Active control of human-induced vibrations on lightweight structures via electrodynamic actuator dynamics inversion

Abstract

The active vibration absorber represents an effective means to mitigate excessive vibrations in low-damping structures. Nevertheless, the dynamics of the actuators employed in such systems may negatively affect their performance and stability restricting their operational frequency range. This article presents an application of dynamics inversion techniques to the force control of electrodynamic proof-mass actuators employed as active vibration absorbers in lightweight pedestrian structures. The dynamics inversion approach is applied to enhance the classical direct velocity feedback scheme. Additionally, a novel method relying on dynamics inversion is presented: the Broadband Force Cancellation Algorithm. This procedure consists in estimating, in real-time, the equivalent force acting on the system to later apply it back to the structure with the opposed sign. The effectiveness of the proposed methods is assessed via numerical simulations carried over a realistic model of an existing lightweight footbridge and an electrodynamic proof-mass actuator. Two load cases are analyzed: a fixed swept-sine force and a walking load. Both cases account for the actuator-structure interaction. The human-structure interaction is considered in the latter scenario due to its importance when dealing with lightweight pedestrian structures. Simulation results demonstrate that the dynamics inversion techniques effectively cancel out actuator dynamics leading to an excellent tracking of the reference force output by the suggested or other vibration control algorithm. The proposed schemes are proved promising since they substantially outperform the widespread direct velocity feedback approach. In particular, the Broadband Force Cancellation Algorithm minimizes the action of the external forces within their estimation frequency range, thus being especially suited to tackle broadband excitations, important in lively pedestrian structures, whereas the velocity feedback methodology performs best at the structural resonant frequencies.

Keywords

active vibration control actuator-structure interaction dynamics inversion human-induced vibrations human-structure interaction lightweight structures

1. Introduction

Inertial vibration absorbers have been extensively used to minimize vibration serviceability problems arising from the human-induced vibrations occurring in modern lightweight, low-damping structures such as footbridges or long-span floors.

A common approach to attenuate excessive vibrations consists in employing the (passive) tuned vibration absorber (TVA) (Elias and Matsagar, 2017; Yang et al., 2022). The TVA is tuned to a specific mode, circumstance which may lead to detuning problems when structural or TVA properties are time-varying. Techniques like multi-TVA (Casciati and Giuliano, 2009; Li et al., 2010; Bortoluzzi et al., 2015; Van Nimmen et al., 2016; Huang et al., 2020; Liu et al., 2021) or semiactive TVAs (Soria et al., 2017; Moutinho et al., 2018; Wang et al., 2021; Wang et al., 2021; Barrera-Vargas et al., 2022) have been proposed to overcome detuning issues.

The active vibration absorber (AVA) represents an effective alternative for vibration cancellation when a higher attenuation is required, more harmonics need to be damped simultaneously or the structural properties variation cannot be neglected. The main components of an AVA system are the actuator(s), the controller, and the sensor(s) (usually accelerometers). Several actuation technologies have been used in the context of human-induced vibrations. Some examples include fixed (Casado et al., 2013; Zhang and Ou, 2015; Mao and Huang, 2019) or deployable (Goorts et al., 2017; Goorts and Narasimhan, 2019) electromagnetic proof-mass actuators (PMAs), pneumatic actuators (Bleicher et al., 2011), and rotating-mass actuators (Terrill et al., 2020; Terrill and Starossek, 2022; Zhang et al., 2022).

A widespread group of AVA implementations focuses on a collocated actuator-sensor pair, leading to a single-input single-output (SISO) scheme. The most representative approach within this category is the direct velocity feedback (VF), which consists in exerting on the structure a damping force equal to the velocity of the point affected by a control gain. This algorithm holds the advantage of unconditional stability when the actuator and sensor dynamics are neglected and does not require a structural model, with the subsequent enhanced stability and robustness features (Wang et al., 2018; Díaz et al., 2021). Nonetheless, when the dynamics of the system are considered, the unconditional stability assumption no longer holds and a limit to the control gain exists. Several techniques have been proposed to increase the stability margin of VF including: the Acceleration Feedback Control (Diaz and Reynolds, 2010), the Inner Control Loop Approach (Díaz et al., 2012a), the Integral Resonant Control (Díaz et al., 2012b), the Virtual Mass Technique (Mao et al., 2020), the adaptive reduction of VF gain (Ahmadi, 2020), or the Independent Modal Space Phase-Lead VF Control (Xue et al., 2022). The SISO VF schemes may perform poorly when the target structure exhibits closely-spaced modes. The SISO limitations may be overcome by multiple-input-multiple-output (MIMO) strategies (Pereira et al., 2014; Camacho-Gómez et al., 2018; Wang et al., 2019).

Another group of AVA vibration control algorithms (VCAs) relies on a precise structure model to estimate the controlled modes states, upon which a gains matrix is applied to calculate the control command(s) (Hudson and Reynolds, 2012). Some examples of model-based controllers (MBC) include the Pole Placement method (Chang and Yu, 1998; Nyawako and Reynolds, 2015; Belotti et al., 2020), the Pole-Zero assignment (Richidei et al., 2022), the Modified Independent Modal Space Control (Etedali, 2017), the Linear Quadratic Regulator and the Linear Quadratic Gaussian Control (Goorts et al., 2017; Goorts and Narasimhan, 2019; Chang, 2020; Moradi et al., 2021), and the $H_{\infty}$ and $H_{2}$ methods (Stavroulakis, et al., 2006; Wu et al., 2006).

In this work, the application of dynamics inversion (DI) techniques to the force control of a Commercial-Off-The-Shelf (COTS) electrodynamic PMA is presented. The DI constitutes a model-based, structured procedure whose aim is to find a PMA system inverse which, upon implementation in the controller, allows for an approximate cancellation of the PMA dynamics (Kwatny and Blankenship, 2000; Ramírez-Senent, et al., 2021). Consequently, a very accurate tracking of the target force output by any VCA is achieved. As to deal with the modeling errors and enhance overall stability and robustness, the DI is complemented with a parallel Three Variable Controller (TVC) (Yao et al., 2016). The proposed PMA DI procedure holds the following advantages: (i) it has been developed for systems with multiple inputs in which some of them are known, thus being able to cope with the actuator-structure interaction (ASI) phenomenon (Goorts et al., 2017; Goorts and Narasimhan, 2019; Díaz et al., 2021; Zhang et al., 2022), (ii) can be employed when the PMA features non-linear dynamics, (iii) may be applied to improve any force-based VCA, and (iv) can be easily used in MIMO schemes. Afterward, two control algorithms incorporating DI are presented: the Direct Velocity Feedback with DI (VFDI) and the Broadband Force Cancellation algorithm (BBFC). The former approximates the classical VF scheme to the ideal case via PMA DI, with the subsequent improvements in stability and performance. The latter constitutes a novel MBC approach based on estimating the equivalent external force exerted on the structure, which is then negated and imposed back to the system in real-time. Therefore, the BBFC aims at minimizing the external actions on the structure within the equivalent force estimation frequency range and is especially suited to diminish vibration amplitude at both resonant and non-resonant frequencies which may be excited by the higher harmonics of human actions or by broadband loads acting on lightweight pedestrian structures. Moreover, BBFC does not require the definition of performance specifications or cost functions as in the mentioned MBCs and the extension of BBFC to a MIMO scheme is straightforward. Despite both the DI and the BBFC implementations require, respectively, a precise model of the PMA and the considered structure, these can be obtained with the suggested set-up.

The effectiveness of the proposed VCAs is assessed and compared to that of a classical VF scheme through realistic simulations carried over models of an existing glass fiber reinforced polymer (FRP) footbridge and a PMA. These simulations comprise a fixed swept-sine force, applied at the midpoint of the edge of the bridge where the PMA is located, to illustrate the theoretical VCAs performance, and a real-life walking excitation. The latter model includes the human-structure interaction (HSI) phenomenon, which cannot be overlooked when dealing with vibration cancellation in lively structures (Ahmadi et al., 2019, 2021; Díaz et al., 2021; Barrera-Vargas et al., 2022).

The remainder of this paper is organized as follows. Section 2 details the implemented model. Section 3 describes the DI procedure and its application to the BBFC and the VFDI algorithms. Simulation results are discussed in section 4. Finally, conclusions are outlined in section 5 along with immediate subsequent works.

2. System description, modeling, and identification

2.1. Structure

The structure under study is a simply supported FRP footbridge with a length of 10 m and a width of 1.5 m (Figure 1(a)) designed to meet ultimate limit states and a long-term deflection lower than $L / 200$ . However, the vibration serviceability limit state is not fulfilled, and it is intended to be met by integrating vibration absorbers such as AVAs (Gallegos-Calderón, et al., 2021).

Figure 1.

Footbridge under study. (a) Photograph. (b) Set-up schematic.

The relationship between the acceleration at a certain point (node) of the structure, $i$ , and the forces exerted at other locations (denoted by $q$ ) is determined by the product of the $i$ -th row of the accelerance matrix of the structure and the forces vector (Preumont and Seto, 2008) and reads in Laplace domain:

s^{2} Z_{s i} (s) = \sum_{q = 1}^{N} \sum_{m = 1}^{N} \frac{Φ_{i m} Φ_{q m} s^{2}}{{m_{m} s}^{2} + c_{m} s + k_{m}} F_{q} (s) = \sum_{q = 1}^{N} G_{s, i q} (s) F_{q} (s),

(1)

where

N

is the number of considered modes and

s

stands for the Laplace variable.

Z_{s i} (s)

represents the Laplace transform (LT) of the structure displacement at node

i

s^{2} Z_{s i} (s)

is the LT of the acceleration, and

F_{q} (s)

denotes the LT of the force exerted at node

q

. Moreover,

Φ_{j k}

is the jth component of the kth eigenvector and

m_{m}

c_{m}

, and

k_{m}

signify the modal mass, damping coefficient, and stiffness of mode

m

. Finally,

G_{S, i q} (s)

represents the transfer function (TF) between the force at node

q

and acceleration at node

i

. The vertical motion of the two midpoints of the footbridge edges, as shown in Figure 1(b), is considered in this work since the BBFC algorithm requires monitoring two points to ensure the observability of the dominant modes (subsection 3.3.1). The VFDI only needs an estimate of PMA attachment point velocity (point 1). The acceleration at points 1 and 2 due to the swept-sine load is modeled with equation (1). The calculation method for the accelerations due to the walking excitation is explained in subsection 2.2.

A preliminary Finite Element Analysis (FEA) was conducted to estimate the footbridge modal shapes (Figure 2). A simply supported configuration was considered; therefore, the vertical and lateral translations of the left corners and the vertical, lateral and longitudinal translations of the right corners were constrained (Figure 1(b)). The first four modes take place at 7.329 Hz: first vertical bending mode, 10.425 Hz: first lateral-torsion mode (mainly lateral), 15.507 Hz: second lateral-torsion mode (mainly torsional) and 25.916 Hz: second vertical bending mode. Figure 3 shows the vertical components of the unity-normalized modal shapes at the edges. The TF at point 1, $G_{s, 11}$ , was identified through an experimental modal analysis to accurately estimate the resonant frequencies, yielding lower frequencies than in FEA due to the PMA mass (see Table 1). The PMA was installed on point 1; therefore, exerting bending and torsion solicitations (Figure 1(b)). The excitation was a random voltage with a constant frequency content between 1 and 200 Hz. The PMA mass and the point 1 accelerations were measured, and the excitation force was calculated (subsection 2.3). Figure 4 shows the experimental and fit results. The approximation was optimized up to 20 Hz, the frequency range of interest, since only modes below that value are likely to be excited by human actions (Wei et al., 2019). Consequently, the structural model accounts for the first four modes which fall within the 20 Hz span. The parameters obtained in the fitting process ( $m_{m}$ , $c_{m}$ , $k_{m}$ , $m = 1, \dots, 4$ ) are shown in Table 1 together with the resonant frequencies $f_{m} = \sqrt{k_{m} / m_{m}}$ .

Figure 2.

FEA Modal shapes at 7.329 Hz, 10.425 Hz, 15.507 Hz, and 25.916 Hz.

Figure 3.

Vertical components of modal shapes at footbridge edges. (a) Point 1. (b) Point 2.

Table 1.

Numerical simulations parameters.

