The modal approach assumes that

The modal approach assumes that different vibrational modes of the distributed object are controlled independently. Ref. [2] first proposed to apply this approach to controlling vibrations in elastic systems, and this concept was further developed in Ref. [3]. The problem of arranging the sensors and actuators on the object, as well as the problem of separating the eigenmodes when they order Ro3306 are measured by the sensors and when they are controlled by the actuators must be solved to implement the modal control algorithm. The main methods of solving these problems involve either distributed sensors and actuators as modal filters [4,5], or arrays of discrete control elements [6–10].
Modal control of a cantilever beam Let us consider the problem of suppressing beam vibrations by modal control, choosing a cantilevered Bernoulli–Euler beam as a model object. Beam vibrations are excited by an external disturbance applied at a given cross-section; this disturbance varies by a polyharmonic law. Control actions, generated by the piezoelectric actuators, can be applied in r cross-sections in order to reduce the amplitude of the steady-state vibrations. Control actions are formed using signals from piezoelectric sensors located in n cross-sections. The mathematical model of a beam with a constant cross-section can be written in the form of the following system of differential equations: where u is the transverse displacement (deflection) of the beam; φ is the rotation angle of the tangent to the midline of the beam; Q is the transverse force; M is the bending moment; x is the longitudinal coordinate measured from the left end of the beam; ρ is the mass per unit length of the beam; I is the moment of inertia of the cross-section; E is Young\'s modulus of the beam material; μ0 is the distributed exciting moment, are the distributed control moments. The system is reduced to a single differential equation:
The boundary conditions reproduce how a beam of length l is fixed:
The initial state of the system corresponds to a rectilinear configuration of the beam:
To describe the control action, let us use the model of the piezoelectric actuator [11], constructed in the form of a thin rectangular plate glued along the beam between the cross-sections x= and x= (Fig. 1). According to this model, applying an electric voltage V to the electrodes of the actuator is equivalent to applying a pair of moments in the cross-sections x= and x=, equal by magnitude and opposite in direction. The moment\'s magnitude is given by the expression where e31 is the piezoelectric constant of the material, b is the width of the piezoelectric layer, is the distance between the midlines of the beam and the actuator. The control moments in the model (see Eq. (1)) can be represented in the form where δ is the delta function. Ref. [2] proposed a sensor model made in the form of a thin rectangular plate of piezoelectric material glued along the beam between the cross-sections x=d and x=D. In this model, the electric voltage acting as the sensor\'s signal is proportional to the difference in the angles of rotation of the cross-sections x=d and x=D: where is the distance between the midlines of the beam and the sensor, C is the capacity of the sensor. In constructing the control algorithm, let us accept the hypothesis that the sensor measures the curvature of the beam in the cross-section x=(D+d)/2 (for brevity, we shall write that the sensor is fixed in this cross-section). The construction of the modal control algorithm assumes that the deflection is represented as a series in eigenmodes: where are the eigenmodes satisfying the orthogonality conditions The principal coordinates (t) must be found as the solutions of the system
Here is the kth eigenfrequency of the elastic beam. The term in the right-hand side of the first equation in (3) governs the external disturbance. The term is the result of the transformation of the control action: