The problems of reduction of structural vibration induced by external force have constituted a great source of interest in recent years [1-4]. These excitations can be natural such as earthquakes, wind etc. or due to mechanical action. In both cases, this can lead to premature destruction of the system. One of the most famous examples illustrating these remarks is found in the Tacoma Narrows Bridge (USA, 1940) which was destroyed following an excitation by the wind. Several solutions have then been considered for this purpose. They range from simple archaic solutions such as destruction of these structures, to more modern solutions using techniques of vibration control which offer the advantage of being both less costly and more effective. Therefore, investigating vibration control of a plates become important in structures and mechanical systems design. To cope with these disturbances, many vibration control techniques have been developed such as sandwich control [5-8], electro-mechanical control [9-11], piezoelectric control [12,13], magneto rheological control [14], opto-electromechanical control [15,16].

In this paper, we are interested in the enhancement of viscoelastic control for a rectangular plate subjected to periodic impulsive excitation. The device is a three layers viscoelastic core subjected to localized periodic impulsive excitation. It is almost two rectangular plates of the same surfaces, hinged at both ends and possibly be made of different materials. The system is composed of two plates: One so-called main plate is subjected to impulsive external excitations located at point of contact, is coupled to another said dynamic absorbing plate by a viscoelastic coupling system consisting of identical springs and dampers with spring constant k and damping coefficient c respectively as shown in Figure 1. Therefore, several configurations of the coupling system can be used. We consider here the case of uniform distribution viscoelastic coupling.

In Ref. [17], Zhang et al. proposed to stabilize an elastic plate using viscoelastic boundary condition as feedback control. Hongpan et al. [18] investigated vibrations control of plate through electro-magnetic constrained layer damping which consists of, electromagnet layer, permanent layer and viscoelastic damping layer. Moreover, Aida et al. [6,7] have modelled the dynamic and optimized control by sandwich of a plate (and beam) under sinusoidal periodic excitation. They have shown

that the effectiveness of this control is more effective as the number of springs and shock absorbers becomes higher. However, model using periodic sinusoidal excitation does not relate enough to processes occurring in real time. Therefore, as a first approach to overcome this issue localized periodic impulsive excitations are considered. Additionally, we identify and define the characteristics of the dynamic absorbing plate over the dynamic responses of the main plate.

The second part of this paper deals with the derivation of the equations of motion of the main plate with hinged ends subjected to localized impulsive excitations. In the third part, analytical amplitudes of vibration are determined. Also, comparison between analytical and numerical amplitudes of vibration is provided. The fourth part is reserved to various configurations leading to the enhancement of the control process. We conclude our work in the last section.

Sine the most path of the energy of the system is embedded in the first mode of vibration, from Equation (9) we then expressed

where

;;

;

; and;

and;

and

; and.

The resolution of Equation (13) is not straightforward. Hence to overcome such a situation, it useful to assume new variables as following:

Adding the two set of Equations (13) gives the following

Subtracting the two set of Equations (13) yields to the following :

The resolution of the two differential Equations (16) and (17) permits to find the solutions of (13) through the following expressions

Thus, Equations (16) and (17) can be rewritten as follows

where

,;,

The solutions of Equation (20) are then derived and given by

in which and represent the constants of integration, are the particular solutions of Equation (20), and are the vibration frequencies expressed as

The shape of the external excitation suggests a treatment at intervals in oder to determine the particular solutions [20]. So, while considering the realistic case of excitation for which the pulse is not actually instantaneous as advocated by the Dirac delta function, it seems worthy to consider the three following main in order to well characterize the dynamics of the plates.

• Phases I:

(The dynamics responses are considered between the impacts of order and)

Here,; therefore :

• Phases II:

Here, one rather has; so :

• Phase III:

(The dynamics response are considered after all N impacts)

This phase of the process can be discerned as an element of the first phases for, corresponding to the dynamics of the plates between the impact of order N and the frictional impact of order. The particular solutions of this phase are given by Equation (23). Therefore, the solutions and are finally obtained and given as

The particular solutions and are given by expressions (21) according to the state of vibration phase. The constants of integration, , and are determined by the initial conditions of a current phase of the movement that are different from the final condition of the previous phase. Thus, the knowledge of the initial conditions of the first phase of the movement, that is also the initial conditions of the problem, permits to get closer to those of the other phases and therefore to determine the solutions of the problem at all times.

In the uncontrolled case (), it comes the following:

.

The two plates then become uncoupled and the solutions are now given by

in which, , and are the new constants of integration.

The control is efficient when the maximum value of is lower that the one of. This criteria enable to foresee the effectiveness of the control strategy.