Parameter	Value	Parameter	Value	Parameter	Value
$c_{1}$ [Ns/m]	8.2835 × 10²	$g_{s d 1}$ [m/V]	1.0000 × 10⁻³	$K_{v, V F D I}$ [V/m/s]	1.0437 × 10⁵
$c_{2}$ [Ns/m]	1.8370 × 10⁴	$g_{s d 2}$ [m/V]	1.0000 × 10⁻³	${\hat{K}}_{e}$ [N/A]	1.7778 × 10¹
$c_{3}$ [Ns/m]	1.7229 × 10³	$ζ_{h}$ [-]	2.0000 × 10⁻²	$L_{c}$ [mH]	1.8150 × 10¹
$c_{4}$ [Ns/m]	3.0915 × 10⁴	$ζ_{h p}$ [-]	2.0000 × 10⁻¹	${\hat{L}}_{c}$ [mH]	1.8770 × 10¹
$c_{a}$ [Ns/m]	1.8634 × 10²	$ζ_{d f}$ [-]	3.0000 × 10⁻¹	$m_{1}$ [kg]	5.4725 × 10²
${\hat{c}}_{a}$ [Ns/m]	1.8042 × 10²	$k_{1}$ [N/m]	1.0936 × 10⁶	$m_{2}$ [kg]	8.4824 × 10³
$f_{1}$ [Hz]	7.1148e0	$k_{2}$ [N/m]	3.3658 × 10⁷	$m_{3}$ [kg]	7.7640 × 10²
$f_{2}$ [Hz]	1.0025 × 10¹	$k_{3}$ [N/m]	5.0337 × 10⁶	$m_{4}$ [kg]	7.1204 × 10³
$f_{3}$ [Hz]	1.2815 × 10¹	$k_{4}$ [N/m]	9.3247 × 10⁷	$m_{a}$ [kg]	3.0400 × 10¹
$f_{4}$ [Hz]	1.8213 × 10¹	$k_{a}$ [N/m]	3.9666 × 10³	${\hat{m}}_{a}$ [kg]	3.0570 × 10¹
$f_{h}$ [Hz]	2.3000e0	${\hat{k}}_{a}$ [N/m]	3.9713 × 10³	$m_{h}$ [kg]	7.0000 × 10¹
$g_{a a}$ [m/s²/V]	3.9240e0	$K_{a}$ [V/m/s²]	1.0000 × 10⁻⁵	$R_{c}$ [Ω]	1.3520e0
$g_{a r d}$ [m/V]	1.0000 × 10⁻²	$K_{a v}$ [V/V]	1.8000 × 10¹	${\hat{R}}_{c}$ [Ω]	1.4010e0
$g_{s a 1}$ [m/s²/V]	3.9240e0	$K_{d}$ [V/m]	1.0000 × 10⁻⁴	$ω_{n d f}$ [rad/s]	3.1416 × 10³
$g_{s a 2}$ [m/s²/V]	3.9240e0	$K_{v}$ [V/m/s]	2.0000e0	$ω_{n h p}$ [rad/s]	3.1416e0

Figure 4.

Experimental and modeled structure TF. (a) Magnitude. (b) Phase.

2.2. Human actions and human-structure interaction

A Mass-Spring-Damper-Actuator (MSDA) system attached to the structure has been used to model the human actions and the HSI phenomenon (Ahmadi et al., 2019, 2021; Díaz et al., 2021; Barrera-Vargas et al., 2022). A human is defined by its equivalent mass, $m_{h}$ , natural frequency, $f_{h}$ , and damping ratio, $ζ_{h}$ . The force developed by the legs, $F_{l}$ , is modeled as an actuator. The values employed for, $m_{h}$ , $f_{h}$ , and $ζ_{h}$ have been chosen according to Dougill et al., (2006) and are listed in Table 1. Figure 5 shows the MSDA system scheme.

Figure 5.

Human actions on the structure. (a) MSDA system. (b) Free-body diagram.

The total force transmitted by the human to the structure, $F_{t h}$ , is:

F_{t h} = k_{h} (z_{h} - z_{s i}) + c_{h} ({\dot{z}}_{h} - {\dot{z}}_{s i}) + F_{l},

(2)

where

k_{h}

and

c_{h}

are the stiffness and the damping coefficient of the MSDA model (related to

f_{h}

and

ζ_{h}

z_{h}

is the human’s mass displacement, and

z_{s i}

is the structure displacement at human’s position,

i

. The human’s mass (

m_{h})

equilibrium equation is:

m_{h} {\ddot{z}}_{h} = - k_{h} (z_{h} - z_{s i}) - c_{h} ({\dot{z}}_{h} - {\dot{z}}_{s i}) - F_{l} .

(3)

Comparing equations (2) and (3), it can be concluded that:

F_{t h} = - m_{h} {\ddot{z}}_{h} .

(4)

From the LTs of equations (3) and (4) the following relationship is obtained:

F_{t h} (s) = \frac{m_{h} s^{2}}{(m_{h} s^{2} + c_{h} s + k_{h})} F_{l} (s) - \frac{m_{h} (c_{h} s + k_{h})}{(m_{h} s^{2} + c_{h} s + k_{h})} s^{2} Z_{s i} (s) .

(5)

The TF on the first term of the right-hand side of equation (5) relates the force developed by the human’s legs to the part of the force transmitted to the structure, due to the human’s action itself ( $G_{H}$ in Figure 10). The product $G_{H} (s) F_{l} (s)$ equals the force that the pedestrian would exert on a rigid structure, $F_{h} (s)$ . Moreover, the TF on the second term relates the acceleration of the structure to the fraction of the transmitted force, due to the HSI ( $G_{H S I}$ in Figure 10). A signal acquired in an instrumented treadmill has been employed to model $F_{h}$ (Figure 6). The adopted pace rate is 2.4 Hz to excite the structure first mode with the third harmonic of the load. It has been considered that the human walks at a constant velocity $v$ = 1.6 m/s.

Figure 6.

Instrumented treadmill walking signal. (a) Time domain. (b) Frequency domain.

In order to calculate the acceleration at point 1 due to the walking load, it will be assumed that the human’s trajectory coincides with the edge where the PMA is installed (Figure 1(b)). The $p$ -th modal equation reads:

m_{p} {\ddot{η}}_{p} + c_{p} {\dot{η}}_{p} + k_{p} η_{p} = \sum_{q = 1}^{N} Φ_{q p} F_{q},

(6)

where

η_{p}

denotes the

p

-th modal coordinate. If eigenvectors are unity-normalized at the location of the target node (

Φ_{1 p} = 1

for

p = 1, \dots, N

), its acceleration is calculated by adding up the modal accelerations. Considering that

F_{q}

takes the value

F_{t h}

if the human’s position coincides with node

q

and zero otherwise, when the number of considered nodes increases, the right-hand side of equation (6) tends to:

\int_{0}^{L} \int_{- W / 2}^{W / 2} Φ_{p} (x, y) δ (x - v t, y - y_{0}) F_{t h} (t) d x d y = Φ_{p} (v t, y_{0}) F_{t h} (t),

(7)

in which

L

and

W

are the length and width of the footbridge,

δ

is Dirac’s delta, and

y_{0}

is the distance from the footbridge longitudinal plane of the moving load trajectory (Figure 1). Consequently, the acceleration at point 1 can be obtained by: (i) spatially modulating

F_{t h}

with each mode shape, (ii) solving for each mode, and (iii) adding up the modal contributions. The acceleration at point 2 has been obtained similarly, assuming that the vertical components of mode shapes 1 and 4 are symmetric with respect to the longitudinal plane, while those of modes 2 and 3 are asymmetric. To account for the HSI, the structure acceleration at the instantaneous pedestrian’s position has been calculated by multiplying each modal coordinate by the value of the mode shape at the instantaneous position,

Φ_{p} (v t, 0.75)

and summing up the modal contributions.

2.3. Proof-mass actuator

The PMA considered to implement the AVA system (Figure 7) is an APS 400 electrodynamic (Lorentz-force based) shaker featuring a frequency range from DC to 200 Hz and a rated force, velocity and stroke of 445 N, 1 m/s and 158 mm, respectively (APS Dynamics, Inc., 2022a). The shaker is driven by an APS 125 amplifier with a rated power of 500 VA (APS Dynamics Inc., 2022b). It is worth noting that this COTS PMA system has been chosen due to its availability, prior successful utilization in simpler AVA systems and flexibility for rapid prototyping purposes. Nevertheless, the PMA parameters do not exactly match the ones ideally required for the application: frequency range is higher than required, while force and displacement might be lower. For permanent implementation, a more cost-effective and application-adapted PMA should be considered.

Figure 7.

PMA on the footbridge.

According to Preumont and Seto (2008), the PMA can be modeled as an MSDA system. Figure 8(a) shows the scheme of the PMA acting on the structure. Proceeding similarly as in subsection 2.2, it can be concluded that the force transmitted to the structure by the PMA, $F_{t a}$ , is:

{F_{t a} = - m}_{a} {\ddot{z}}_{a} = k_{a} (z_{a} - z_{s i}) + c_{a} ({\dot{z}}_{a} - {\dot{z}}_{s i}) - F_{a}

(8)

where

k_{a}

and

c_{a}

are the PMA stiffness and its damping coefficient,

z_{a}

is the PMA mass displacement,

z_{s i}

is the structure displacement at PMA installation point, and

F_{a}

is the electromagnetic force developed by the PMA, which can be expressed as:

F_{a} = K_{e} i_{a},

(9)

in which

K_{e}

is the electromagnetic constant and

i_{a}

is the current through PMA coil. Figure 8(b) shows the equivalent circuit of the PMA. The current is governed by:

u_{a} = R_{c} i_{a} + L_{c} \frac{d i_{a}}{d t} + K_{e} ({\dot{z}}_{a} - {\dot{z}}_{s i}),

(10)

where

u_{a}

is the voltage applied across PMA terminals and

R_{c}

and

L_{c}

are the circuit equivalent resistance and inductance. Furthermore,

K_{e} ({\dot{z}}_{a} - {\dot{z}}_{s i})

is the back-electromotive force developed in the circuit. The voltage output by the power amplifier reads:

u_{a} = K_{a v} u_{c},

(11)

in which

K_{a v}

is the voltage gain and

u_{c}

is the control voltage output by the controller.

Figure 8.

PMA scheme. (a) Mechanical part. (b) Electrical part.

The parameters of the PMA have been identified experimentally. For that purpose, the following expression has been derived from the LTs of equations (8) to (11):

F_{t a} (s) = \frac{- K_{e} K_{a v} m_{a} s^{2}}{(m_{a} s^{2} + c_{a} s + k_{a}) (R_{c} + L_{c} s) + K_{e}^{2} s} U_{c} (s) - \frac{m_{a} (c_{a} s + k_{a}) (R_{c} + L_{c} s) + K_{e}^{2} s}{(m_{a} s^{2} + c_{a} s + k_{a}) (R_{c} + L_{c} s) + K_{e}^{2} s} s^{2} Z_{s i} (s) .

(12)

The TF of the first term on the right-hand side of equation (12) relates the control voltage to the part of the force transmitted to the structure due to PMA action itself ( $G_{A}$ in Figure 10), whereas the second term TF relates the structure acceleration to the fraction of the force transmitted to the structure due to the interaction between the motion of the PMA and the structure ( $G_{A I}$ in Figure 10). This latter term corresponds to the ASI.

The shaker has been installed on a rigid floor to estimate its TF experimentally. The amplifier has been fed with a constant amplitude swept-sine voltage from 0.1 to 100 Hz and the acceleration of the PMA mass has been measured. With this test set-up, the experimental data must be fit to the first term of equation (12). Figure 9 shows the experimental and modeled TF. As it can be appreciated, the modeled TF accurately approximates the experimental one up to 20 Hz, progressively diverging from that frequency on, due to non-modeled PMA dynamics. Nonetheless, the attained model is considered enough for this work purposes since the dominant modes of the structure and the external actions frequency range fall within this interval and an overall linear behavior is assumed. Table 1 lists the obtained PMA parameters.

Figure 9.

Experimental and modeled TF $G_{A} (s)$ .

2.4. Electrical noise

Electrical noise has been added to the sensors measurements and to the controller voltage (Figure 10). White noise with a peak value $u_{n p}$ = 0.02 V has been assumed. The instantaneous electrical noise value has been affected by each sensor gain (unity for the output) and added to the magnitude values. The system sensors are (Figure 1(b)): (i) PMA mass acceleration, (ii) PMA mass relative displacement with respect to point 1, (iii) point 1 acceleration, (iv) point 1 absolute displacement, (v) point 2 acceleration, and (vi) point 2 absolute displacement. The sensors measurements have been denoted by ${\ddot{z}}_{a, m}$ , $z_{a r, m}$ , ${\ddot{z}}_{s 1, m}$ , $z_{s 1, m}$ , ${\ddot{z}}_{s 2, m}$ , and $z_{s 2, m}$ and the value of the controller output by $u_{c, m}$ . The considered ranges have been ±2 g for the accelerometers, ±100 mm for the PMA mass displacement sensor, and ±10 mm for the structure displacement sensors. The resulting gains of the sensors are denoted respectively by $g_{a a}$ , $g_{a r}$ , $g_{s a 1}$ , $g_{s d 1}$ , $g_{s a 2}$ , and $g_{s d 2}$ and are shown in Table 1.

