Active Control of A Piston-Type Absorbing Wavemaker with Fully Reflective Structure

Multiple reflections of the waves between structure and wavemaker in hydraulic flumes could change the frequency content of the desired incident wave or result in resonance. A prominent approach to avoid multiple reflections is active control of the wavemaker. This paper proposes a simple and practical active control algorithm for piston-type wavemaker. The block diagram of the control system is presented in real time domain. It is shown that there is no need to use any transfer function or filter in the feedback and feed forward loops and the use of constant gains can yield acceptable results. In the operating frequency range (0.25–2 Hz), it is revealed that the proposed system is very effective at suppressing the excitation of resonant sloshing for regular wave. In the case of irregular waves, it is depicted that the experimental waves agree quite well with the desired wave elevation in frequency domain. In addition, comparison of the results obtained both with and without absorption discloses the good characteristics in time domain.


Introduction
To investigate the wave-structure interactions, researchers have developed numerical and experimental wave flumes. In a numerical wave flume, a semi-infinitely or bounded, constant depth, long channel has been considered as a domain. Assumed waves are generated by an oscillating boundary and an artificial beach or porous region has been considered at the other end to restrict reflection (Clément, 1996). For small wave amplitudes the linear velocity potential theory has been used to simulate the wave propagation (Yueh and Chuang, 2013). However, in the case of finite or large wave amplitudes the second order Stokes equations (Senturk, 2011, Khait andShemer, 2019) and/or nonlinear Navier−Stocks equations have been used (Anbarsooz et al., 2013;Lin and Liu, 1999). However, it was shown that the different theories did not exhibit the same level of accuracy, especially for steep and deep water waves (Saincher and Banerjee, 2015).
An experimental wave flume is a long and narrow physical tank with a wavemaker of some kind (Stagonas et al., 2011;Belden and Techet, 2011) at one end to generate twodimensional wave trains. The generated waves could be reflected due to the presence of a test device or end wall of the flume. Reflected waves could violate the frequency contents of the desired incident waves or cause resonance in the system. A very common technique to reduce the reflections from the end wall of the flume is a beach with a constant slope. However, it would consume a large portion of the test domain. Other methods to reduce the reflected waves include the use of transversal bars, horsehair, and wire screen (Ouellet and Datta, 1986). However, these types of absorbers cannot eliminate any waves reflecting from the test device back to the wave maker.
The most recent effective approach for eliminating the reflective waves is the use of absorbing wavemakers whose paddles are controlled so as to generate the desired incident waves and absorb reflected waves at the same time. Milgram (1970) used wave elevation gauge as a hydraulic feedback and built an absorption system in a flap-type wave maker. Hirakuchi et al. (1990) developed a piston-type absorbing wavemaker with a front mounted wave gauge as a hydraulic feedback. Christensen and Frigaard (1994) employed two wave gauges and developed an active absorption system. They used digital filters to estimate the absorption transfer function. Schäffer and Jakobsen (2003) have also proposed digital filter to approximate the absorption transfer function, but they did not discuss the fitting technique. Thereafter, Yang et al. (2016) suggested iterative reweighted least-squares algorithm for approximating the absorption transfer function. They demonstrated the performance of the designed controller for the regular and irregular waves. Yang et al.(2015) considered some delays in the control loop and designed an absorption system for a pistontype wavemaker with delay compensation.
Another choice of the hydrodynamic feedback is to use the oscillatory part of the wave force acting on the paddle. Owing to integral properties of the wave force it is less susceptible to small local disturbances than the wave gauges. Salter (1981) used the force on the wave paddle as the hydrodynamic feedback in conjunction with a recursive filter and developed an absorbing wavemaker. Spinneken and Swan (2009a) presented a mathematical model for operation of an absorbing wavemaker based on the force-feedback control and also provided an experimental verification of their new theory (2009b). Then, they proposed forcefeedback control to absorb directional waves (2012). One can find a comprehensive review of the absorbing wavemakers published by Schäffer and Klopman (2000). Although some active control of absorbing wavemakers have been reported recently (Yang et al., 2016, Higuera et al., 2013, De Mello et al., 2013, Spinneken and Swan, 2012, a practical block diagram in real time is not described in those papers. The literature review shows that most researchers have investigated the performance of active absorbers in the frequency domain. Some have even considered the effect of evanescent waves in the absorber design and proposed a complex structure for the absorber (Hirakuchi et al., 1990). This paper proposes a simple and practical control algorithm to absorb reflected waves in regular and irregular wave experiments. Initially the specifications of the equipment used in the laboratory tests are presented. Without employing absorption loop, the mathematical model of the coupled system is derived with a full reflective structure. Vibrational properties of the system is extracted and compared with the experimental results. Then, the structure of the proposed active absorber is presented as a block diagram in real time domain. Performance of the designed controller is evaluated with regular and irregular wave experiments where a fully reflective structure is placed across the channel.

