Structural health monitoring of single degree of freedom flexible structure having active mass damper under seismic load

Technical Paper
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Abstract

In this research, the application of active mass damper (AMD) has been experimentally tested using electromagnetic uniaxial shake table. The characteristic of an active damper has been presented. A state-feedback controller has been introduced for single flood active mass damper (AMD) using numerical modeling. The model was built and experiment was performed in laboratory having AMD to discuss the effect of model parameters and the development of parameters by inputting the data obtained from most severe earthquake on Oct 8 2005 in Pakistan. The system model, control design and observer were tested on shake table having 46 cm × 46 cm dimensions in real time vibrations. The shake table used has capability of running with powerful actuator having scaled accelerograms of real time earthquakes. The setup to operate the controller on real experimental work was discussed. It was observed that about 40% reduction in vibration can be achieved using active mass damper.

Keywords

Seismic simulation Active mass damper Shake table test 

Introduction

The structures vulnerable to Earthquakes must be evaluated in design properly due to effect of stability. About 87,000 peoples were dead and 138,000 persons were injured due to 7.6 magnitude earthquake on Oct 8, 2005 [1]. Epicenter was near Muzaffarabad; the city was wrecked along with Balakot and few other towns [2]. However, same magnitude earthquake will not affect, and damages are minor in developed countries like USA and Japan. Strong motion recording stations of Kashmir earthquake were located at Nilore, Abbotabad, and Murree under Micro Seismic Simulation Program, MSSP as shown in Fig. 1 [3]. The high-rise building in Islamabad was collapsed and rose serious questions about the construction and design practices prevailing in the country. In summary, economic loss of over five billion dollars was suffered due to this earthquake [4, 5].
Fig. 1

Pre–post seismicity of epicentral area of 2005 Pakistan earthquake

The use of active and passive control system, e.g., tuned mass dampers, viscous dampers etc. is increased now a day to prevent earthquake damages in high-rise buildings using controller, e.g., linear quadratic regulators [6, 7, 8, 9, 10].

In this research, the application of active mass damper (AMD) has been experimentally tested using electromagnetic uniaxial shake table. The characteristic of an active damper has been presented. A state-feedback controller has been introduced for single floor active mass damper (AMD) using numerical modeling [11, 12, 13, 14, 15]. Shake Table Test of earthquakes signal obtained from strong motions Centre at Nilore and Abbotabad during Oct 8, 2005 earthquake obtained from Pakistan Atomic Energy Commission (PAEC) and the observed damping was compared for active control cases [16].

Research performed before 2005 earthquake

The earthquake data obtained from PAEC was not used previously for any vibration control experiments, although many researchers suggested various control techniques to be used in buildings and bridges [17, 18]. The probability of reducing stress has been investigated on single floor experimental system. To input the desired earthquake loading various simulations were produced using SIMULINK/MATLAB. Various cases were analyzed, and the module was amplified to obtain desirable results [19, 20, 21].

Approach

Active mass damper system

The upper part of the structure fitted with the shaft and support used to function with cart having mass which is controllable. The structure is moved along the same direction of the cart. Particularly, it is an accurate form of solid aluminum cart operated by DC motor having gearbox [22, 23, 24].

Once the motor started, the output shaft generated the torque by gear and track function which results in force control to run the cart. A multi-I/O panel is used to develop controller same as real time control function. SIMULINK is used to develop the controller of the system.

