br Assuming the curvature of the beam shape

Assuming the curvature of the beam shape to be constant along the length of the piezoelectric actuator, we can take that where the coefficient =e31(–) is the same for all actuators, is the electrical voltage applied to the ith piezoelectric actuator. Thus, we assume that each actuator is fixed in a certain cross-section and is capable of changing the curvature of the beam in accordance with the control signal . Taking into account expressions (2) and (3), the goal of the control is in reducing the amplitudes of the steady-state vibrations of the first m principal coordinates (t), k = 1, 2, …, m by selecting the control signals . Combining r control signals into a vector U, we can write the modal control algorithm: where ×1 is the vector combining the buy dhpg signals; × is the diagonal gain matrix; ×× are the modal matrices to be selected (from now on the subscript will indicate the dimension of the matrix object). The modal matrix × serves as the mode analyzer, converting the vector of measured signals Φ×1 into a vector of estimates of the first m principal coordinates :
The matrix × sets the gain factors that transform the vector of estimates of eigenmodes into the vector of desired disturbances acting on each of the modes :
The modal matrix is the mode synthesizer, transforming the vector of desired disturbances acting on the modes into the vector of control signals sent to the actuators ×1:
In general, the elements of the matrix × do not necessarily have to be constants and may be functions of a complex variable: (s). With a diagonal matrix ×, the control system includes m loops, each corresponding to a special eigenmode of the beam\'s vibrations and having its own transfer function (s). A combination of sensor signals with the coefficients given by the jth row of the matrix × is fed to the input of the jth modal control loop, while the output signal of the loop is distributed between the actuators in the proportions set in the jth column of the matrix ×. Notice that control algorithm (4) can use the readings from an excessive number of sensors in n beam cross-sections to calculate the control signals for r≤n actuators. In this case it is possible to control no more than m harmonics (m = 1, 2, …, r). Let us introduce the modal matrices ×× and fill them in the following manner. The row of the matrix × consists of the values –d2/dx2, k = 1, 2, …, m in the point where the actuator is fixed, and the number of rows is determined by the number r of actuators. The row of the matrix × consists of the values –d2/dx2, k = 1, 2, …, m in the location where the sensor is fixed, and the number of rows is determined by the number n of the sensors. Let us represent the vector of sensor signals by a term including the first m harmonics and a vector containing higher harmonics: taking the coefficient to be the same for all sensors. The first m equations of the infinite system (3) taking into account control (4) and expansion (5) can be written in matrix form as: where Λ×=diag 2 is the diagonal matrix of squared natural frequencies of the elastic beam; × is the vector of external disturbances, combining the components ; The vector Δ contains only the higher harmonics but not components of the vector ×1. To achieve modal control, it is necessary to obtain the diagonal structure of the matrix this would allow to independently control the m lower harmonics of the elastic beam, provided that the higher harmonics are stable. The diagonal structure can be obtained by choosing the matrices × and ×. Below we propose a procedure for determining the modal matrices × and ×, including an identification experiment on a real object.
Procedure for determining the modal matrices First, it is necessary to obtain an estimate of r lower eigenmodes of the beam\'s vibrations experimentally or by calculation. The obtained estimates of the vibrational modes allow to mount the piezoelectric elements in the sections of the beam where the contribution of the modes is significant. After the sensors and actuators are installed, we can proceed to determining the modal matrices × and ×. The identification procedure is carried out for an open-circuit system and includes the following steps.