design parameters optimization of a particles impact damper

Research ArticleVolume 5 Issue 4 - June 2018DOI: 10.19080/CERJ.2018.05.555667

Civil Eng Res JCopyright © All rights are reserved by Cherif Snoun

Design Parameters Optimization of a Particles Impact Damper

Cherif Snoun*1 and Moez Trigui2

1Mechanical Laboratory of Sousse, University of Sousse, Tunisia2Mechanical Engineering Laboratory, University of Monastir, Tunisia

Submission: February 16, 2018; Published: June 25, 2018

*Corresponding author: Moez Trigui, Mechanical Laboratory of Sousse (LMS), National Engineering School of Sousse (ENISO), University of Sousse, Cité Erriadh, BP 264, 4023, Sousse, Tunisia, Tel : ; Fax : +216 73 500 415; Email:

IntroductionPassive mitigation of vibrations is an important research field

in many areas of engineering. Unlike the traditional methods of the passive control of vibration such as tuned mass damper, Particles Impact Damper (PID) is considered one of the passive means to provide high damping to the vibrating structure. Moreover, PID is simpler to construct, and can be used for application at elevated temperatures without a significant consequence on its competence [1]. The PID process consists in introducing a small mass or particles free to vibrate unidirectionally between two end stops of an enclosure attached to the vibratory structure. When the system moves, the particles undergo a cyclic motion and collide with the stops. As a result of these collisions, the amplitude of vibrations of the primary structure is reduced through momentum transfer by collision and kinetic energy converted to heat. Additional energy dissipation can also occur due to frictional losses and inelastic particle-to-particle collisions [2,3].

In the literature and in order to characterize the passive damping of PID, two main approaches were sensed. The early works, which have been the subject of several analytical and experimental studies, have a propensity to characterize a PID attached to a vibratory primary structure. Moez et al. [4]

investigated an experimental study of a vertical impact damper attached to a clamped-free beam under free excitation. The authors illustrated the effect of some PID parameters on the vibratory behaviour of the structure. They proved that, with a PID, a very high value of loss factor can be achieved compared with intrinsic material damping of the majority of structural metals. Ahmed et al. [5] studied an analytical and experimental model of a honeycomb beam filled with damping particles. The authors developed a cell level model of the considered honeycomb. In this investigation, the Discrete Element Method (DEM) is used to model the dynamics of the particles in conjunction with the motion governing equations of the beam and the cell-walls. Recently, another method tends to study PID alone without the primary structure. The advantageous of those approaches is the possibility to use of the characterized damping in other structures in order to predict its dynamic responses [6,7].

Despite the increasing use of PID in mechanical systems, the modelling of PID remains a difficult task due to a number of problems [8-10]. One of the principal complexities in using PID is the remarkable nonlinear behaviour making it complicated to design [11-13]. The design complexity is explained by the large

Civil Eng Res J 5(4): CERJ.MS.ID.555667 (2018) 00131

Abstract

Particles impact damping is one of the promising technologies used for the passive mitigation of vibrations. Through the dynamic impact and friction between particles, particles impact damper is able to reduce or eliminate vibratory energy under kinetic shape.

In this paper, the dynamic behaviour and efficiency of this process to reduce vibrations are highlighted. Using a simple analytical model, a clamped-free beam coupled to a particles impact damper, the influence of some system parameters is investigated. The proposed model is, then, validated through a comparison of simulated responses with experimental results established in a previous work. It is noticed that the nonlinear behaviour and the large number of their design parameters make the determination of its participation in damp very complicated. For this reason, it is crucial to seek optimal design parameters of the particles impact damper. In this context, an optimized method based on a genetic algorithm is proposed. The obtained results demonstrate the satisfactory side of this approach to identify the optimal parameters of the considered particles impact damper.

Keywords: Vibration; Passive damping; Loss factor; Optimization; Genetic algorithm

Abbreviations: PID: Particles Impact Damper; DEM: Discrete Element Method; SUS: Stochastic Universal Sampling; PPX: Precedence Preservative Crossover; OF: Objective Function; GA: Genetic Algorithms

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How to cite this article: Cherif S, Moez T. Design Parameters Optimization of a Particles Impact Damper. Civil Eng Res J. 2018; 5(4): 555667. DOI: 10.19080/CERJ.2018.05.555667.00132

Civil Engineering Research Journal

number of parameters such as the enclosure geometry, the shape and material of particles, the amount of free space (gap size or volume fraction) given to the particles, the level of displacement and acceleration of the primary structure. Furthermore, the efficiency of a PID depends on those parameters, and then, the optimization will be a necessary assignment.

In this context, the use of Genetic Algorithms (GA), which is considered as a powerful and broadly applicable stochastic and optimization researches applied for diversity problems, is an interesting way to optimize PID parameters. In recent years, GA has been gaining popularity as an appropriate tool for solving combinatorial problems which grows exponentially in complexity as the number of parameters rises. Furthermore, GA is a heuristic technique that can produce quick and cost effective solutions to problems that would, otherwise, take a prohibitive amount of time [14]. In addition, GA is attractive because it’s easy to develop with efficiency results [15]. In this work, the PID parameters are optimized using GA. The studied PID is attached to a clamped-free beam and studied under free vibrations.

This paper is organized in two main sections. The first one presents the model of a beam treated with a PID under free vibration. In the second section, an optimization of PID parameters with GA is presented.

Presentation of the Studied SystemThe well-known one degree of freedom Friend’s analytical

model [3] coupled to PID in its extremity is used in this work. The simplicity of this model allows the development of analytical expressions in order to better understand the PID role. Figure 1(a) shows the enclosure containing particles and attaching to the end of the clamped-free beam. The clamped-free beam is assumed to be uniform in the cross-section, having a mass bm , young modulus E, and cross-sectional area moment I. The first bending mode of this structure is characterized by an important deformation at its free extremity where the enclosure is attached. Figure 1(b) shows the adopted equivalent model of the beam and the enclosure. The continuous beam is modelled as a discrete single degree of the freedom system, having a reduced stiffness K and a reduced mass

rM computed at the tip of the beam with:

Figure 1: (a) A schematic of the clamped-free beam and the enclosure, (b) The equivalent single-degree-of-freedom model adopted.

33 /K EI L= (1)

0.24M m mr b c= + (2)

where mc , represents the mass of the enclosure.

To complete the analogy between the continuous system and the equivalent one, it is assumed that all particles move as an Impact Mass (IM) m. Therefore, there are periods times when the impact mass moves in contact with the beam, and in other periods it moves separately. Moreover, the equivalent mass M of the system and the corresponding undamped natural frequency ω are two-valued functions of time.

In the first case, the lumped mass moves in contact with the beam

M M mr= + (3)

/ ( )1 K M mrω ω= = + (4)

In the second case, the particles move separately from the beam

M Mr= (5)

/2 K Mrω ω= = (6)

The equation of the reduced damping coefficient of the system is

/2

bC K Mψ

π= (7)

Where, bψ is the intrinsic material damping of the beam [3].

Equations of motionTaking into account that there are times when IM moves in

contact with the beam, and at other times it moves separately, the




reduced system is governed by the following movement equation2

2( )d x d xM c Kx f t

d td t+ + = (8)

By dividing the equation (8) by M , it becomes2

22

( )2d x d x f txd t Md t

ξω ω+ + = (9)

Where, ( )f t is a two-valued function of time: ( ) 0f t = if the IM is in contact with the beam and ( )f t mg= − if the IM moves separately from the beam .

ξ is the damping ratio of the beam given by

/ 4bξ ψ π= (10)

The resolution of equations (9) lead to two solutions, 1( )x t and 2( )x t which represent the displacement of the beam in the two cases explained above

0 1( ) cos( ) sin( )1 0 1 0 11

v tx t x t x t e

ξωηω ηξ ηω

ηω

− = + +

(11)

0 2( ) cos( ) ( ) sin( )2 0 2 0 22

v tmg mg mgx t x t x t eK K K

ξωηω ηξ ηω

ηω

− = + − + + −

(12)

Where, 21-η ξ= , 0x and 0v represent respectively the initial displacement and initial velocity of the reduced mass. The

term /mg K represents the static deflection of the beam due only to the IM.

Impact mass motion At the beginning of one cycle of the system motion, which

is defined as the duration between two successive peaks of the velocity, the reduced mass and the IM which are in contact move together and its motion is expressed by equation (11). This contact remains as long as the acceleration of reduced mass rM exceeds the acceleration due to gravity g (Figure 2(b’, b’’)). Otherwise, separation occurs when

2

2

( )1-d x t

gd t

> (13)

After separation, the motion of Mr is given by ( )2x t equation (12) and the IM moves under the influence of gravity (Figure 2(a’, a’’)), its motion is expressed by

21( ) ( - ) - ( - )2

x t x v t t g t tp s s s s= + (14)

Where, ts is the instant of separation, xs and vs represent respectively the position and the velocity of the IM at time ts. Figure 2 represents the numerical simulation of displacement, velocity and acceleration of the system under free vibration. It is observed that two sorts of impact are credible (Figure 2):

Figure 2: Dynamic responses of the beam treated with PID, (a): h= 5 mm; (b): h = 8 mm.

a) The first impact of the IM will be with the enclosure ceiling. This case occurs if sv is sufficient or when the value of clearance h is sufficiency small.

b) The IM undergoes its first impact with the enclosure floor and by this mode of the particles motion, one impact per cycle or two impacts per cycle are possible.

In the present work, the impacts of the IM with the floor or ceiling of the enclosure are considered inelastic collisions. Consequently, the velocities of m and after impact depend on the mass ratio /m Mrµ = and the coefficient of restitution R. This coefficient, which is defined as the ratio of the differences in velocities before and after the collision, is given by




- 2- -- 2

- p

p

x xR

x x

+ + =

with 0 1R< < (15)

Where, - , x xp p+

represent the velocities of m respectively before and after impact, 2 2

- ,x x+ represent the velocities of Mr respectively before and after impact.

The Application of the conservation of linear momentum gives- -

2 2x x x xp pµ µ + ++ = + (16)

Using equations (15, 16), the velocities of m and Mr after impact are given respectively by

( ) ( )1 - -1 -21x R x R xp pµ

µ + = + + +

(17)

( ) ( )1 - -1- 12 21x R x R xpµ µ

µ+ = + + +

(18)

Since the computation of the velocities after the impact of particles and reduced mass are done, it will be possible to determine the instant and position of the second impact and the process continues.

Expression of the loss factor

The loss factor ψ is defined as the ratio of the stored elastic energy per maximum stored elastic energy during the cycle (Figure 3)

2 2

2

- 1x xi ixi

TT

ψ = +=∆

(19)

Figure 3: Velocity cycle of the treated system (clearance h = 4mm).

Where, ∆Τ is the kinetic energy dissipate per cycle, T is the maximum kinetic energy during the cycle and xi and 1xi + represent respectively the structure velocity for the (i)th and the (i+1)th cycle.

In order to implement our approach, the different stages

which have been described in the previous sections, the MATLAB software is used. The beam dimensions are length L = 250 mm, width l = 30 mm and thickness 2e mm= . The steel beam material is specified with a mass density 37840 /kg mρ = ; young modulus

112.1*E e Pa= and Poisson ratio ν = 0.33.

Figure 4: Flow chart describing the motion of the beam treated by PID.




The motions of the structure and the particles are calculated through a cycling process of small time step. The structure vibrates causing the impact of the IM, which will dissipate energy from the structure. In the end of each time step, the position and velocities of the IM and structure are updated. The cycle of the structure and IM motion calculation is repeated for the next time step. The iterative procedure for the numerical calculation of the motion for the primary structure and the IM is summarized in the flow chart

illustrated in Figure 4.

The efficiency of the model suggested to reduce the structure vibration is clearly shown in Figure 5. Without the particles damper (blue line in the figure), the value of loss factor is an order of 0.02 while for IM 7m g= and clearance 4h mm= (red line), the value of loss factor is an order of 0.45 meaning that the structure vibration had mitigated almost in the middle of the initial vibration (without damper).

Figure 5: The loss factor evolution of the system versus dimensionless acceleration.

Influences of PID Parameters The mathematical model was used to carry out parametric

study. Three parameters are considered to evaluate the effects of PID on the system response. The studied parameters are the value of IM m , clearance h in the enclosure and the coefficient of restitution R .

Influence of the impact mass amount mThe study of IM effect on the loss factor is investigated. Three

masses are considered 3.5, 7 and 10g. These values are chosen

in such a way that they do not bring too much variation on the natural frequency of the beam, on the one hand, and in order to validate the model through comparisons of the results of the simulation with the experimental results obtained by M et al. [4].

The variation of the loss factor for the three considered cases shows that the maximum of the damping contribution increases with the value of IM (Figure 6). These results are explained by the mechanism of dissipation of the damper. Indeed, the dissipated kinetic energy is higher if the mass of impact is greater; an increase in mass corresponds to a proportional increase in kinetic energy.

Figure 6: Influence of the impact mass IM on loss factor evolution.

Influence of the coefficient of restitution RDuring a collision, some of the kinetic energy of the IM is

converted into dissipated energy as heat. The variation between

the kinetic energy before and after the collision, perfectly elastic or inelastic, determines the coefficient of restitution.




In order to simulate and analyze the collisions that occur between the IM and the casing walls, three coefficients of restitution are considered 0.1, 0.3 and 0.7. The IM and the clearance

are kept constant: 37*10m kg−= and 5h mm= . Figure 7 shows the variation of the loss factor for the three considered cases.

Figure 7: Influence of the restitution coefficient R on loss factor evolution.

It is noticed that the simulated loss factor varies from 0.34, corresponding to R = 0.1, to 0.45 for R = 0.7, which summarizes that the maximum value of loss factor increases with the elasticity of the collision: the shock is elastic (R tends to 1).

Influence of the enclosure clearance hTo illustrate the effect of clearance on the evolution of loss

factor, some simulations were performed for the clamped-free beam with the enclosure containing the lead particles. The IM

was kept at 0.7g and the coefficient of restitution was kept 0.7 for the different value of force excitation amplitude. Eight values of clearance were considered from 1 to 8mm. Figure 8 shows the evolution of the loss factor with the respect to the dimensionless acceleration for the eight cases of the considered clearance. The analysis of the computed results allows us to note that compared to the system without particles a high value of the loss factor is attained. Its value varies from 0.33 corresponding to a clearance of 1mm to 0.48 for the one of 8mm.

Figure 8: Influence of the clearance h on loss factor evolution.

The loss factor presents a maximum for different values of the dimensionless acceleration. Moreover, for the two first cases, a peak corresponding to the acceleration denoted by Ac in the figure is observed, whereas for a clearance of 7mm, the curve form of the loss factor tends to be flattened from Ac and constitutes an interval in which the maximum value of the loss factor is constant. This interval is denoted by Z on the figure.

Actually, for a fixed clearance h and for the few first cycles, the value of acceleration is sufficiently large such that, upon leaving the floor, the impact of the particles occurs with the ceiling of

the enclosure; within this arrangement two impacts of IM per cycle take place. Similarly, if the value of acceleration is small, the particles don’t reach the ceiling, the impact occurs with the floor which favours only one impact per cycle. The value of acceleration which separates the two regimes’ impacts corresponds to Ac.

In order to validate the developed model, a comparison with experimental results of our previous work was conducted M et al. [4].

Table 1 shows a comparison of the maximum loss factor values and the non-dimensional acceleration values (for which the




maximum loss factor value is reached Ac) obtained numerically with those of the experiment.

Table 1: Comparison between numerical and experimental results for different values of clearance.

Mask 1 Mask 2

1 2

2 1

2 2

Parent 1 Parent 2

0.008 0.008

0.01 0.007

0.2 0.7

Chromosome h m R

Child 1 0.008 0.007 0.7

Child 2 0.008 0.01 0.007

Clearance (mm)

Simulation Experimental

Maximal value of loss factor Ac Maximal value of

loss factor Ac

1 0.33 0.8 0.18 1.6

2 0.4 1.2 0.25 2

3 0.43 1.7 0.32 2.2

4 0.45 2 0.35 2.6

5 0.46 2.1 0.38 2.8

6 0.47 2.4 0.4 3

7 0.47 2.7 0.4 3.2

8 0.48 2.9 0.45 3.2

For the eight cases of the considered clearance, the experimental results show a slight shift of cA and the value of maximum loss factor with those obtained numerically. This shift value is more remarkable in the low values of clearance.

