Both numerical and experimental studies

Both numerical and experimental studies have shown that gluing the piezoelectric elements onto the beam only insignificantly affects the object\'s natural frequencies and eigenmodes. This allowed to generate a calculated estimation of the modal matrices based on the data obtained:
Next, the modal matrices and θ were determined experimentally in accordance with the proposed identification procedure. Each of the two columns of the matrix was obtained as a result of processing the SAR 405 signals in the resonant modes generated by vibration of the piezoelectric stack actuator with the first and the second natural frequencies:
Then we were able to calculate the modal matrix T:
The modal matrix θ was determined by measuring the amplitude and phase of resonant vibrations of the beam\'s upper point using a laser vibrometer. The resonant modes with each of the natural frequencies were generated by an excitation from the first or the second actuator. The resulting matrix has the following form:
From here we can calculate the modal matrix F:
Next, we checked the quality of mode separation in accordance with the fourth step of the above-described identification procedure. The checking revealed that good separation of the modes was achieved both in measurements and in control. To illustrate the quality of mode separation, Fig. 4 shows different amplitude and frequency characteristics of an open-loop system in the frequency range containing two lower natural frequencies of the beam. To obtain the amplitude-frequency characteristic, the amplitude of the measured steady-state harmonic signal is divided by the amplitude of the harmonic exciting signal . Fig. 4a shows the amplitude-frequency characteristics , obtained when the beam vibrations were excited by the ith actuator and the signal was measured by the jth sensor. It can be seen that each of these amplitude-frequency characteristics contains two pronounced resonant peaks, since each of the actuators excites, and each of the sensors responds to both the first and the second frequencies of the beam\'s vibrations. Fig. 4b shows the amplitude–frequency characteristics of the open system, corresponding to the modal control loops. Here the beam was simultaneously excited by two actuators in the proportions given by the ith column of the matrix F, and the resulting signal was a combination of sensor signals with the coefficients given by the jth row of the matrix T. It can be seen that only the first resonant peak is present in the amplitude–frequency curve 11 corresponding to the first control loop, only the second peak is present in the amplitude–frequency curve 22 corresponding to the second loop, and resonant peaks are virtually absent in the amplitude–frequency curves 12 and 21 reflecting the mutual influence of the loops. This result indicates a high quality of separation of the first and second modes using modal matrices (8) and (9). Within the framework of the experiment, we created a modal control system allowing to suppress the forced resonant vibrations of the beam with the first and second natural frequencies. For this purpose, the modal control law (4) was achieved in the controller: where the diagonal structure of the matrix provided separate control of the vibrations with the first and the second natural frequencies, and selection and adjustment of the transfer functions k1(s), k2(s) maintained the stability of the closed system and the most effective suppression of the vibrations at the corresponding resonant frequency. Fig. 5 illustrates the efficiency of the control system with an oscillogram of the vibration velocity of the upper end of the beam
The experiment was organized as follows. First, when the control system was switched off, resonant vibrations were induced in the beam by the vibration of the piezoelectric stack actuator: Fig. 5a corresponds to the resonance with the first natural frequency, Fig. 5b to the resonance with the second natural frequency. The system was then closed as shown in Fig. 2. The time when the control system was switched on can be clearly seen in Fig. 5. As a result, the amplitude of the beam\'s vibrations significantly decreased: the decrease was 79% for the first resonance and 88% for the second resonance.