Figure 10.

Control system block diagram.

3. Control methodology

The proposed control methodology is explained next. Figure 10 shows the conceptual block diagram corresponding to the proposed control architecture.

3.1. Actuator dynamics inversion

The aim of the DI is to cancel out the internal dynamics of the actuator so that the total force transmitted by the PMA, $F_{t a}$ , closely matches that calculated by a specific control algorithm, $F_{t a, r}$ ( $r$ stands for reference). For that purpose, a state-space model of the form $\dot{{x}} = [A] {x} + [B] {u}$ , ${y} = [C] {x} + [D] {u}$ of the PMA has been derived. The selected state variables are $z_{a}$ , ${\dot{z}}_{a}$ , and $i_{a}$ . The output of the system is $F_{t a}$ . The following state-space representation is obtained from equations (8) to (11):

{\begin{array}{c} {\dot{z}}_{a} \\ {\ddot{z}}_{a} \\ d i_{a} / d t \end{array}} = [\begin{array}{c} 0 & 1 & 0 \\ - \frac{k_{a}}{m_{a}} & - \frac{c_{a}}{m_{a}} & \frac{K_{e}}{m_{a}} \\ 0 & - \frac{K_{e}}{L_{c}} & - \frac{R_{c}}{L_{c}} \end{array}] {\begin{array}{c} z_{a} \\ {\dot{z}}_{a} \\ i_{a} \end{array}} + [\begin{array}{c} 0 & 0 & 0 \\ \frac{k_{a}}{m_{a}} & \frac{c_{a}}{m_{a}} & 0 \\ 0 & \frac{K_{e}}{L_{c}} & \frac{K_{a v}}{L_{c}} \end{array}] {\begin{array}{c} z_{s 1} \\ {\dot{z}}_{s 1} \\ u_{c, D I} \end{array}},

(13)

F_{t a} = [\begin{array}{c} k_{a} & c_{a} & - K_{e} \end{array}] {\begin{array}{c} z_{a} \\ {\dot{z}}_{a} \\ i_{a} \end{array}} + [\begin{array}{c} {- k}_{a} & - c_{a} & 0 \end{array}] {\begin{array}{c} z_{s 1} \\ {\dot{z}}_{s 1} \\ u_{c, D I} \end{array}} .

(14)

The procedure followed to perform the DI is a particularization of the Structure Algorithm to linear systems in which some of the system inputs are known (Kwatny and Blankenship, 2000; Ramírez-Senent et al., 2021). Therefore, the output equation is differentiated with respect to time successively, until the inputs ( $u_{c, D I}$ ) required for the system to achieve the desired output ( $F_{t a, r}$ ) can be factored out. Differentiating equation (14) yields:

{\dot{F}}_{t a, r} = [\hat{C}] [\hat{A}] {\hat{x}} + [\hat{C}] [\hat{B}] {\hat{u}} + [\hat{D}] {\hat{\dot{u}}},

(15)

where hats have been employed to denote the estimates of magnitudes. The required control voltage can be factored out from equation (15):

u_{c, D I} = \frac{{\dot{F}}_{t a, r} - [\hat{C}] [\hat{A}] {\hat{x}} - \frac{{\hat{k}}_{a} {\hat{c}}_{a}}{{\hat{m}}_{a}} {\hat{z}}_{s 1} - (\frac{{\hat{c}}_{a}^{2}}{{\hat{m}}_{a}} - \frac{{\hat{K}}_{e}^{2}}{{\hat{L}}_{c}} - {\hat{k}}_{a}) {\hat{\dot{z}}}_{s 1} + {\hat{c}}_{a} {\ddot{z}}_{s 1, m}}{- {\hat{K}}_{e} K_{a v} / {\hat{L}}_{c}} .

(16)

Consequently, the calculation of the control voltage implies estimating the actuator state-space model, the system state vector, the structure displacement, velocity, and acceleration at the PMA location and the target force time derivative.

The actuator state-space model has been identified repeating the procedure outlined in subsection 2.3. The obtained PMA parameters are shown in Table 1 denoted with a hat. The state vector has been estimated by implementing an observer which makes use of the matrices $[\hat{A}]$ , $[\hat{B}]$ , $[\hat{C}]$ , and $[\hat{D}]$ . As to estimate the structure displacement and velocity at the target point, a continuous-time Kalman Filter (KF) has been employed. The KF includes the PMA equations of motion to improve the final estimate of structure kinematics by incorporating additional information. The KF model reads:

{\begin{array}{c} {\dot{z}}_{a} \\ {\ddot{z}}_{a} \\ {\dot{z}}_{s 1} \\ {\ddot{z}}_{s 1} \end{array}} = [\begin{array}{c} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}] {\begin{array}{c} z_{a} \\ {\dot{z}}_{a} \\ z_{s 1} \\ {\dot{z}}_{s 1} \end{array}} + [\begin{array}{c} 0 & 0 \\ 1 & 0 \\ 0 & 0 \\ 0 & 1 \end{array}] {\begin{array}{c} {\ddot{z}}_{a, m} \\ {\ddot{z}}_{s 1, m} \end{array}} - [\begin{array}{c} 0 & 0 \\ 1 & 0 \\ 0 & 0 \\ 0 & 1 \end{array}] {\begin{array}{c} w_{a} \\ w_{1} \end{array}},

(17)

{\begin{array}{c} z_{a r, m} \\ z_{s 1, m} \end{array}} = [\begin{array}{c} 1 & 0 & - 1 & 0 \\ 0 & 0 & 1 & 0 \end{array}] {\begin{array}{c} z_{a} \\ {\dot{z}}_{a} \\ z_{s 1} \\ {\dot{z}}_{s 1} \end{array}} + {\begin{array}{c} v_{a} \\ v_{1} \end{array}},

(18)

where

w_{a}

w_{1}

v_{a}

, and

v_{1}

represent the instantaneous noise value (subsection 2.4) of PMA and point 1 accelerometers and displacement sensors. The displacement and velocity approximations calculated by means of the KF are denoted by

{\hat{z}}_{s 1}

and

{\hat{\dot{z}}}_{s 1}

(Figure 10).

Finally, to calculate the reference force time derivative, ${\dot{F}}_{t a, r}$ , a second order differentiation filter has been utilized:

G_{d f} = \frac{ω_{n d f}^{2} s}{s^{2} + 2 ζ_{d f} ω_{n d f} s + ω_{n d f}^{2}},

(19)

where

ω_{n d f}

and

ζ_{d f}

are the filter natural angular frequency and damping ratio (Table 1).