In the case of the system with the particles, the offset is more visible and shows the effect of the impacts and the friction between the particles that have been neglected in the modelling as well as the air-mass impact friction that appears only in the experiment. These differences arise from the hypotheses considered in the modelling of the system.

The model of this research work is, thus, able to predict the dissipation of kinetic energy due to impacts of the IM with the walls of the enclosure, taking into account the effect of variations in the design parameters of PID.

Application of the GA in PID optimization

The stages of the GAIn general, genetic algorithms are the optimization techniques

used for the resolution of nonlinear systems. This method starts with the evolutionary principle of Darwin’s natural selection, and

as its name indicates, it simulates the reproduction of living beings in order to find a solution to a given problem. In this section, a description of the GA formulation in the case of PID optimization is given. The different stages of GA can be structured in the flow chart represented in Figure 9.

Initial population and chromosome representation: The algorithm starts with a set of solutions called the initial population. It consists of 10 chromosomes which represent each solution in the initial population generated randomly. Every chromosome consists of 3 parameters (m, h and R) then all the stages of the GA evoked in Figure 9 are automatically started. The number of chromosomes of the initial population improves the quality of the solution. For that reason, the size of the initial population was chosen 10. Table 2 shows examples of the chromosomes of the initial population.

Figure 9: Flow chart of the proposed Genetic Algorithm.

Table 2: Computation of the loss factor with different chromosomes.

Chromosome h m R Loss Factor

1 0.004 0.007 0.4 0.3794

2 0.008 0.007 0.7 0.4843

3 0.007 0.007 0.6 0.4719

4 0.008 0.01 0.2 0.4958

5 0.006 0.0035 0.4 0.2684

6 0.001 0.004 0.3 0.0901

7 0.004 0.0035 0.1 0.1439

8 0.002 0.008 0.5 0.1017

9 0.005 0.007 0.6 0.4375

10 0.002 0.005 0.6 0.0994




Evaluation of chromosomes: The second step of GA is the evaluation of the chromosome. The solutions from one population are taken and used to generate a new population using an Objective Function (OF). The hope is that the new population will expose a better performance than the previous one. In this case, the perfect

chromosome would correspond to find the best combination of parameters h, m and R allowing to reach a maximal value of the loss factor. Table 3 shows the result of evaluation of the three considered chromosomes.

Table 3: Probability of selection of chromosomes considered.

Chromosome h m R Objective Function Probability in %

1 0.004 0.007 0.4 0.3794 12.76

2 0.008 0.007 0.7 0.4843 16.29

3 0.007 0.007 0.6 0.4719 15.87

4 0.008 0.01 0.2 0.4958 16.68

5 0.006 0.0035 0.4 0.2684 9.02

6 0.001 0.004 0.3 0.0901 3.03

7 0.004 0.0035 0.1 0.1439 4.84

8 0.002 0.008 0.5 0.1017 3.42

9 0.005 0.007 0.6 0.4375 14.71

10 0.002 0.005 0.6 0.0994 3.34

Based on the experimental studies of DIP realized by M et al [4], the maximal value distinguished from the loss factor is an order of 0.5. Therefore, the value of the OF would belong to this interval:

0 0.5OF≤ ≤ (20)

Selection of individual solution: The selection consists in choosing individuals that can survive and create a new optimized generation. The methods of selection can be distinguished in two categories:

a) The selection is made according to a probability assigned to every individual as in a method called Stochastic Universal Sampling (SUS).

b) The methods which require only the value of capacity of every individual: (the local selection, the selection of truncating and the selection by tournament).

Within the framework of our study, the roulette wheel selection

genetic operator was chosen. This method consists in associating a probability of selection Pi with each individual chromosome in proportion to its performance [12]. It is defined by

1

p

ii M

ii

OFPOF

=

=

∑ (21)

Where iOF is the fitness of individual i and pM represents the size of the population.

