertial mass: (a) a simple numerical differentiation of the displacement, (b) a kinematic Kalman filter (KKF) using the displacement and the acceleration, 36,37 , and (c) a leaky-integrator filter (LIF) using only the acceleration. 38 In the first case, differentiating a signal has the drawback that the noise levels can potentially be increased at the output, so this method is directly discarded. Also, it needs to measure the displacement, and it is easier to measure accelerations. On the other hand, KKF is a more robust method to estimate the velocity, but it has the disadvantage that both displacement and acceleration must be measured. Finally, an LIF is also a robust procedure for a harmonic signal (as usual for resonance response), and it only requires the acceleration to be measured. In addition, for the ON/OFF control law, only the velocity phase is actually important. The results of KKF and LIF have been experimentally compared, with similar results, making the LIF the most convenient method for the velocity estimation experimentally. Thus, the following integrator filter is adopted: in which s = j , being the angular frequency and I and I are, respectively, the natural circular frequency and damping ratio of the integrator filter. The parameter I should be chosen sufficiently smaller than S to avoid significant phase shift at frequencies of interest. The parameters selected were f I = 0.5 Hz and I = 1%. Then, G(s) is transformed into the discrete domain using the zero-order hold with a sampling frequency of 1000 Hz, resulting in where z is the discrete variable. Thus, this filter obtains the estimation of the velocity for frequencies greater than I .

Experimental tests
In this case, passive and semiactive vibration control is applied to the cases presented in Section 4.1. The purpose of these tests is to prove the results obtained numerically for tuning the semiactive device. The damper current applied is 0.11 A for passive and 0.0 A for semiactive, because they have shown to be the optimal values for the nominal case. Figure 15 shows the experimental results obtained for the three cases of Table 5. It shows the frequency domain response, magnitude (Figure 15a-c), and phase (Figure 15d-f) between the structure acceleration and the excitation force for the structure without control and those controlled with the MR-TMD and with the MR-STMD. With respect to the magnitude, the MR-STMD shows a much better performance when out of the nominal case, although it also behaves better in the nominal case. It is really interesting to look at to the phase plots. The semiactive control law tries to correct the

FIGURE 15
Experimental vibration results in frequency domain comparing the uncontrolled structure, MR-TMD, and MR-TMD for the three cases of Table 5. The dashed red line in the phase plots indicates 90 • delay between the structure acceleration and the control force, which means perfect tuning. MR, magneto-rheological; TMD, tuned mass damper; STMD, semiactive tuned mass damper phase to keep the inertial mass velocity tuned to the structure acceleration (this results in a 90 o delay between the structure acceleration and the control force). Note that the phase correction in Case 3 is less effective than in Case 1. That is, when > 1, the STMD frequency is higher than that of the structure and the semiactive control law is more effective, because it is easier to brake the inertial mass of the control device in this case (as opposed to < 1). This result matches the results obtained in Tables 2 and 4 from the optimization process for broad frequency-band vibration. It was demonstrated that the STMD frequency should be tuned at higher frequencies of the nominal structure frequency to cope with broad frequency-band vibrations. Table 6 shows the reduction improvements obtained in the experimental tests. The improvement of the semiactive device with respect to the passive is also shown for each case. It can be seen that, although the semiactive device always improves the performance of the passive one, the greatest difference is obtained for Case 1, in which the control device has a greater frequency than the structure. Finally, it should be mentioned that these results have come out as expected, according to the results obtained in the simulations carried out including the MR damper model.

CONCLUSIONS
This paper presents a methodology for the tuning of STMDs to reduce vibrations in broadband-frequency-varying structures. The methodology is clearly geared to implementation because several practical issues are carefully considered. The methodology is convenient to both structures with an uncertain mode that may change over time or a structure with several close-in-frequency modes.
The following conclusions can be drawn: • For the sensitivity analysis: a. The STMD is much less sensitive to structure parameter uncertainties than the TMD, thus showing a better performance for a given frequency band. b. Regarding the damper force saturation, for the passive case, the force requirements are merely to dissipate the transferred energy, whereas the peak force needed is lower. In the semiactive case, higher requirements are needed.
• For the optimization analysis: a. The STMD should be tuned at a higher frequency with a lower damping ratio than the TMD. That is, the TMD needs more damping to get a certain level of robustness. The higher frequency tuning of the STMD is due to the fact that it is easier for the semiactive controller to decelerate the inertial mass when its frequency is higher than the frequency of the structure. b. If a model of an existing MR damper is included instead of a linear-viscous damper, the performance is clearly degraded, although the STMD remains more robust and effective. c. Table 7 shows a summary of all the results obtained for the different cases jointly: TMD, STMD, MR-TMD, and MR-STMD. The aspects mentioned in the previous items are clearly appreciated in this table.
• For practical implementation: a. To carry out practical implementations, three elements have been added to the semiactive control law: a switching off function, a low-pass filter, and an integrator filter. These elements increase the stability and reliability of the control system. b. The effect of time delay for the MR damper to apply the semiactive control law can be considered negligible because the structure dynamics are much slower than the MR damper response, which always happen in lively low-frequency structures.  c. The experimental results support the previous numerical results. The phase control degrades faster for structures with a higher frequency than that of the STMD, which is why the tuned frequency of STMD should be higher than the nominal case.
Two future works are derived from this paper. The first is the implementation of this methodology on an in-service structure with several problematic modes in a frequency range. The second is to introduce the MR parameters into the optimization process. This will allow the best MR damper for a particular application to be found.