Analytical model

Active mass damper analytical model is developed shown in Fig. 2. The beam, floor and cart are moving in the model. The potential and kinetic energies are gravitational potential energies and elastic potential energy as the cart is in constant elevation, therefore, the potential energy would be elastic potential energy which were derived as follows:
$$T = \left( {\frac{1}{2}M_{\text{c}} + \frac{1}{2}\frac{{J_{\text{m}} K_{\text{g}}^{2} }}{{r_{\text{mp}}^{ 2} }}} \right)\left( {\frac{\text{d}}{{{\text{d}}t}}x_{\text{c}} (t)} \right)^{2} + M_{\text{c}} \left( {\frac{\text{d}}{{{\text{d}}t}}x_{\text{f}} (t)} \right)\left( {\frac{\text{d}}{{{\text{d}}t}}x_{\text{c}} (t)} \right) + \left( {\frac{1}{2}M{}_{\text{c}} + \frac{1}{2}M{}_{\text{f}}} \right)\left( {\frac{\text{d}}{{{\text{d}}t}}x_{\text{f}} (t)} \right)^{2}$$
(1)
$$V = \frac{1}{2}K_{\text{f}} x_{\text{f}} (t)^{2}$$
(2)
Fig. 2

Analytical model of active mass damper

Where cart mass (Mc) = 0.68 kg, floor mass (Mf) = 0.7 kg, Kf is floor stiffness constant which is 6EI/l3 = 520 N/m, xf(t) is floor position at time ‘t’, xc(t) is cart position at time ‘t’, E = 200 GPa, I = 3.599 × 10−12 m4, Jm is cart rotor moment of inertia = 0.038 10−5 kg m2, Kg is cart gear ratio = 3.71, rmp is pinion radius = 0.00625 m, Fc is linear force applied to cart and the Lagrangian is defined as
$$L \, = \, T \, {-} \, V.$$
(3)
Therefore, proceeding to the two equation of motion, the results are defined as follows:
$$\ddot{x}_{\text{c}} (t) = \left( {\frac{{K_{\text{f}} x_{\text{f}} (t)}}{{M_{\text{f}} }}} \right) + \left( {\frac{{\left( {M_{\text{c}} + M_{\text{f}} } \right)F_{\text{c}} }}{{M_{\text{f}} M_{\text{c}} }}} \right)$$
(4)
$$\ddot{x}_{\text{f}} (t) = - \left( {\frac{{K_{\text{f}} x_{\text{f}} (t)}}{{M_{\text{f}} }}} \right) - \frac{{F_{\text{c}} }}{{M_{\text{f}} }}$$
(5)
Equation (4) represents acceleration of cart and Eq. (5) represents the acceleration of floor. The two equations of motion are linear and were used to formulate state space matrices A, B, C and D and state vector X and output vector Y:
$$\frac{\partial }{\partial t}X = AX + BU\;{\text{and}}\;Y = CX + DU$$
(6)
where state space matrix variables are as follows:
$$\begin{aligned} A = \left[ {\begin{array}{*{20}c} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & {280} & { - 20} & 0 \\ 0 & { - 340} & 6 & 0 \\ \end{array} } \right] ; \quad B = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 3 \\ { - 1} \\ \end{array} } \right] \hfill \\ C = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 \\ 0 & { - 340} & 6 & 0 \\ \end{array} } \right]; \quad D = \left[ {\begin{array}{*{20}c} 0 \\ { - 1} \\ \end{array} } \right]. \hfill \\ \end{aligned}$$

The analytical model shown in Fig. 2. The total mass of structure is 3 kg, the height is 500 mm, natural frequency is 2.6 Hz, and linear stiffness is 520 N/m.

Observer system

Observer system with full order is given as follows
$$\frac{\partial }{\partial t}X_{\text{o}} = AX_{\text{o}} + BU + G(Y - Y_{\text{o}} )$$
(7)
and
$$Y_{\text{o}} = CX_{\text{o}} + DU.$$
(8)
It can be seen from above equations that the state observer has for inputs the system’s input(s) and output(s) and calculates as outputs of the states estimate. The observer is basically a replica of the actual plant with corrective term G(Y − Yo) multiplied by the observer gain matrix. The obtain estimated state vector can then be used for state-feedback law, as expressed Eqs. (9) and (10)
$$\left( {U = V_{\text{m}} } \right) = - KX_{\text{o}}$$
(9)
$$\frac{\partial }{\partial t}X_{\text{e}} = \left( {A - GC} \right)X_{\text{e}}$$
(10)
and
$$X_{\text{e}} = \, X - X_{\text{o}} .$$

Xe is the estimation error it will asymptotically go to zero if (A − GC) is stable.

Seismic excitation

The earthquake data was obtained from two stations, i.e., from Nilore and Abbotabad, and the results are not free field. The nearest station to the epicenter was at Abbotabad, and Nilore station was close to Islamabad city. Figure 3 shows the strong motion obtained from Abbotabad in EW direction and 0.231 g was the PGA observed and Fig. 4 was obtained from Abbotabad in NS direction and 0.197 g was the PGA in this direction.
Fig. 3

Acceleration records Abbotabad EW

Fig. 4

Acceleration records Abbotabad NS

Figure 5a shows strong motion obtained from Nilore station in EW direction with a PGA of 0.023 g and Fig. 5b shows the values in NS direction having PGA of 0.026 g.
Fig. 5

a Acceleration records Nilore EW DIR. b Acceleration records Nilore NS DIR

The faulting mechanism solution indicated that it was thrusting fault and damages were recorded to structures of the area.

As there was a limit ± 7.62 cm maximum stroke of the shaking table. To simulate an earthquake on the shaking table system, the ground motion data was scaled to ± 3 cm. The gravitational acceleration with units (g) has been recorded and the method was produced in MATLAB which was used to compute the desired positions such that the measured accelerations yielded on the STII are equivalent to the recorded values.

Figure 6a–d represents the output displacement obtained for the scaled position; these displacements produce equivalent acceleration values of the recorded data on the shake table. The significant ranges of values were the time steps near the PGA values.
Fig. 6

a Scaled displacement (Abbotabad EW DIR). b Scaled displacement (Abbotabad NS DIR). c Scaled displacement (Nilore EW DIR). d Scaled displacement (Nilore NS DIR)

Simulation

The simulation has been developed and the state feedback has been operated to start cart movement on model top. Simulations are designed in the Simulink diagram as shown in Fig. 7. After collecting individual contributions to the state gain for a mass is kept on the cart. Within the simulation, modification can be made to alter the damping case from active damping to passive case [25].
Fig. 7

Seismic excitation Simulink model

Vibration control

The response of the model was analyzed after the controller is used using various input signals of earthquake excitations. First input was a periodic impulse at the bottom of the model to check damping of structure then second inputs were the seismic scaled waves of Oct, 2008 mentioned earlier. The displacement of top for passive and active are shown in Fig. 8.
Fig. 8

Displacement when subjected to periodic excitation PMD vs AMD

Figure 9a–c shows the floor displacement with the control having earthquake signals. A significant decrease in displacement has been observed using active mass damper. The friction due to bearings in the system will be used in further studies to control the displacement. The seismic data of the region will be used which are frequently encounters natural disasters.
Fig. 9

a Top displacement of the structural model PMD vs AMD (Abbotabad NS DIR). b Top displacement of the structural model PMD vs AMD (Nilore EW DIR). c Top displacement of the structural model PMD vs AMD (Nilore NS DIR)

Conclusion

Application of shake table test for presenting the earthquake engineering and structural dynamic concept is convenient. The mathematical and experimental aspects have been demonstrated with hands on experiment using real time earthquake ground motions having active mass damper. Public awareness of seismic hazards using the Shake Table experiment may be used. The earthquake ground motions were used to analyze the structure using controlled response of the buildings. The use of active mass damper is an effective tool to control the severe damage to the structure. After various experimental work, the control model for real time earthquake has been proposed using active mass damper. The experiments were successful. The model will be expanded for future research.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Civil Engineering, Faculty of EngineeringIslamic University of MadinahMadinahKingdom of Saudi Arabia
  2. 2.National University of Sciences and Technology (NUST)IslamabadPakistan

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