This method could be imagined to be similar to a Roulette wheel in a casino. Usually, a proportion of the wheel is assigned to each of the possible selections based on their OF value. This could be achieved by dividing the fitness of a selection by the total fitness of all the selections, there by normalizing them to 1. Then, a random selection is made similar to how the roulette wheel is rotated. The best individuals have more chances to be chosen. Figure 10 represents the ten chromosomes and their distributions according to their performances. For example, the individual four has 17% of chance to be selected.

Figure 10: Probability of chromosomes selection.

Crossover genetic operator: Once the most efficient chromosomes are selected, the crossover operation can begin. In this work, the precedence preservative crossover (PPX) method

was chosen. This technique starts with randomly generated masks with the same length as the parents. Those masks, which represent progenies, define the order in which operations are successively


https://en.wikipedia.org/wiki/Probability



established by parent 1 and parent 2. This procedure is repeated until some predetermined success condition is satisfied. After a removal operation is chosen, corresponding genes will be deleted from both parents, and then, this operation of selection will be awarded to the descendant. This stage is repeated until the values of the parameters h , m and R of both parents are browsed and emptied and the offspring will be created.

After selection, the best chromosomes and the crossing are capable to occur and give descendants according to probability cP (in general it is between 0.6 and 0.9). First, a random value A of

cP such as 0 1A< < is, then, generated:

a) If cP A> , the phenomenon of production will take place,

b) If not, we keep the same characteristics of the parents to the children.

For example: if the value of A generated randomly such as A= 0.23, both masks generated have the same size as the parents.

And both parents selected from the crossover genetic operator are:

Mask 1 is dedicated to the creation of the first child and mask 2 to the second child. The child 1, corresponding to mask 1, is going to inherit from the parent 1 his first characteristic. Then, it is a question of eliminating this value of parent 2 as well as parent 1. Afterwards, the second value of mask 1, which is 2, means that the child 1 does not inherit of parent 1 but from parent 2. Then, parent 2 is going to pass its parameter which is 0.007 to child 1.This method is repeated as long as the random generation of A

is lower than cP , until the complete creation of the children.

The second stage of the crossover genetic operator is repeated until we obtain a new population of the same size as the initial population.

Mutation genetic operator: After the crossover operation, the population is ready to be mutating with a given probability mP which is generally between 0.01 and 0.1. The mutation operator plays the role of a disruptive element; it introduces ‘‘noise’’ in the population following a probability mP to introduce and guarantee the diversity within the population. There are several methods of the mutation genetic operator. In this work, the mutation-exchange or swap mutation is chosen with mP = 0.01. In this method, the probability of mutation mP is generally between 0.01 and 0.1. If mP A> ( A represents a randomly generated number), then the chromosome is ready to undergo a mutation. Otherwise, it keeps its properties and does not undergo change.

Let us take here, just for information purposes, an ascending chromosome of both parents after the crossover genetic operator. We generate a random number A = 0.001, it is clear that mP A> , by applying the method of mutation, both loci to exchange posts are 2 and 3. Thus, we obtain at the end;

Generation of optimal solution: This procedure, previously detailed, is repeated until some predetermined success condition is satisfied. The considered condition in this study was to find the best combination of parameters h , m and R allowing to get a maximal value of loss factor. The optimization results are:

Figure 11: Dynamic responses of the structure treated with optimal PID. (a) Loss factor, (b) Displacement, (c) Velocity, (d) Acceleration.




Figure 11 represents the numerical simulation of the optimal results, loss factor, displacement, velocity and acceleration of the system under free vibration, where the crossover probability cP is equal to 0.8 and the mutation probability

mP is equal to 0.01.

It should be noted that the quantity of energy dissipated during a cycle depends inevitably on the energy supplied by IM before the impact with the floor and the ceiling of the enclosure. Indeed, the more the value of the energy carried out by the mass of impact is important, the more the reduction in the velocity of the structure is important. Figure 11(c) shows clearly the reduction in the velocity of the structure of an initial equal value in 7.5m/s until affecting the value 0.3m/s during two cycles and it is due to the big value of the energy of the IM which takes off, during the first cycle, from the floor of the enclosure and comes to strike its ceiling. The shock is, then, more marked and translates an important quantity

of dissipated energy. The maximal value of loss factor, which is 0.48, found by the technique of GA optimization (Figure 11).

