ABSTRACT

This paper examines modelling of the dynamics of a plate by plate type dynamics vibration absorber subjected to a localized periodic impulsive excitation. An analytical solution of the modal equation is proposed and validated using direct numerical simulation of the basic equations. The basics equations are solve numerically using fourth order Runge Kutta algorithm. Various types of dynamic absorbing plate are tested to optimize the control efficiency. Particular attentions have been paid on the effects of localization of external forces on the dynamics response of the system under control. Ours findings demonstrate that a good achievement of control strategy should follow the above mentioned analysis.

1. Introduction

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.

2. Mathematical Modelling of the System

2.1. Modelling of Localized Periodic Impulsive Excitation

This type of load on a plate corresponds for example to the actions of generators or some robots. Both machines can be regarded as a mass device M, located on the plate by its coordinates. These devices in addition to their own weight per unit area, impose at the point of the plate, at regular time intervals equal to T, a surface force whose intensity (I) may change over time. So, the external excitation is a combination of these two efforts :

(1)

where,

(2)

while is defined literally by several models.

In general, the impact force is modelled as a rectangular pulse or as a Gaussian function whose integral over the duration of the impact is equal to the impulse of impact. The duration of the impact depends on the nature of contact between the exciter and the plate, the material they are made off and their dimensions. Therefore, minimizing the impact force is equivalent to minimizing the impulse of impact [19]. The case of rectangular plate is address in this study.

This model of excitation assume a force occurring at regular intervals of time equal to T and can be described as follows:

(3)

with.

N stand for the total number of impact while represents the duration of the pulse.

The presence of the Dirac delta functions and denote the local character of the impulsive force. If the pulse duration becomes too small as to be neglected, this model can be replaced by the model of the Dirac delta function giving by :

(4)

2.2. Equations of Motion of the Plate under Control

The main plate (see Figure 1) which is under the same boundary conditions with the dynamic absorbing plate is attached to this one by springs and dampers distributed uniformly (see Ref. [6]).

The equations of motion of flexural vibration for the main and dynamic absorbing plates with uniform flexural rigidity and uniform mass per unit area are expressed as

(5)

where the deflections of the main and dynamic absorbing plates are denoted by and respectively. and are their mass per unit area, and their damping coefficients, and their flexural rigidity, k and c are the spring constant and the damping coefficient of connecting springs and dampers between the main and dynamic absorbing plates, respectively. I is the intensity of the periodic impulsive load, T the exciting period, P the weight per unit area of the device creating the impulsive load, is the Dirac delta function and coordinates of the load position.

Let’s denote and the densities, and the thicknesses, and the Young’s modulus, and the Poisson’s ratios of the main and dynamic absorbing plates respectively. It comes whereas:

, , , and

While taking into account the boundaries conditions, the following approximate functions can be used for the solutions of equations of the system (5):

(6)

(7)

in which and are unknown functions of the time of the main and dynamic absorbing plates respectively and the normalized eigenfunction of mode of the both plates given by :

(8)

is a natural mode with n and m nodal lines lying the xand y-directions, respectively, including the boundary as the nodal lines. Substituting Equations (7) and (8) into Equation (5), according to the orthogonality of the eigenfunction gives

(9)

and are natural frequencies of modes (n,m) of the main and dynamic absorbing plates respectively and are derive as follows:

(10)

(11)

In non dimensional form, Equation (5) becomes,

(12)

where

,;

,;,

;

,;,

;,

where l a reference length.

3. Analytical Solution of the Modal Equations

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

(13)

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:

(14)

(15)

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

(16)

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

(17)

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

(18)

(19)

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

(20)

where

,;,

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

(21)

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

(22)

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 :

(23)

• Phases II:

(The dynamics response are considered during the impacts of order)

Here, one rather has; so :

(24)

• 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

(25)

(26)

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

(27)

(28)

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.

4. Numerical Analysis

4.1. Validation of the Analytical Results

We compare graphically, the analytical and numerical solutions for several values of parameters of the problem. As an example, we consider the main and dynamic absorbing plates identical made of steel with the following parameters :

Density of material

Young’s modulusPoisson’s ratiosDamping coefficientWidth a = 0.9 mLength b = 1.5 mThickness h = 0.002 m.

These quantities permit to find the dimensionless parameter and the free frequencies of the two plates as.

The fourth order Runge Kutta algorithm is used to computed the numerical solutions. Figures 2 and 3 display the time histories of the maximum amplitude of vibration in other to validate the result of analytical investigation. These figures show an excellent agreement between numerical and analytical results. The slight shifts observed for each impact are probably caused by sudden numerical fluctuations due to the discontinuity of the external excitation.

4.2. Enhancement of Control Process

Finding accurate condition that allows optimization of the control strategy is of interest. Thus we have at one’s disposal, several possibilities to choose the type of absorbing plate and among which the following: 

• Steel plate:,

• Tungsten plate:, Kg/m³

• Aluminium Plate (7075): ,

• Concrete plate (E20): ,

• Wood plate (plywood): ,

Each of these plates has thickness and their coefficients of dissipation, chosen such that the dimensionless parameter β₂ is always as β₂ = β₁ = 0.105. The initial conditions and the external force of excitation

are the same that in the section (4.1); but with and

4.2.1. Control Efficiency

The dynamic absorbing plate here is made of steel whose thickness is. The evolution of amplitude plotted in Figure 4 shows that the amplitude of vibration under control is lower than the one without control. Then control is indeed efficient for the control

parameters chosen. The possibility of the efficiency of the control being already pointed out for particular values of parameters of control. For that aim, we investigate separately their effects by considering other type plate having different physical characteristics compared to the one studied above.

4.2.2. Effect of the Control Parameters

Here, we analyze how much the nature of absorbing plate impacts the control process. All the absorbing plates used here are taken such that the thickness.

1) Effect of viscous coupling

Figure 5 shows that the control is efficient for all values of C since the amplitude of the control system is less than the one of the uncontrolled system (C = 0). This effect is particularly noticeable for higher values of C. In addition, this damping appears to depend on the material of the dynamic absorbing plate, and increases with the Young’s modulus.