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  • First stage To determine the matrix an external disturbance

    2018-10-24

    First stage. To determine the matrix ×, an external disturbance is used to successively generate forced resonant vibrations of the beam corresponding to m lower natural frequencies. Let the bending moment applied in the cross-section x=x0, be the external action, then model (6) of an open-circuit model takes the form
    Resonant modes are used as a selective transformation, since beam vibrations in one eigenmode, corresponding to the natural frequency , prevail over the others in each of them. Therefore, we can assume that the signal of the ith sensor in the jth resonant mode is determined by the expression
    Accepting that there is proportional damping in the system, the equation for the jth principal coordinate can be written as: where is the damping coefficient, is the amplitude of the disturbance. The steady solution has the following form: which allows to estimate the steady-state signal from the ith sensor by the expression where . From here we can obtain the AICAR manufacturer for the components of the matrix :
    Measuring the amplitudes and the phases of the signals from each of the n sensors in the jth resonant mode allows to obtain the jth column of the matrix up to a constant if we accept that Φ. This estimate is sufficient for separating the vibrational modes in modal control loops. Second stage. If the actuators and their corresponding sensors are installed in the same sections of the beam, then the matrix × is written out from the matrix ×. In practice, this condition is not always satisfied, so the algorithm for determining the matrix θ experimentally is given below. In the absence of external disturbances, model (6) of the open-circuit system is represented in the form
    Let us excite resonant vibrations of the beam only by the ith actuator, feeding it a harmonic voltage with a jth natural frequency
    All vibrational modes will be present in a steady-state process caused by damping in a real system. However, the vibrations in the jth eigenmode will be dominant, while the contribution from the remaining modes will be negligibly small. The equation for the jth principal coordinate upon excitation of vibrations by the ith actuator can be written in the form
    Thus, the jth column of matrix θ indicates the contribution of each actuator in exciting the jth vibrational mode of the beam. The steady-state solution of the equation has the following form:
    Let us assume that telophase is possible to measure the deflection in some cross-section x* by means of an external device. The estimate of the deflection in the resonant mode with the jth natural frequency is given by the formula
    Let us express the components of the matrix θ through the measured quantity :
    In order to obtain an estimate for the jth column of the matrix θ, resonant vibrations of the beam with a jth natural frequency must be successively excited by each of the r actuators. The amplitude and the phase of steady vibrations need to be measured for all r experiments in one resonant mode in the same cross-section of the beam. Measurements should be made with an external device, for example, a laser vibrometer, in a cross-section located as far as possible from the actuators in order to minimize the effect on the resulting inaccuracy in reproducing the resonant eigenmodes. The measurements allow to obtain the jth column of the matrix θ with an accuracy up to a constant, if we accept that . This estimate is sufficient for separating the modes in modal control loops. Third stage. In the particular case when , the desired modal matrices are chosen to be inverse to the eigenmode matrices:
    Whether the eigenmode matrix can be inverted depends on the arrangement of piezoelectric elements along the beam; this is related to the properties of observability and controllability [12]. In practice, this case means that the number of controlled modes is pre-determined and equal to the number of sensor–actuator pairs. It is often not known in advance how many forms will have to be involved in the modal control algorithm for achieving an acceptable degree of vibration suppression. For this reason, the number m of the modes which are intended for controlling the object can be first specified in order to minimize the number of identification experiments. Then the modal matrices will take a rectangular form and will be determined by a pseudo-inversion operation: