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Studies on vibration control of stay cables in cable-stayed bridges have been keenly interested by researchers and engineers in designing new bridges and assessing in-service bridges. Mitigation of undesired cable vibration can be achieved by attaching a damping device. A Viscous Mass Damper (VMD) is composed by the arrangement a rotational Viscous Damper (VD) and an inertial mass element in parallel, which has been used to control seismic in many buildings in Japan; however, it has not been studied to apply in field of cable vibration control. This paper proposes the application of VMD to robustness in suppressing cable vibration. Oscillation parameters of the cable-VMD system are investigated in detail using an analytical formulation of the complex eigenvalue problem and compared with Viscous Damper (VD). Asymptotic formulas to calculate the complex eigenvalue solutions and the damping ratios of the cable with VMD installed in the proximity of the cable end are proposed, discussed and compared with the exact solutions. Further, the influence of elastic support on the VMD effectiveness of a cable-VMD system is analyzed and examined. Lastly, a case study is provided to justify the proposed methodology. The results of this study show that VMD can improve the cable damping ratio more efficiently than VD where the maximum damping ratio of VMD is larger than that of VD, and the increased damping ratio is higher when the relative modes are higher. The paper also provides necessary insights into the dynamics of the cable-VMD system and a useful tool for selecting optimal parameters of VMD for stay cables.

Careful design of vibration control for flexible structures is demanded due to the presence of multiple resonances in the frequency response of the dynamic system. Besides, bonding the piezo-actuators to the vibrating beam may excite the strong resonances. Therefore, the design of feedback control is not trivially selected and the stability of the control architecture should be ensured [6]. The possible control methods to deal with nonlinear vibrations are (see Chap. 3 of [6]) linear velocity control (integral acceleration feedback control), (2) PID-type control, (3) Feedback linearization, (4) adaptive control, and (5) robust control [6,7,8,9]. In velocity control, the input control is selected as integral of the system acceleration and hence a damping term is added to the dynamic equation of the smart beam that reduces the amplitudes of the resonances. The well-known PID-type control is extensively used in stabilization of second-order dynamic systems. Under specifically tuned gains, the closed-loop dynamics can be reduced to the first-order system with acceptable tracking or regulation. However, these gains are limited by stability conditions and hence PID control works well within the low-frequency region below the cut-off frequency. Therefore, using a feedforward term with the feedback PID can improve the system stability and makes the system work with infinite control bandwidth [10]. However, if unmodeled dynamics exist adaptive or sliding mode control approaches are good choices. Sliding mode control selects a sliding surface in terms of position/velocity errors and a signum function plays an important role in control architecture. Due to discontinuous behaviour of signum function, alternative continuous functions are used such as tanh function but with bounded error due to approximation. Higher-order sliding mode control is suggested to solve the discontinuous problem but the complexity of computations may result, see [11] for more details. On the other hand, adaptive control associated with Slotine-Li approach is a powerful control law for the control of nonlinear rigid/flexible body systems [12]. It integrates both the feedforward and feedback terms under uncertainty. The key idea of adaptive control is to linearly parameterize of the dynamic equation such that the left-hand side of the equation of motion is decomposed in terms of regressor matrix that is a function of state variables and uncertain constant parameters vector. Regressor-based adaptive control deals with uncertain constant parameters and a robust sliding term or robust learning algorithms are required to compensate for modelling error and disturbances [12]. One the other hand, adaptive approximation control is a powerful tool to control the nonlinear dynamic system with time-varying disturbances. The uncertainty is approximated by weighting and basis function matrices and then the weighting matrices are updated based on Lyapunov theory. For more information on this topic, the reader is referred to [13,14,15].

Due to the nonlinear elastic restoring force vector \({{\varvec{\upeta}}}\), Eq. (8) has nonlinear behaviour and requires advanced nonlinear control methods to regulate the vibration motion, this is well treated in the next section.

The proposed controller aims at suppressing the vibration motion of the target beam. The input control is provided by the piezo-actuator while the beam deflection is sensed indirectly via the sensor voltages (or equivalently modal amplitudes). Consequently, this section is focused on regressor-free adaptive control. The key idea is to approximate the uncertain coefficients/terms of the equation of motion by using linear combinations of basis functions such as fuzzy, neural, or orthogonal basis function approximators, etc. Then the unknown coefficients of the basis functions are updated based on Lyapunov theory. Now let us represent the dynamic coefficients/terms of Eq. (8) in terms of the weighting coefficient and orthogonal basis function vectors as follows.

The control law described in Eq. (15) is fully decentralized due to the diagonal matrix of the weighting matrix and hence it is a powerful tool to suppress vibration of multi-mode shapes for the vibrating beam.

Now let us come back to the problem of vibration control of vibrating beam described in Fig. 2. As abovementioned, the first two mode shapes are considered for control purposes with dynamics described in Eq. (8). It is assumed that dynamic coefficients, nonlinear stiffness and the excitation forcing terms are unavailable and hence adaptive approximation control is selected as a powerful tool for solving the problem. The control architecture used for vibration suppression is based on Eq. (15a, b) with zero initial conditions for weighting coefficient vectors and amplitudes. The Chebyshev polynomials are used as an approximator with (\(\beta = 11\)). In effect, increasing the number of terms may not increase the accuracy of tracking/regulation control for the desired references. The key idea of conventional adaptive control is to track the desired reference trajectory with ensured stability while the estimators for uncertainty may not exactly approach to the real value. The control law described in Eq. (15a) consists of a feedforward term and a feedback PD term. The feedback and adaptation gains are listed in Table 1. From Figs. 3 and 4, it is noted that the proposed controller can suppress the vibration of the smart beam very well. In effect, the feedback PD gain plays an important role in tracking and reject disturbances while the feedforward term can reduce the resonance amplitudes within transient regions. The input voltage controls for the two piezo-actuators are shown in Fig. 5. We assumed that the actuators are strong enough that no saturation would occur while in reality, the saturation problem should be considered well for safety.

Modelling and nonlinear vibration control of smart beam with large deflection. Large deflection results in a nonlinearly coupled cubic stiffness term that changes the dynamics behaviour of the vibrating beam.

Abstract:Due to the shortage of natural sand from rivers and seas, artificial sand production from large stones or rocks is being increased. However, this sand manufacturing process is dangerous and causes several social problems such as a high level of unwanted vibrations or noises. This study investigates the vibration characteristics of sand and screen units in an artificial sand production plant whose operation is multiple with several actuators different exciting frequencies. As a first step, vibration levels are measured at the sand and screen unit positions using accelerometers in time and frequency domains. The measurement is carried out at two different conditions: activating only the sand unit and operating entire facilities such as a stone crusher. Vibration signals acquired from several locations of the sand and screen units of the plant are collected and analyzed from waveforms and spectrums of the signals. We identified that the vibration acceleration level of the screen unit is higher than that of the sand unit. In addition, it is found from the acceleration signals measured at the plant office and shipping control center (which are far away from the plant location) that the beating phenomenon arose due to close driving frequencies for several sand units. In this work, the vibration caused from the beating is significantly reduced by adjusting the driving frequencies for the sand units so that they are sufficiently scattered to avoid the beating.Keywords: artificial sand plant; stone crusher; screen unit and sand unit; beating phenomenon; vibration measurement and reduction

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