Vibrational properties of the wavemaker
One of the main features of the active absorption system is that it helps suppressing wave flume resonance. So, before presenting the absorber structure, the resonance conditions of the system should be identified. Two-dimensional motion of an inviscid, incompressible fluid in a finite channel with full reflector at its end is shown in Fig. 1. The coordinate y is taken vertically upward with its origin at the bottom of the channel, while x is taken horizontally with the mean position of a piston-type wave paddle at x = 0.
On the assumption of small wave amplitudes, the wave motion can be described with the associated velocity potential, where it should satisfy the following Laplace equation.
where x 0 denotes the paddle amplitude, g is the acceleration of gravity, and ω is the forcing frequency of the paddle. To find the natural frequencies and mode shapes of the unperturbed coupled system one should freeze the paddle motion. With this assumption one can find the following natural frequencies and mode shapes of the system.
However, to find the frequency response of the system, a harmonic motion is assumed for the paddle movement. So, the solution of Eq. (1) while imposing the boundary conditions (2) gives the following velocity potential: where, employing the free surface and bottom condition of the wave tank, the wave number k 0 and the eigenvalues k n (n=1,2,3,…) are the roots of the following relations.
In Eq. (4), A n , B n , A 0 , and B 0 are the unknown modal amplitudes of the fluid oscillation. In order to find these unknown modal amplitudes the boundary conditions (2) are employed. Substituting the obtained velocity potential in to the first boundary condition and making use of the orthogonality of trigonometric functions yield the following relations: Saeed MAHJOURI et al. China Ocean Eng., 2020, Vol. 34, No. 5, P. 730-737 731 Using the end or reflector side boundary condition, one can find the remaining modal amplitudes as: 2.2 Absorption procedure In the present research the wave elevation in front of the paddle is used as a hydraulic feedback and a control algorithm is proposed and implemented to absorb the reflected waves. Fig. 2 shows the block diagram of the proposed control system where the digital, analogue, and interface parts of the circuit are specified.
It should be pointed out that the wave elevation control is equivalent to the velocity control of the paddle. Therefore, controlling of the paddle velocity does not necessarily guarantee the position control of the paddle. So, during the wave elevation control, possible drift in the position of the paddle may saturate the paddle stroke (Schäffer, 2001). To avoid the paddle drift an axillary closed loop position control of the paddle is employed. It is evident that the drift compensation should be operated with very low velocity to avoid any disturbance of the wave absorption loop. To implement the axillary drift compensation, a displacement sensor (LT-M-400S) measures the position of the paddle and its output is used as a feedback signal. Mean position of the paddle 'x d =0' is used as the desired input and a proportional control with gain K 4 is used to regulate the paddle. In the experiments, its value starts from zero and gradually increases. The minimum value of K 4 that prevents the paddle drifts is selected as its final value. Structure of the drift compensation loop is depicted in Fig. 2 with red signals.
In the main control loop, the measured wave elevation η 0 which includes the generated, reflected, and re-reflected waves is used as a feedback signal. The wave gauge output is a weak signal that passes through a conditioner. The conditioner gain is specified as K 3 . The desired wave form η d is provided as a data file and applied through a computer. It is worth noting that the type of the parameters that are compared in the summation points should be the same. Therefore, the wave gauge output is calibrated with coefficient K 1 . It should be emphasized that K 1 and K 3 values may need to be reviewed each time the wave gauge is calibrated.
In the absorption loop, in addition to the feedback signal a feed-forward signal of the target wave η d is already used as the major part of the required control value. The value of control gain K 2 is crucial in the control system. In other words, to achieve the acceptable absorption rate, its value must be adjusted in each test where the water depth and working frequency change. So, a table should be formed in this regard, and by changing the test conditions, the relevant K 2 should be extracted from the mentioned table or an interpolation procedure should be done. However, the experiments in the present study were performed at a constant water depth (h=0.63 cm) and in a limited forcing period (T=1.2−3 s in regular waves and T p =1.6-2 s in irregular waves). Therefore, by setting a single value for K 2 and tuning it, the desired results are extracted.
In this research it is shown that there is no need to use any filter or transfer function in the feedback and feed-forward loops (Bullock and Murton, 1989) and the use of constant gains can yield acceptable results. Also, the servomotor time constant is order of 1 ms and the sampling period of the communication A/D card is 330 Hz. So, referring to the maximum operating frequency of the system (f max =1 Hz) we have neglected these delays in the controller design process (Yang et al., 2015). In addition, the proposed controller structure is very simple, practical and in real time domain. In other words, it does not require the design of digital filters (Yang et al., 2015, Hirakuchi et al., 1990 and the use of approximate techniques (Yang et al., 2016). Moreover, the structure and implementation of the proposed method seem to be simpler than the forced feedback-based control methods (Spinneken and Swan, 2009a, 2009b, 2012.