3.2. Parallel Three Variable Controller

A TVC feedback controller has been implemented, in parallel with the DI algorithms, to cope with the modeling errors and improve system stability and robustness. TVC has been adopted to provide simultaneous corrections to PMA mass displacement, velocity, and acceleration errors. The voltage output by the TVC is:

u_{c, T V C} = K_{d} (z_{a, r} - {\hat{z}}_{a}) + K_{v} ({\dot{z}}_{a, r} - {\hat{\dot{z}}}_{a}) + K_{a} ({\ddot{z}}_{a, r} - {\ddot{z}}_{a, m}),

(20)

in which

K_{d}

K_{v}

, and

K_{a}

are, respectively, the displacement, velocity, and acceleration control gains (Table 1) and

z_{a, r}, {\dot{z}}_{a, r}

and

{\ddot{z}}_{a, r}

represent the kinematic references. Acceleration reference is calculated by means of

- F_{t a, r} / {\hat{m}}_{a}

and velocity and displacement references by successive integration. The PMA velocity and displacement feedback signals (estimates) are taken from the state observer block.

3.3. Control algorithms

3.3.1. The BBFC algorithm

The BBFC algorithm estimates, in real-time, the equivalent external force acting on the structure (excluding the PMA force) and, thanks to the actuator DI procedure, approximately cancels its action forcing the PMA to exert the same force in absolute value, with the opposed sign. The equivalent force on the structure is defined as the force acting on point 1 which causes the same motion as the actual one. Its value:

F_{t h, e q} = \frac{{\ddot{z}}_{s 1} + \sum_{p = 1}^{N} (k_{p} η_{p} / m_{p} + c_{p} {\dot{η}}_{p} / m_{p})}{\sum_{p = 1}^{N} 1 / m_{p}},

(21)

is deduced from the modal equations of motion and the modal expansion of the acceleration at point 1, assuming

Φ_{1 p} = 1

for

p = 1, \dots, N

(

N = 4

The procedure to estimate the equivalent force is explained next. The force vector acting on the structure can be approximated by:

{\begin{array}{c} {\hat{f}}_{1} \\ {\hat{f}}_{2} \end{array}} = [\hat{M}] {\begin{array}{c} {\ddot{z}}_{s 1, m} \\ {\ddot{z}}_{s 2, m} \end{array}} + [\hat{C}] {\begin{array}{c} {\hat{\dot{z}}}_{s 1} \\ {\hat{\dot{z}}}_{s 2} \end{array}} + [\hat{K}] {\begin{array}{c} {\hat{z}}_{s 1} \\ {\hat{z}}_{s 2} \end{array}},

(22)

where

[\hat{M}], [\hat{C}]

, and

[\hat{K}]

are the mass, damping, and stiffness matrices estimates calculated by means of

[\hat{M}] = {({[\hat{Φ}]}^{t})}^{- 1} [\hat{M}_{\mod}] {[\hat{Φ}]}^{- 1}

[\hat{C}] = {({[\hat{Φ}]}^{t})}^{- 1} [\hat{C}_{\mod}] {[\hat{Φ}]}^{- 1}

, and

[K] = {({[\hat{Φ}]}^{t})}^{- 1} [\hat{K}_{\mod}] {[\hat{Φ}]}^{- 1}

, in which

{[\hat{M}}_{\mod}]

{[\hat{C}}_{\mod}]

, and

{[\hat{K}}_{\mod}]

are the modal structural matrices estimates and

[\hat{Φ}]

is the modal matrix estimate. Since only the two dominant modes (1 and 3) will be accounted for,

{[\hat{M}}_{\mod}]

{[\hat{C}}_{\mod}]

and

{[\hat{K}}_{\mod}]

are the diagonal matrices constituted by

{\hat{m}}_{1}

and

{\hat{m}}_{3}

{\hat{c}}_{1}

and

{\hat{c}}_{3}

, and

{\hat{k}}_{1}

and

{\hat{k}}_{3}

, respectively, and the modal matrix estimate reads:

[\hat{Φ}] = [\begin{array}{c} Φ_{11} & Φ_{12} \\ Φ_{21} & Φ_{22} \end{array}] = [\begin{array}{c} 1 & 1 \\ 1 & - 1 \end{array}],

(23)

where the modal shapes from subsection 2.2 have been employed. The equivalent force on point 1 is then calculated by subtracting the actuator force from the first component of the external forces (equation (22)):

{\hat{F}}_{t h, e q} = {\hat{f}}_{1} - {\hat{F}}_{t a} = {\hat{f}}_{1} + {\hat{m}}_{a} {\ddot{z}}_{a, m} .

(24)

The accelerations in equations (22) and (24) are measured via accelerometers (Figure 1). Structure velocities and displacements are estimated employing a KF, similarly as explained in subsection 3.1, but expanded to account for point 2 displacement and velocity ( $z_{s 2}$ , ${\dot{z}}_{s 2}$ ), point 2 accelerometer ( ${\ddot{z}}_{s 2, m}$ ), its noise ( $w_{2})$ and point 2 displacement transducer noise $(v_{2}$ ).

Finally, a high-pass filter, $G_{h p} (s)$ , with a cut-off frequency of 0.5 Hz is employed to set the PMA target force, $F_{t a, r},$ from the estimated equivalent force, ${\hat{F}}_{t h, e q}$ , minimizing the influence of its low-frequency components:

{F_{t a, r} (s) = G}_{h p} {\hat{F}}_{t h, e q} (s) = \frac{ω_{n h p}^{2} s^{2}}{s^{2} + 2 ζ_{h p} ω_{n h p} s + ω_{n h p}^{2}} {\hat{F}}_{t h, e q} (s),

(25)

where

ω_{n h p}

and

ζ_{h p}

are the filter natural angular frequency and damping ratio (Table 1).

3.3.2. The VFDI algorithm

In order to implement the VFDI, the time derivative of the total force transmitted by the actuator, required by equation (16), must be calculated:

{\dot{F}}_{t a, r} = K_{V F D I} d {\dot{x}}_{s 1} / d t \approx K_{V F D I} {\ddot{x}}_{s 1, m},

(26)

where

K_{V F D I}

is the VFDI control gain. The structure acceleration is measured directly by an accelerometer installed on point 1. The value of the control gain is shown in Table 1. The rest of the measurements required by the implementation of this algorithm are obtained as explained in subsection 3.1. Additionally, a high-pass filter (as in subsection 3.3.1) is used to condition

F_{t a, r}

by attenuating

{\ddot{x}}_{s 1, m}

low-frequency components.

