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An improved model for the damped impact of composite materials applicable to wind turbine blades

Composite materials are used in manufacturing a vast game of products, from usual objects, medical devices, aerospatial parts to military equipment. One of the features considered when a composite material is designed as target aims the energy dissipation with respect to the projectile. The paper presents an improved alternative of a model which is broadly used in the impact of projectile and composite target. The initial model has the weakness that does not present the hysteresis loop. The improvement of the model consists in including a nonlinear damper placed between the projectile and the target, the damping force being simultaneously proportional to the impact velocity and the elastic force. The new model has a hysteresis loop closed in the origin. The proposed model is applied for concrete situations and the results are presented in graphical manner.


Introduction
Composite materials are widely used in developing the products met in everyday life [1] but also in specialized domains, like medical devices [2][3][4][5], military equipment or wind electric systems [6]. Theoretically they can be designed for achieving preferred characteristics but as Abrate highlights [7][8][9][10] the domain of application is narrowed by the impact behavior. The numerous papers dedicated lately to this matter prove the increasing attention produced by this subject.
A number of papers study the impact at low velocities and the impact strength. Abrate [11] underlines that a first step in understanding the effect consist in the development of a model capable to allow the prediction of time evolution of the impact force and of the reaction of the structure. The analysis of impact behavior emphasizes that any impact model is situated between two limit situations: the first case, when the entire energy of the projectile is transferred to the target, and small deformations are produced that finally lead to crack micro initiators; the second case, when the composite, due to its properties, returns the entire impact energy to the projectile. A typical case for the situations mentioned above are the rotor blades of the wind turbines made of composite materials, Mishnaevsky [12]. A recent paper due to Gaudern [13] proofs the destructive effect produced by the erosion upon the rotor blades of the wind turbine. The statement is demonstrated with probes obtained in wind tunnels. To be remarked that the leading edge is the most affected region of the rotor blade. As known, the wind turbine farms must be placed in regions where minimum impact upon the environment is produced. Onshore, it is expected to view wind turbines working quite remote from popular areas, even in agricultural lands or in desert regions. A characteristic of these regions is the presence of dust particles that will be swept up into motion by the large air masses with velocities up to sec / m 13 , [14] and thus becoming actual micro-projectiles on the surface of the turbine. The mentioned facts are good reasons for the conception of models capable to estimate the impact force and lastly to allow for optimum design of rotor blades of wind turbines.

The Abrate spring-mass models for the impact of composite structures
Abrate makes a comprehensive review [11] concerning the representations that study the projectile-composite material impact and identifies several models.  Among these the spring-mass models are present. These models have the advantage of being simple and offering precise solutions for certain impact types, frequently met during the tests where small projectiles are used.
A two degree of freedom model is presented in [11] and consists in a deformable structure (2) collided by a projectile of small dimensions (1), as presented in Fig. 1.
The system is modeled using two masses and three springs. The equations describing the motion of the system are: where 1 M is the mass of the projectile, bs K is the elastic constant of the structure with linear behaviour, m K is the elastic constant corresponding the the nonlinear behaviour of the structure. In equation 1 P is the impact force between the structure and the projectile, expressed as a strongly nonlinear dependency on the deformation Next it is assumed that the impact force P presents a Hertzian form. Thus, for the case of centric impact of two spheres, the impact force has the expression: are the radii of curvature of the contacting bodies and  is the contact stiffness, defined according to Johnson [15] as: In the above relation, 12  and are the Poisson coefficients and the Young moduli, respectively, of the two materials. Next it is considered the impact between a ball-projectile and plane membrane-target. For a complete clamped isotropous circular membrane, the coefficients m K and b K are expressed by the relations: where h is the thickness and a is the radius of the membrane, [16]. To be mentioned that the effective mass of the membrane taking part to the oscillation motion is a quarter of the total mass of it. The equations 1 considered together form a system of nonlinear differential equations , [17] and solving it requires applying a numerical procedure, [18]. The initial conditions are necessary in order to integrate the system and in the present case they are: The next concrete situation was considered for integrating the system: Runge-Kutta IV method, [18] was applied for integrating the system of equations 1.
The absolute displacements of the projectile 1 x , of the center of the membrane 2 x and the relative displacement of the projectile with respect to the membrane are presented in Fig. 2. The deformations from Fig. 2 are represented for a longer period in Fig. 3. It can be remarked that both the projectile and the membrane get an oscillatory motion as a result of at least two harmonics and therefore, the relative displacement of the projectile with respect to the target is a quasi sinusoidal oscillation.
The plot force-relative deformation of the projectile presented in Fig. 4 is of particular attention.   Fig. 4 it can be concluded that the system of differential equation 1 describe the behavior of an undamped system since for a damped action, a closed forcedeformation curve entrapping an area equal to the work of damping should occur.
The fact that the model of a composite material exhibits un-damped behaviour is a major deficiency of the representation.

Proposal of improved model
In order to improve the model proposed by Abrate, a viscous dashpot (damper) is attached to the model, connected in parallel to the spring between the projectile and the membrane. As it can be observed form Fig. 5, the damping constant was considered variable. This requirement was applied by Lankarani and Nikravesh [19], and Flores et al. [20]; for modeling an impact phenomenon with viscous damping, they establish the model based on the remark made by Hunt and Crossley [21], who showed that when the viscous damping depends only on the relative velocity the hysteresis loop is open in origin, this meaning that at the end of the impact, the two bodies attract to each other instead of rejecting.
Based on the observations made by Dubowsky and Freudenstein [22][23], Hunt and Crossley [21] showed that for obtaining a loop closed in origin it is necessary that the damping force should be proportional both to relative velocity and to the contact force exerted when no damping is present.

Fig. 5. Proposed model with damper
Founded on this remark, a modified equation of motion, that considers the damping occurring in the ballcomposite contact, is proposed under the following form: In the relations 8, the parameter  is the characteristic of damping and 2 / 3   is the exponent typical to the Hertzian contact [24][25][26][27].
The system of differential equations 8 was integrated numerically for the parameters 7 and for the same initial conditions 6, for two values of the  constant, that is:    It can be observed that for lower damping there is only an asymptotic point, while for higher damping values there are two asymptotic points. That means that there is a critical value of damping which splits the two aspects.
In Fig. 10 it is presented the phase diagram for critical damping of the system from Fig. 5.
A number of important conclusions result from Figs. 6-8: 1. The proposed model demonstrates damping with the hysteresis loop closing in origin. 2. The damping work is more important in the approaching phase than during the detaching phase. 3. The effect of increasing the damping is a slight increase of the time when the displacements of the projectile and target are equal. 4. The area of the hysteresis loop increases significantly when the damping augments. 5. It is observed that the maximum impact force is not in phase with the minimum relative displacement. When the maximum impact force is attained, the minimum relative displacement is delayed with respect to this moment.