Study on Multi-objective Optimization and Multi-attribute Decision Making of Dynamic Vibration Absorber

LI Xue-bin

(Wuhan 2nd Ship Design and Research Institute,Wuhan 430064,China)

1 Introduction

The dynamic vibration absorbers(DVAs)are a kind of classic engineering device,consisting of a mass,a spring and perhaps a damper,which is attached to a vibrating main system so as to attenuate its undesirable forced vibratory response.Since Frahm’s invention in the beginning of 20 century[1],these devices have been extensively used for attenuating vibrations of different types of machines and structures in many engineering domains,such as ship,automobile,plane,railway and so on.

A comprehensive study on the theory and practice of DVAs is given in the book by Koronev and Reznikov[2].Some research developments on passive,semi-active and active DVAs were presented by Sun et al[3].Classically,the DVA parameters(inertia,stiffness and damping)are chosen(it is said that the DVA is tuned)so as to minimize vibrations with a single harmonic frequency.Thus,the devices are prone to lose efficiency(it is then said that they become mistuned)if the forcing frequency happens to change,even slightly.One possible strategy,for coping with this problem is to use adaptive DVAs,which have self-tuning capabilities[3].Another strategy,towards which much effort has been directed,is to search for a set of optimal parameters so as to guarantee that the vibration level is minimized over a frequency band as much as possible,taking also into account design constraints.

Several authors have investigated different strategies for optimization of DVA parameters by using either time domain or frequency domain-based performance indexes.Design curves were presented by Den Hartog[4]for the case of single-degree-of-freedom(sdof)undamped primary systems subjected to harmonic excitation.Warburton[5]presented the expressions for optimum absorber parameters of undamped sdof primary systems,considering harmonic and white noise random excitations.The effect of damping in the primary system on optimum absorber parameters is also investigated by the same author in reference[6].In reference[7],Wang and Cheng compared four optimization methods(the equal peak method,the minimal variance method,the energy method and the area ratio method),when applied to an sdof system with primary damping.Nishimura et al[8]addressed the optimization of a DVA for mdof systems subjected to random input with a dominant frequency,using an optimization method based on the optimal control theory.In reference[9],Kitis et al proposed an efficient optimal design algorithm for minimizing the vibratory response of mdof systems under sinusoidal loading over several excitation frequencies.The method explores the mass,stiffness and damping matrices provided by an analytical model and incorporates an effective time-saving reanalysis approach to compute the cost function and its derivatives.A modal theory for viscoelastic dynamic neutralizers was developed and associated to different optimization techniques for the optimal design of neutralizers over a frequency band[10-11].More recently,Genetic Algorithm(GA)is applied to optimization problem of sdof vibration isolator mount by Alkhtib et al[12],and ant colony optimization(ACO)method was used to study the vibrating blade DVA and multi-mode DVA[13].

In this paper,a multi-objective optimization problem of DVA parameter design is described.The main focus of this paper is to find the optimal parameter of DVA to simultaneously minimizing the maximum value of the primary system,maximizing the suppression band and relative vibration reduction efficiency.A hybrid approach for multi-objective optimization study of DVA is proposed in the present analysis.In the first stage,a Non-dominated Sorting Genetic Algorithm II(NSGA II)is employed to approximate the set of Pareto solution through an evolutionary optimization process.In the subsequent stage,a multi-attribute decision making(MADM)approach is adopted to rank these solutions from best to worst and to determine the best solution in a deterministic environment with a single decision maker.A DVA example is conducted to illustrate the analysis process in present study.Pareto frontiers are obtained and the ranking of Pareto solution is based on entropy weight and TOPSIS method.

2 System analysis

2.1 Structure model

As shown in Fig.1 the primary system consists of the mass M1,the spring K1amd the damper C1(a list of notation is given in the Appendix).The absorber mass M2is attached by means of the spring K2and damper C2.The harmonic exciting force f（t）=F0sin（ωt）is applied to the primary mass M1and the system is assumed to be in steady state vibration.

The equations of motion are

The normalized amplitude of the steady-state response of the primary mass is given as

where

If DVA is not mounted on primary system,the nondimensional amplitude of primary system can be given by

Two variables,ratio and Eff,are introduced to evaluate the effect of vibration absorption,

If it is assumed that the absorber effect is beneficial when the amplitude ratio is less 0.7,then the frequency bandwidth over which this occurs is called the‘suppression band’.The bandwidth problem was first identified by Roberson[14]who considered the impractical case of a linear plus cubic spring with no damping.A further discussion about bandwidth was given by Hunt and Nissen[15].It should be noted that,amplitude is unity in the definition of bandwidth in reference[5].Therefore,suppression band can be defined as.

where λuand λlare upper and lower frequency,respectively.