4. Numerical simulations

The results obtained in the simulations carried over the target structure are presented in this section. Table 1 shows the employed parameters. A fourth order Runge-Kutta scheme with a time step of 1 × 10⁻⁴ s has been used in all cases.

4.1. Swept-sine excitation

A swept-sine force has been chosen to illustrate the suggested VCAs theoretical features. Excitation frequency varies linearly from 1 to 20 Hz in 30 s. The amplitude has been linearly modulated during the first 5 seconds until a peak value of 200 N is reached to avoid excessive PMA displacements. Figure 11(a) shows the force envelopes obtained for the BBFC case: the equivalent force on the structure ( $F_{t h, e q}$ ), the estimated equivalent force ( ${\hat{F}}_{t h, e q}$ ), the PMA reference force ( $F_{t a, r}$ ), the actual actuator force ( $F_{t a}$ ) and the net force on the structure ( $F_{t h, n e t} = F_{t h, e q} + F_{t a}$ ). The force estimation error remains within reasonable limits reaching a maximum value of 16.41 N (8.20%), since ${\hat{F}}_{t h, e q}$ is approximated with the two dominant modes and noise-contaminated measurements are used for the estimation. Force tracking is very accurate due to the joint action of the DI and TVC, attaining a maximum error of 1.20 N (0.57%). The $F_{t h, n e t}$ curve provides insight on the algorithm performance. It remains bounded at a low level within the whole frequency range, indicating an effective force cancellation. The maximum value occurs at low frequencies due to the distortion introduced by the high-pass filter. Figure 11(b) shows the time histories of the previous forces zoomed around the first resonance and the second anti-resonance. The force exerted by the PMA essentially opposes the equivalent force on the structure at the two example frequencies and within the complete frequency range.

Figure 11.

BBFC algorithm forces. (a) Force envelopes. (b) Force waveforms.

Figure 12 is the counterpart of Figure 11 for the VFDI case. The $K_{V F D I}$ value (6.7842 × 10⁴ Ns/m) has been selected by trial and error ensuring that the PMA displacement remains within reasonable limits (±100 mm). The actuator force control performs well within the whole frequency range, reaching a peak error of −8.07 N (−3.62%); however, due to the nature of VF algorithms, the actuator force only tends to oppose the excitation at the resonances of the structure which affect the control point (valleys of $F_{t h, n e t}$ , Figure 12(a)), changing its amplitude and phase difference with respect to the external force at intermediate frequencies (Hudson, et al., 2016). Figure 12(b) emphasizes this fact by displaying the forces at the first resonance and at the second anti-resonance. In the former case, the PMA force nearly cancels the external force, whereas in the latter, a significant phase shift between $F_{t h, e q}$ and $F_{t a}$ takes place, leading to a poorer force cancellation.

Figure 12.

VFDI algorithm forces. (a) Force envelopes. (b) Force waveforms.

Finally, Figures 13(a) and 13(b) compare the performance of BBFC, VFDI, and a direct VF scheme using the same actuator. The control gain selected value (2.7055 × 10² vs/m) ensures stability. Figure 13(a) shows the acceleration envelopes achieved at points 1 and 2 with the three VCAs and the running vibration dose value (VDV), known as the root-mean-quad method, which quantifies human exposure to vibration. Figure 13(b) displays the closed loop TFs between the external force and the acceleration structure: $G_{C L, 11}$ . The uncontrolled structure TF, $G_{S, 11}$ , denoted as $G_{C L, U}$ has been included for comparison. The performance of the three VCAs is similar at low frequencies, before the first structural resonance; however, the BBFC method achieves a better vibration reduction, due to its force cancellation feature, in comparison with VFDI and VF which increase their efficiency when the excitation matches resonant frequencies. The VFDI outperforms the VF because of the combined action of the DI and the TVC and compensates the resonance of the PMA (Figure 13(b)). Nevertheless, the performance difference is not substantial at low frequencies, higher than PMA resonant frequency, where the dynamics of the actual PMA are close to ideal (Díaz et al., 2021). At the first resonant frequency, the VFDI yields slightly better results than the comparable outcomes of BBFC and VF algorithms. This circumstance is caused by the fact that the BBFC is affected by the force estimate errors, while the VFDI is not. The VF performs almost ideally for such low frequencies. Nevertheless, the VDV values are much better for the BBFC up to this frequency (and in the whole range) due to the lower cumulative effect at frequencies below resonance. For frequencies higher than the first resonance, the BBFC clearly outperforms the VFDI algorithm, being its effect also marked at point 2. The reason is that the BBFC opposes the external force, whereas the VFDI effect is equivalent to adding a high-damping dashpot between point 1 and the ground. The VF algorithm behaves qualitatively like the VFDI, though leading to higher responses for frequencies up to 12 Hz (Figure 13(b)); for increasing frequencies, the behavior difference is more noticeable because of the true PMA dynamics. Table 2 summarizes the swept-sine test results.

Figure 13.

Comparison between AVA algorithms. (a) Accelerations envelopes and VDVs. (b) $G_{C L, 11}$ .

Table 2.

Swept-sine results. Values between parentheses represent percentage reduction over the uncontrolled case.

Parameter	BBFC	VFDI	VF	Uncontrolled
${\ddot{z}}_{s 1, \max}$ [m/s²]	1.5620 × 10⁻¹ (−97.94)	3.4500 × 10⁻¹ (−95.45)	1.1765e0 (−84.48)	−7.5791e0
${\ddot{z}}_{s 2, \max}$ [m/s²]	1.4700 × 10⁻¹ (−97.95)	−7.8880 × 10⁻¹ (−89.00)	−9.2120 × 10⁻¹ (−87.15)	−7.1686e0
${V D V}_{1}$ [m/s^1.75]	1.0840 × 10⁻¹ (−98.33)	4.1980 × 10⁻¹ (−93.53)	1.0898e0 (−83.19)	6.4843e0
${V D V}_{2}$ [m/s^1.75]	1.0706 × 10⁻¹ (−98.32)	6.5740 × 10⁻¹ (−89.67)	8.1350 × 10⁻¹ (−87.22)	6.3634e0

4.2. Walking excitation

The actions of a pedestrian walking at 1.6 m/s along the footbridge edge where the PMA is installed have been simulated to predict the characteristics of the proposed VCAs in a real-life load case. Figure 14(a) displays the forces developed in the BBFC case. The high-pass filter described in subsection 3.3.1 has been replaced, for this case, by two identical filters in series with a cut-off frequency of 0.75 Hz to keep PMA displacements low enough. The obtained equivalent force estimate, ${\hat{F}}_{t h, e q}$ , and the PMA force tracking are very accurate being the respective errors 156.44 N (11.41%) and −37.73 N (−6.05%). The force exerted by the PMA essentially opposes the external force leading to a considerable reduction in the net transmitted force, $F_{t h, n e t}$ . Figure 14(b) shows the point 1 and 2 accelerations and the calculated VDV values. The reduction in the peak accelerations with respect to the uncontrolled case is remarkable as well as in the VDV values: −93.03% and −94.52%, respectively, at point 1.