Results discussionsTo estimate the performance and the robustness of the GA

optimization tools, two criteria of evaluation are considered: the stability of the algorithm and the convergence.

Stability of the algorithm: The stability is a very important element to estimate the proper functioning of GA by keeping the same entered parameters: size of population, the number of generation and the probability of crossover and mutation. Several executions are made for the same probability Pm and Pc to show the stability of the program. Figure 12 shows the evolution of the objective function for two executions with the same introduced parameters.

Figure 12: Stability of the GA for the same parameters introduced.

For the same number of generation N =50, the size of the population 10pM = , mP and cP , the speeds of curves are almost the same. Indeed, the structure of the obtained chromosome does not change radically. A slight modification between chromosomes appears on the speed of the objective function, and it returns to the random aspect of the GA. We can conclude that the algorithm is stable. The obtained solution does not vary outstandingly according to the executions. Besides, the maximal value affected by the objective function is superior to those of the chromosomes of the initial population calculated initially. It justifies the contribution of this method in the improvement of the quality of the population from one generation to another.

Convergence of the algorithmFrom a generation to another one, the objective function of the best chromosome and the whole population is going to evolve and aim towards the optimum. The convergence is the progressive increase towards this optimum.

The implementation of a GA asks for the control of certain parameters. This control has an influence on the convergence of the GA and the results to be obtained. However, there is no rule determined well to control the parameters of GA. They are experimentally chosen. Among them:

a) Probability of crossing: the bigger it is, the more important recombination of the individuals is and the convergence to the optimum becomes effortless.

b) Probability of mutation: the smaller it is, the lower disturbance of the algorithm evolution is.

c) Size of the population: the bigger it is, the slower convergence towards the optimal regions will be.

To see the effect of cP and mP on the quality of the solution, a parametric study has been realized. Figure 13 shows the objective function evolution for different values of probability of mutation and crossing. The size of the population is maintained constant throughout this study. According to this figure, we notice that for:

a) 0.6cP = and 0.1mP = : the convergence towards the final value is made after 45 generations,

b) 0.8cP = and 0.1mP = : The convergence is made just in 20 generations,

c) 0.8cP = and 0.01mP = : the convergence is made in 5 generations.




Figure 13: Convergence of the GA for different values of cP and mP .

It is noticed, that when the crossing probability cP increases, the objective function reaches a maximal value in a staircase form. This result, justifies that the reproduction of the population is capable to gives successful descendants. However, by decreasing the value of the probability of mutation mP , the algorithm converged very quickly on a solution which approaches well the optimal solution.

We can conclude that these parameters have no effect on the quality of chromosomes but they have an effect on the time of the calculation and the moment of convergence. The reserved solution has been chosen as the following probability: 0.8cP =and 0.01mP = .

ConclusionIn this paper, a model of a primary structure treated with PID

in a vertical vibrating system was developed. Then, the simulated responses were compared with experimental results investigated in a previous work. By invoking the concept of an equivalent single degree of the freedom system, an approximate analytical solution is provided to estimate, with reasonable accuracy, the response of the primary system under free vibration. A parametric study on the influence of some system parameters on its loss factor was conducted. The results of this study can be summarized as follows:

a) The maximum value of the loss factor of PID is amplified with the increase of the mass of particles. These results can be explained by the value of the dissipated kinetic energy which becomes higher when the impact of particles is larger.

b) The maximum value of the loss factor decreases with the elasticity of the collision. Indeed, if the shock is soft, the velocity after impact becomes smaller and the energy dissipated will be high.

In addition, it was revealed that for a fixed clearance, a maximum of loss factor occurs when particles strike the enclosure twice by cycle in opposite directions. The good agreement between the simulated and the experimental results, following a comparison, shows clearly the effectiveness of PID to reduce the vibratory energy due to the impacts of the particles with the stops of the enclosure.

In the second part, the optimization of PID design parameters is studied by GA. The stability and speed of the developed algorithm are studied. Several executions of the algorithm by varying different parameters have shown that the results found are stables. By this technique, the optimum values of the particles’ mass, the restitution coefficient and the clearance allowing the maximization of the loss factor of the treated system were established.

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