In this paper,three parameters related to vibration absorption,A（λ）,Band and Effare taken into count.A optimization model with there 3 objectives functions can be written as

2.2 Optimization model

In many realistic problems,several goals must be simultaneously satisfied to obtain an optimal solution.However,sometimes these multiple objectives,which must be simultaneously satisfied,are conflicting.The multiobjective optimization method is a common approach to solve this type of problem.The general multi-objective problem(MOP)is formulated as follows:

where solution x is a vector of discrete decision variables,Ω is the finite set of feasible solutions.The image of a solution x∈Ω is the point z=f（x）in the objective space.A point z dominates z’,for at least one j.A solution x dominates x’if the image of x dominates the image of x’.A solution x*∈Ω is a non-dominated(or efficient)solution if there is no x∈Ω such that z=f（x）dominates z*=f（x*）.The solutions that are non-dominated within the entire search space are denoted as Pareto-optimal solutions and constitute the Pareto-optimal set or Pareto-optimal frontier.

In this research study,to find the Pareto-optimal solutions,the fast and elitist NSGA-II approach proposed by Deb et al in 2002[16]is used.In this approach,to sort a population of size N according to the level of nondomination,each solution must be compared with every other solution in the population to determine if it is dominated.At this stage,all individuals in the first non-dominated front are found.In order to find individuals of the next front,the solutions of the first front are temporarily discounted,and the above procedure is performed again.The procedure is repeated to find subsequent fronts[16].For details about the NSGA-II approach,refer to Ref.[16].

In optimization studies that include multi-objective optimization problems,the main objective is to find the global Pareto-optimal solutions,representing the best possible objective values.In the present study,the central focus is to find the global Pareto-optimal solutions;The normalized amplitude of the steady-state response of the primary mass,bandwidth of system and vibration absorption are considered the objective functions to be simultaneously minimized.Furthermore,the mass,stiffness,and damping ratio of the base isolators are the decision variables to be evaluated through the multiobjective optimization process.

2.3 Decision model

The hybrid approach involving multiobjective optimization and MADM procedures is listed in Fig.2

Once solutions lying on the estimated Pareto-optimal set are found,it is usually required to choose one of them for implementation.Moreover,the choice of a solution over the other entails additional knowledge,e.g.experts’preferences.From a decision maker’s perspective,the choice of a solution from all Pareto optimal solutions is called a posteriori approach and it requires a higher-level decision-making approach,which is to determine the best solution amongst a finite set of Pareto optimal solutions with respect to all relevant attributes.Multiple attribute decision making(MADM)techniques are generally employed in posterior evaluation of Pareto optimal solutions to choose the best one amongst them.

In this study,higher-level decision-making model is considered as an MADM problem in which,having a finite set X of alternatives and a consistent family A of K attributes on X,one wishes to rank the alternatives of X from best to worst and determine a subset of alternatives considered to be the best with respect to A[17].

The classical MADM model is described as follows:

A large number of methods have been developed for solving multiple attribute or multiple criteria problems[18-19].In this paper,the concept of the approach used for solving the problem is based on the technique for order preference by similarity to ideal solution(TOPSIS)[18].This is because(a)the concept is rational and comprehensible,(b)the involved computation is simple,(c)the concept is capable of depicting the pursuit of the best performance of design solution for each evaluation criterion in a simple mathematical form,and(d)the concept allows objective weights to be incorporated into the comparison process.The concept of TOPSIS is that the most preferred alternative should not only have the shortest distance from the positive ideal solution,but also have the longest distance from the negative ideal solution.This concept has also been pointed out by Zeleny[20],who refered to the positive and negative ideal solutions as the ideal and anti-ideal solutions,respectively.

Almost all MADM methods require predetermined information on the relative importance of the attributes,which is usually given by a set of normalized weights.There are many techniques to elicit attribute weights,referring Sen and Yang’s review[21].Objective weights of criteria importance,measured by the average intrinsic information generated by a given set of alternatives through each criterion,reflect the nature of conflicting criteria and enable the incorporation of inter-dependent criteria[20].

Shannon’s entropy concept[22]is used in the present analysis.The entropy method is based on information theory,which assigns a small weight to an attribute if it has similar attribute values across alternatives,because such attribute does not help in differentiating alternatives.Entropy is a measure of uncertainty in the information formulated using probability theory.It indicates that a broad distribution represents more uncertainty than a sharply peaked one.‘If all available alternatives score about equally with respect to a given attribute,then such an attribute will be judged unimportant by most decision makers’(Ref.[22],p.187).In other words,such an attribute should be assigned a very small weight.

To determine objective weights by the entropy measure,the decision matrix in Eq.(10)needs to be normalized for each attributeas

The value of Pijis highly dependent uponAs a consequence,a normalized decision matrix representing the relative performance of the alternatives is obtained as

The amount of decision information contained in Eq.(4)and emitted from each attributecan thus be measured by the entropy value ejas

The degree of divergence(dj)of the average intrinsic information contained by each attributecan be calculated as

where djrepresents the inherent contrast intensity of the attribute Aj.The more divergent the performance ratingsfor the attribute Aj,the higher its corresponding dj,and the more important the attribute Ajfor the problem.This reflects that an attribute is less im portant for a specific problem if all alternatives have similar performance ratings for that attribute.If all the performance ratings against an attribute are the same,the attribute can be eliminated for the given situation on which a decision is to be based,because it transmits no information to the decision maker[20].

The objective weight for each attributeis thus given by

The weighted normalized value vijis calculated as

After determining performance ratings of the alternatives and objective weights of the attribute,the next step is to aggregate them to produce an overall performance index for each alternative.This aggregation process is based on the positive ideal solution(A+)and the negative ideal solution(A-),which are defined,respectively,by

The overall performance index of an alternative is determined by its distance to A+and A_.This distance is interrelated with the attribute weights[20]and should be incorporated in the distance measurement.This is because all alternatives are compared with A+and A_,rather than directly among themselves.In TOPSIS,the attribute weights mainly serve as a channel through which the attribute with different performances can be brought together.The decision matrix is weighted by multiplying each column of the matrix by its associated attribute weight.

Separation(distance)between alternatives can be measured by the n-dimensional Euclidean distance.The separation of each alternative from the ideal solution is given as

Similarly,the separation from the negative ideal solution is given as

The relative closeness to the ideal solution of alternative Xjwith respect to A+is defined as

Choose an alternative with maximum Cjor rank alternatives according to Cjin descending order.It is clear that an alternative Xjis closer to A+than to A_as Cjapproaches 1.

In order to facilitate the presentation of the case study in Section 3,the steps required by the proposed approach are given as follows:

Step 1 Establish multiple objective problem model with its parameters and constraints.

Step 2 Employ NSGA II method to generate the feasible design space in multi-objective optimization environment.

Step 3 Carry out design filtering and select Pareto optimal solutions according to multiple objective optimization definition.

Step 4 Identify the selection attributes with types(cost or benefit)of them and list all possible Pareto optimal solutions.

Step 5 Calculate the relative importance of attributes by using Eq.(15).

Step 6 Construct the normalized rating,and weighted normalized rating of the decision matrix.

Step 7 Calculate positive ideal solutions and negative ideal solutions.

Step 8 Order or rank the Pareto optimal solutions according to the overall ranking values and select the alternative with the maximum overall ranking value as the best solution.

3 Case study

A numerical optimization example is studied here.There are 4 decision variables in this model.The upper and lower bounds of decision variables are

Tab.1 Parameters for NSGA II in computation

Parameters used by NSGA II approach are listed in Tab.1.

At the end of this run,5 266 different designs were obtained in solution space with 4 842 of them being feasible designs,which comply with optimization constraints.Then,4 862 feasible designs were filtered in design space to obtain only designs that belong to a Pareto-optimal set.There are 870 solutions in this set.Solution space with three objectives and evolution of these objectives are shown in Fig.3.

The results of scatter plots of 3 objectives are shown in Figs.4～6.Pareto frontiers are given in these scatter plots.

Edwin Jaynes was interested in estimates of probability distributions which were unbiased in that they were as uniform(flat)as possible subject only to specified constraints.He proposed that maximizing the entropy of the distribution subject to these constraints provided just such an estimate,the maximization ensuring the flattest possible distribution[23].These flat distributions are said to be minimally discriminating;in Jaynes’case minimally discriminating between the relative likelihoods of different events or states.The same principle may be applied to the calculation of weight distributions and a review is provided by Jessop[24].This method is therefore proposed as a way of finding weights and so making selection decisions which may be said to be unbiased.In this example,there is no preference introduced.The size of decision matrix R is 870*3.The weights of three attributes are calculated by entropy method and shown in Fig.7.The weights of maxA（λ）,Band and ratio are 0.693,0.063 and 0.244,respectively.This characteristic is related to the mathematical model itself.

According to TOPSIS procedure,the first 5 alternatives with highest ranking are given in Tab.2.The corresponding design variables are also listed.The No.873 solution is selected are the‘best’one amongst all alternatives.It is labeled as point G in Figs.3-6.From Fig.4 we know that point G is located in Pareto frontier of Effand maxA（λ）scatter plot.

The response and vibration absorption effect of final decision(Design G)are shown in Fig.8.

Tab.2 Ranking results of alternatives(first 5 alternatives)

4 Conclusions

A hybrid approach for multi-objective optimization of DVA is obtained with NSGA II method and MADM techniques.From the numerical example we know that multi-objective optimization using the NSGA II approach is a powerful method to design the parameters of the DVAs.A higher level decision is always needed in practical multi-objective problems.The results of TOPSIS with entropy method in weights computing are reasonable and unbiased.