Figure 14.

Walking test results. BBFC algorithm. (a) Forces. (b) Accelerations and VDVs.

Figure 15 is the counterpart of Figure 14 for the VFDI case. Again, the force tracking is very precise attaining a maximum error of −0.91 N (−0.20%). The reduction in the acceleration response with respect to the uncontrolled case is noticeable (Table 3), but lower than in the BBFC case, both in peak and VDV terms: 81.46% and 91.06%, respectively, at point 1. This is so, because the overall advantages of the BBFC method over the VFDI at frequencies other than the first resonance outweigh those of the VFDI for the actual, multi-component, analyzed load. Lastly, Figures 16(a) and 16(b) show the acceleration response and VDVs at points 1 and 2 for the VF and uncontrolled cases. Despite the fact that the VF approach leads to a considerable vibration attenuation (62.66% in peak acceleration and 80.76% for VDV at point 1), its performance is far from that of the algorithms incorporating DI. Table 3 summarizes the results of the walking load tests.

Figure 15.

Walking test results. VFDI algorithm. (a) Forces. (b) Accelerations and VDVs.

Table 3.

Walking excitation results. Values between parentheses represent percentage reduction over the uncontrolled case.

Parameter	BBFC	VFDI	VF	Uncontrolled
${\ddot{z}}_{s 1, \max}$ [m/s²]	−1.5910 × 10⁻¹ (−93.03)	4.2350 × 10⁻¹ (−81.46)	8.5280 × 10⁻¹ (−62.66)	−2.2837e0
${\ddot{z}}_{s 2, \max}$ [m/s²]	−1.5350 × 10⁻¹ (−94.25)	−3.0230 × 10⁻¹ (−88.67)	−3.7570 × 10⁻¹ (−85.92)	−2.6680e0
${V D V}_{1}$ [m/s^1.75]	1.0330 × 10⁻¹ (−94.52)	1.6850 × 10⁻¹ (−91.06)	3.6260 × 10⁻¹ (−80.76)	1.8848e0
${V D V}_{2}$ [m/s^1.75]	1.0220 × 10⁻¹ (−94.83)	1.8680 × 10⁻¹ (−90.55)	2.5720 × 10⁻¹ (−86.98)	1.9759e0

Figure 16.

Walking test accelerations and VDVs. (a) VF. (b) Uncontrolled case.

5. Conclusions

In this article, DI techniques have been applied to electrodynamic-actuator–based AVA systems for human-induced vibrations. The DI algorithm has been derived from an identified state-space model of the PMA system. DI implementation implies estimating the displacement, velocity and acceleration at the PMA installation point; therefore, the acceleration and the absolute displacement of that point must be measured. The proposed architecture is completed with a parallel TVC which improves overall stability and robustness. The DI has been applied to the direct VF technique deriving the VFDI algorithm and to a novel approach: the BBFC algorithm. The BBFC consists in estimating, in real-time, the equivalent external force acting on the structure and imposing it back, negated, so that external actions are approximately canceled. The force estimation requires measuring the displacement and acceleration at another point of the structure and relies on a structural model which can be experimentally identified with the suggested configuration. The effectiveness of the proposed techniques has been assessed via numerical simulations carried over a realistic model. Two loading scenarios have been analyzed: a fixed swept-sine force at the midspan of the footbridge and a walking load considering the HSI phenomenon. Simulation outcomes indicate that DI leads to an excellent tracking of the force reference calculated by the suggested VCAs achieving maximum control errors of 1.20 N (0.57%) and −8.07 N (−3.62%), respectively, for the BBFC and the VFDI in the swept-sine test and of −37.73 N (−6.05%) and −0.91 N (−0.20%) in the walking load case. In addition, the equivalent force estimate carried out in the BBFC algorithm is quite reasonable, reaching maximum discrepancies of 16.41 N (8.20%) and −156.44 N (11.41%), respectively, for the swept-sine and walking tests. Results for the swept-sine load show that the BBFC algorithm clearly outperforms the VFDI in the whole frequency range, excepting for a narrow band near the first resonant frequency, mainly due to structural modeling errors. Nevertheless, the overall attenuation with respect to the uncontrolled case is better in the BBFC case, both in terms of peak accelerations (97.94% vs 95.45% at the control point) and VDV values (97.95% vs 93.53% at the control point). Consequently, it can be concluded that the BBFC algorithm minimizes the vibration levels in the structure within the whole excitation frequency range, whereas the VFDI approach performs well at resonances affecting the control point. The results obtained for the walking excitation are better for the BBFC than for the VFDI case for all the evaluated parameters, achieving reductions with respect to the uncontrolled case, at the control point, of 93.03% against 81.46% in peak accelerations and of 94.52% against 91.06% in VDVs. In both scenarios, the new algorithms incorporating DI yield an excellent attenuation in comparison with the classical VF approach, which attains reductions in comparison with the uncontrolled case, at the control point, of 84.48% and 62.66% in the peak accelerations for the swept-sine and walking cases, respectively, and of 83.19% and 80.76% in the VDVs. Future works include the evaluation of the stability and robustness of the proposed schemes, the addition of constraints to avoid PMA saturations and the actual implementation in the footbridge analyzed herein.

Footnotes

Acknowledgments

Christian Gallegos-Calderón thanks the Secretariat of Higher Education, Science, Technology and Innovation of Ecuador for the PhD scholarship CZ02-000167-2018.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by Grant RTI2018-099639-B-I00 funded by MCIN/AEI/ 10.13039/501100011033 and by “ERDF A way of making Europe”.

ORCID iDs

Jose Ramirez-Senent

Christian Gallegos-Calderon

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