Experimental equipment
The piston type wave maker is installed in a wave flume being 25 m long, 1.3 m high and 1 m wide. The flume has been fabricated as a modular structure comprising of 14 segments. The length of each segment is 1.8 m and the side walls of the first five segment are made of steel plate and the 9 remaining segments have secure glass walls. The wave maker is located 1.8 m away from the one end and the reflector plate is installed 14.2 m away from the wave maker paddle (see Fig. 3). In order to minimize the sloshing effects on the paddle driving force, an energy dissipating rubble-stone beach was set at the rear part of the wave maker.
The paddle is driven by a 3 kW electrical AC servomotor (ASD-B2-3 023 servo driver and ECM servomotor) and a ball screw mechanism, which is controlled by a computer. A resistance type displacement sensor (LT-M-400S) measures the paddle displacement and a capacitive type wave gauge was mounted on the front side of the stainless steel paddle to measure the wave elevation. With an A/D and D/A convertor (μDAG-Lite) the controller output is sent to the ser-vomotor and the paddle displacement and the wave elevation signals are transferred to the computer. Experimental and theoretical results are presented in the next section.

Results without absorption loop
In this section, the natural frequencies and mode shapes of the wave tank are evaluated taking into account the full reflector plate. In addition, due to harmonic motion of the wave maker, frequency response of the system is extracted. For some special cases the experimental results are presented and compared with the numerical results.

Theoretical results
For the present experimental setup, Fig. 4 demonstrates the first twelve natural periods of the system and Fig. 5 illustrates the first three associated mode shapes. With regard to the operating range of the paddle periods (up to 4 s) it seems that the first two periods will not be excited. However, the system may be driven near the higher periods. In these cases the resonance condition may occur and consequently the tests could not be continued any more. To show the resonance condition, frequency response of the system is depicted in Fig. 6. It is seen that in resonance conditions the wave elevation could be magnified dramatically.    Saeed MAHJOURI et al. China Ocean Eng., 2020, Vol. 34, No. 5, P. 730-737

Experimental results
By referring to frequency response (Fig. 6), specific experiments have been done where some of them are closer to the resonance conditions. For harmonic paddle movement with period 1.2 s and amplitude 40 mm, Fig. 7a shows the wave elevation in front of the paddle. It can be seen that when the reflected waves reach the wave paddle (at about t=21 s), the resultant wave is magnified and consequently the test is turned off by force at t=103 s. Frequency content of the wave is illustrated in Fig. 7b. Magnification of the wave elevation is attributed to the closeness of the driving period to the 11th natural period of the system. Another experiment near the resonance condition is done where the forcing period is set as 3 s. Referring to Figs. 4 and 6 it is inferred that the forcing period is very close to the 4th natural period of the system. Fig. 8 shows the wave elevation and its frequency content. It is seen that the reflective waves reach the wave paddle at about t=13 s. Time history shows that the re-reflective waves contaminate the total incident wave after that time. This can be inferred with looking to the frequency contents of the incident waves (Fig. 8b) where some super harmonics come into view.