The approach introduced in the present analysis can make the isolation system more effective.The method can also be used in other kinds of DVAs parameter design.

[1]Frahm H.Device for damping vibrations of bodies[P].US.Patent 989,958,1911.

[2]Koronev B G,Reznikov L M.Dynamic vibration absorbers.Theory and Technical Applications[M].New York:Wiley and Sons,Ltd,1993.

[3]Sun J Q,Jolly M R,Norris M A.Passive,adaptive vibration absorbers-a survey[J].Transactions of the ASME,1995(17):234-242.

[4]Den Hartog J P.Mechanical Vibrations[M].New York:McGraw-Hill,1956.

[5]Warburton G B.Optimum absorber parameters for various combinations of response and excitation parameters[J].Earthquake Engineering and Structural Dynamics,1982(10):381-401.

[6]Warburton G B,Ayorinde E O.Optimum absorber parameters for simple systems[J].Earthquake Engineering and Structural Dynamics,1980(8):197-217.

[7]Wang Y Z,Cheng S H.The optimal design of dynamic absorber in the time domain and the frequency domain[J].Applied Acoustics,1989(28):67-78.

[8]Nishimura H,Yoshida K,Shimogo T.Optimal dynamic vibration absorber for multi-degree-of-freedom systems(Theoretical consideration in the case of random input)[J].JSME International Journal,1989(32):373-379.

[9]Kitis L,Wang B P,Pilkey W D.Vibration reduction over a frequency range[J].Journal of Sound and Vibration,1983(89):559-569.

[10]Bavastri C A,Espidndola J J,Teixeira P H.Hybrid algorithm to compute the optimal parameters of a system of viscoelastic vibration neutralizers in a frequency band[C]//Proceedings of the MOVIC’98.Zurich,Switzerland,1998.

[11]Espidndola J J,Bavastri C A.Reduction of vibrations in complex structures with viscoelastic neutralizers:A generalized approach and a physical realization[C]//1997 Proceedings of the ASME Design Engineering Technical Conferences.Sacramento,CA,U.S.A.,1997.

[12]Alkhatib R,Nakhaie Jazar G,Colnaraghi M F.Optimal design of passive linear suspension using genetic algorithm[J].Journal of Sound and Vibration,2004(275):665-691.

[13]Viana F A C,Kotinda G I,Rade D A,Steffen Jr V.Tuning dynamic vibration absorbers by using ant colony optimization[J].Computers and Structures,2007,doi:10.1016/j compstruc.,2007.05.009.

[14]Roberson R E.Synthesis of a non-linear dynamic vibration absorber[J].Journal of the Franklin Institute,1952:205-220.

[15]Hunt J B,Nissen J C.The broadband dynamic vibration absorber[J].Journal of Sound and Vibration,1982(83):573-578.

[16]Deb K,Agrawal S,Pratap A,Meyarivan T.A fast and elitist multi-objective genetic algorithm:NSGA-II[J].IEEE Transactions on Evolutionary Computation,2002(6):182-197.

[17]Vincke P.Multicriteria Decision Aid[M].Cambridge,UK:Wiley,1992.

[18]Hwang C L,Yoon K.Multiple Attribute Decision Making-Methods and Applications:A state-of-art Survey[M].New York:Springer-Verlag,1981.

[19]Olson D L.Decision Aids for Selection Problems[M].New York:Springer,1996.

[20]Zeleny M.Multiple Criteria Decision Making[M].New York:McGraw-Hill,1982.

[21]Sen P,Yang J B.Multiple Criteria Decision Support in Engineering Design[M].London:Springer,1998.

[22]Shannon C E,Weaver W.The Mathematical Theory of Communication[M].Urbana:The University of Illinois Press,1947.

[23]Jaynes E T.Information theory and statistical mechanics[J].Physics Review,1957,106:620-630;108:171-190.

[24]Jessop A.Entropy in multiattribute problems[J].Journal of Multi-Criteria Decision Analysis,1999(8):61-70.

Appendix:Notation

A:displacement amplitude ratio with DVA;

A:attribute of alternative;

A’:displacement amplitude ratio without DVA;

Band:suppression band;

c1c2:viscous damping constants;

Eff:relative vibration reduction efficiency;

ratio:ratio of A/A’;

F0:amplitude of excitation force;

R:decision making matrix;

f（t）:harmonic excitation force;

x1,x2:primary and absorber mass displacement;

X1,X2:amplitude of primary and absorber mass displacement;

xst:static deflection of primary mass;

α:frequency ratio,ω2/ω1;

μ:mass ratio,m2/m1;