Regular waves
In this section performance of the absorbing controller in eliminating the spurious re-reflection of outgoing waves and suppressing the wave flume resonances is demonstrated. Fig. 9 shows the wave elevation in front of the paddle when the desired wave period is 1.2 s and its amplitude is set as 40 mm. Comparison of time histories with and without activating the absorption loop shows that the resonance is suppressed and stable standing wave is pro-duced ' Fig. 9a'. It is worth noting that to prevent the initial shock wave, the desired wave amplitude gradually increases during the first two cycles and reaches its final value. On the other hand, the output of the absorption loop controller is zero or very low level signal until the first reflective wave reaches the paddle. In other words, before that time, the response of the open loop and closed loop systems are similar. However, at the re-reflection moment the absorption controller reacts and prevents the incident wave from diverting. In Fig. 9a, around t=21 s, the first two mentioned cycles reach the paddle position. At that moment, the controller absorbs the reflected waves and prevents them from being reflected again. However, due to the presence of a single wave gauge, the controlled incident and reflected waves are not separated and the wave gauge data will actually be the sum of these two waves. Therefore, due to the phase difference between these waves, the amplitude of the resulting wave is smaller than that of the incident wave. Frequency contents of the waves are compared in Fig. 9b. Another resonance condition which was depicted in previous section (Fig. 8) is tested with imposing absorption loop. The results are shown in Fig. 10. It is shown that the growing wave amplitude is controlled excellently.
Comparison of the results in time and frequency domains shows that the absorption algorithm eliminates the spurious re-reflection of outgoing waves effectively. It is worth noting that the wave gauge data include the incident and reflected waves. Therefore, in the steady state conditions, depending on the phase difference between the incident and reflected wave, the resulting wave amplitude may be larger (Figs. 9a and 10a) or smaller than that of the desired wave.

Irregular waves
To investigate the performance of the absorption loop in irregular wave experiments, three irregular waves with different peak wave periods T p and significant wave heights H s are considered. In other words, the JONSWAP spectrum (Sheng and Li, 2017) is used as the desired irregular wave spectrum. Wave elevations and frequency spectrum of the waves with and without absorption loop are compared. Figs. 11a and 11b show the time histories of the desired in-cident wave and the measured wave elevations with and without utilizing the absorption loop. For more transparency the wave elevation data without absorbing algorithm is displaced in vertical axis deliberately. Fig. 11b presents the zoomed version of Fig. 11a. It is observed that the wave with absorption is well comparable with the incident wave and the wave elevation without absorption is very different from the incident wave. Fig. 11c shows the frequency spectra of the wave elevations where the peak wave frequency is set as 0.55 Hz or T p =1.8 s (H s =0.06 m). It is inferred that the spectra with absorption loop agree well with the desired spectra.
Irregular wave with peak wave periods 2 s (H s =0.08 m) is also presented in Fig. 12. Referring to the zoomed time history, we can see that with absorption loop the generated waves are consistent with desired waves whereas without the absorption, the produced waves deviate dramatically ' Fig. 12b'. Comparison of the wave's spectra highlights the effects of the proposed controller in absorbing reflected waves and eliminating the unwanted re-reflections. It is also shown that the re-reflections can add numerous peaks to the incident wave spectrum. As mentioned earlier, the wave   gauge records the sum of the incident and reflected waves. Because of this, the wave spectra and time history of the measured wave with absorption are slightly different from the desired values.

Conclusions
This paper introduces an efficient piston-type active absorbing wave-maker with a front-mounted wave gauge. The structure of the control algorithm was presented as a block diagram in the real-time domain. It was shown that there is no need to use any digital filter in the case of relatively low operating frequency range (0.25−2 Hz) and the use of constant gains can yield acceptable results. For regular waves, it was shown that the proposed algorithm could effectively suppress the resonant sloshing condition and it is possible to produce pure standing waves. For irregular waves, the performance of the wave-maker is evaluated for peak waves at dominant periods of 1.8 s and 2 s. It was shown that the proposed absorption algorithm can generate waves having a good correlation in the frequency content and phase angle with the intended wave. In addition, the time history of the desired and incident waves was also compared.