- Email: [email protected]
Passive vibration control via unusual geometries: the application of genetic algorithm optimization to structural design
Journal of Sound and Vibration (1995) 185(3), 441–453
PASSIVE VIBRATION CONTROL VIA UNUSUAL GEOMETRIES: THE APPLICATION OF GENETIC ALGORITHM OPTIMIZATION TO STRUCTURAL DESIGN A. J. K Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, England (Received 28 April 1994, and in final form 8 August 1994) In the majority of aerospace structures, vibration transmission problems are dealt with by the application of heavy, viscoelastic damping materials. More recently, interest has focussed on using active vibration control methods to reduce noise transmission. This paper examines a third, and potentially much cheaper method: that of redesigning the load bearing structure so that it has intrinsic, passive noise filtration characteristics. It shows that very significant, broadband noise isolation characteristics (of around 60 dB over a 100 Hz band) can be achieved without compromising other aspects of the design. Here, the genetic algorithm (GA), which is one of a number of recently developed evolutionary computing methods, is employed to produce the desired designs. The problem is set up as one in multi-dimensional optimization where the geometric parameters of the design are the free variables and the band averaged noise transmission the objective function. The problem is then constrained by the need to maintain structural integrity. Set out in this way, even a simple structural problem has many tens of variables; a real structure would have many hundreds. Consequently, the optimization domain is very time consuming for traditional methods to deal with. This is where modern evolutionary techniques become so useful: their convergence rates are typically less rapidly worsened by increases in the number of variables than those of more traditional methods. Even so, they must be used with some care to gain the best results. 7 1995 Academic Press Limited
1. INTRODUCTION
Structure-borne noise and vibration control is an aspect of design where there are few reliable design techniques available and one that is relevant to almost all vehicles and many other lightweight engineering structures. Moreover, structures such as cars, aeroplanes, ships, suspension bridges, etc., all suffer from exposure to noise and vibration sources. These sources often excite unwanted structural resonances which can cause damage or the transport of vibrational energy to distant parts of the structure where they cannot be tolerated. For example, the engines in cars, ships and aircraft always vibrate to some degree and, despite isolation treatments, excite broadband motions of their mounting points. Since most structures are quite stiff and have inherently low damping characteristics the subsequent resonant motions may well be relatively large. This vibrational energy can then flow through the structure and cause significant motions in, say, cabin panels. These in turn radiate noise into the working environments of crew or passenger spaces. Vibrations of this kind may also effect sensitive equipment or lead to fatigue failures in light-weight alloy structures. The most common treatment for such problems is to coat the structural elements with heavy viscoelastic damping materials with consequent weight and cost penalties. 441 0022–460X/95/330441 + 13 $12.00/0
7 1995 Academic Press Limited
442
. .
Moreover, the effectiveness of such treatments diminishes with the vibration levels which makes continuously improving noise and vibration targets difficult to meet. Clearly, if the vibrational energy could be contained near to the points of excitation there would be a reduced need for damping treatments and, additionally, they could be concentrated in regions where they were most effective. This is precisely the aim of the vibration isolators used between most pieces of equipment and their supporting structure. However, it is difficult and expensive to isolate large pieces of structure in this way, although this is done in modern ship design where entire deckhouses are sometimes placed on resilient mountings. Of course, some pieces of structure, such as the wings on an aircraft, cannot be mounted in this way. The consequence of this problem is the desire for some kind of widely applicable, generic structural filter design capability that could be used to build desirable characteristics into a structure, retaining their ability to carry static loads while blocking higher frequency motions. Recently, much effort has been expended on so-called ‘‘active’’ vibration control measures for use in this area. These employ ‘‘anti-noise’’ to cancel out unwanted vibrations and so block noise propagation. However, they are inevitably expensive to install and maintain, and passive solutions would be preferable if they could be found. One passive approach that has been extensively studied, mainly for use in aircraft fuselages, makes use of the vibration characteristics exhibited by structures with geometric periodicity. Such structures possess so-called ‘‘stop’’ bands which are regions of the frequency domain where natural frequencies do not occur and where travelling waves are very rapidly attenuated by constructive reflections [1, 2]. These characteristics become increasingly complex as the nature of the periodicity becomes more complex and also as the structural type changes from one- through two- to three-dimensional. It has in practice, so far proved impossible to design real structures with the desired characteristics by using this behaviour. Moreover, such success as there has been, has been found to be extremely sensitive to the accuracy of manufacture and subsequent modifications, the desired stop bands being easily disrupted or shifted by the addition of extra structural features, changes in payload, etc. Clearly, there is no point in adopting such methods if they work only for a single cargo configuration or if they preclude even minor modifications through the life of the structure concerned. In short, to be of practical use, noise isolation design techniques must be widely applicable, yielding robust designs, even if this is achieved at the expense of ultimate theoretical performance. As part of a recent programme of work concerned with improving noise prediction methods, statistical energy analysis (S.E.A.) was applied to simplified aerospace structural models. S.E.A. is a method dating back to the early 1960s [3] which is the subject of much current research [4]. It is used to predict the flow of energy around vibrating structures and is seen by many as providing the best long term means for analyzing their noise performance. During the work being undertaken the design of anomalous structural configurations became of interest [5]. The primary reason for this was the desire to quantify the tails in the statistical descriptions of the responses of structures with random physical properties. It became clear that the most direct way of finding configurations where energy flowed well or badly was to employ optimization techniques to maximize or minimize the flows. This proved difficult when using classical methods because of the many variables involved and a G.A. approach was adopted instead. The flow of vibrational energy around a complex structure is dominated by the many resonances exhibited by such structures and also the large number of physical parameters needed to specify typical structural designs (in the previous case 40, many hundreds in a full car, ship or aircraft structural model), i.e., the G.A. is well suited to such problems. This approach, combined with S.E.A. methods, showed that dramatic reductions in energy flows between coupled structures could be achieved, and moreover such reductions could be made
443
to hold over significant frequency bandwidths while at the same time remaining insensitive to minor structural modifications. When the geometries that gave rise to these reductions were studied it was seen that they showd characteristic periodic fluctuations in properties (i.e., alternating light and heavy sections), although these were not so precise as those that could be adopted when using classical periodic structure theories. In short, although only at a very preliminary stage and applied to a very idealized model, this finding suggested that the possibility of designing the highly desirable filtering characteristics sought during the earlier programme of work by using periodic structures need not be abandoned. Instead, designs produced and analyzed in the normal way might be modified by using G.A. optimizers seeking to retain the desired structural characteristics while at the same time incorporating robust noise filtration features. Moreover, this approach should be applicable to a wide range of structural noise problems. This paper is a sequel to this earlier work and is concerned with the flow of energy around a network of 40 coupled beams, of the sort that may be found in satellite booms (and girder bridges, tower cranes, etc.). Here the vibration analysis is carried out by using matrix receptance methods based on the Green functions of the individual beam elements, which are set up to calculate the forces and velocities at the joints in the structures [6, 7]. This approach enables the energy flows around the structure to be calculated directly and efficiently and this, in turn, allows for the solution of the thousands of configurations that must be considered when dealing with designs consisting of many elements. 2. THE STRUCTURE
Figure 1 indicates the structure to be optimized. This consists of 40 Euler–Bernoulli beams all having the same properties per unit length. Here, EA is taken as 69·87 MN, EI as 12·86 kNm2 and the mass per unit length as 2·74 kg/m. the beams are all either 1 m or 1·414 m long and the joints at (0, 0) and (0, 1) are taken to be pinned to ground; all other joints are free to move. The structure has been chosen to be two-dimensional for simplicity. It is excited by a point transverse force halfway between (0, 0) and (1, 0) and, for this study, the aim is set as the minimization of the vibrational energy level in the right-hand end vertical beam (which in practice might carry an instrumentation package or similar). The damping of the structure is fixed so that the normal modes of the uncoupled beam elements all have a constant bandwidth of 20 s−1 . 3. THEORY
The forced vibrational performance of such a structure may be readily computed by using finite element methods but this is rather cumbersome when evaluating the response of a
Figure 1. Initial structural design.
444
. .
Figure 2. Initial frequency response (band average = 0·33 × 10−6 ).
structure to broadband forcing, especially when very many different configurations will need to be considered during optimization. Instead, the well-known characteristics of Euler–Bernoulli beams combined with receptance methods are used. Such methods have been studied for many years [8] and, as has already been mentioned, may be set up to solve directly for the energetic quantities of interest [6, 7]. Using this approach leads to the problem being defined in terms of the unknown displacements and forces at the joints. The displacements at the ends of each member meeting at a joint are equated and the joint coupling forces and moments summed to zero. The effects of external driving and the joint forces on any beam are specified in terms of the Green functions of the elements which, in turn, are based on the modes of the elements in isolation. This approach requires some care to ensure convergence of the Green functions but allows the selection of various models for the elements to be made with great ease. The references show that energy flows calculated in this way agree very well with those based on finite elements. Figure 2 shows how the vibrational energy level of the end beam varies with frequency for this base design. As can be seen from the figure, the energy level exhibits the many peaks and troughs characteristic of a complex, lightly damped structure. Moreover, there is a large response in the region 100–200 Hz. The calculation of a single frequency point in this plot requires the solution of some 260 complex simultaneous equations and this takes around 20 s on a Sun Sparc 10 workstation; i.e., the calculation of a frequency band averaged takes
Figure 3. Optimized design with limits of 25% on all 18 joints, 1000 evaluations over five generations.
445
Figure 4. Frequency response of optimized design with limits of 25% on all 18 joints, 1000 evaluations over five generations (band average = 0·21 × 10−7; initial frequency response shown dotted).
around seven minutes when using the 21 point rule adopted here. Even so, this is considerably faster than can be achieved by using a finite element approach with the same level of accuracy. 4. THE OPTIMIZATION PROBLEM
The aim of this study is set as the reduction of the frequency averaged response of the end beam in the range 150–250 Hz by the use of optimization: i.e., to trim down the upper half of the major response peak. In practice, such a requirement would reflect known sensitivities of the payload or known excitation frequencies. To meet design requirements the optimization is constrained to keep the end beam unchanged in length and position with respect to the fixed points. Further, to meet structural requirements, all of the joints within the structure must be kept within fixed distances from their original positions. This ensures that no beam is too long or short and also restricts the overall envelope of the structure. The free variables in the problem are thus set as the x and y co-ordinates of the 18 mid-span joints: i.e., 36 variables in all.
Figure 5. Optimized design with limits of 225% on all six mid-span joints, 1000 evaluations over five generations.
446
. .
Figure 6. Frequency response of optimized design with limits of 225% on six mid-span joints, 1000 evaluations over five generations (band average = 0·31 × 10−7; initial frequency response shown dotted).
5. THE OPTIMIZER
Optimization problems of the sort posed here are characterized by having many variables, highly non-linear relationships between the variables and the objective function, and an objective function that has many peaks and troughs. Moreover, as has already been noted, any one configuration is time consuming to evaluate. In short they are difficult to deal with. The search for methods that can cope with such problems has led to the subject of evolutionary computation. Techniques in this field are characterized by a stochastic approach to the search for improved solutions, guided by some kind of evolutionary control strategy. There are three main methods in use today: (1) simulated annealing [9], where the control strategy is based on an understanding of the kinetics of solidifying crystals; (2) genetic algorithms [10], where the methods of Darwinian evolution are applied to the selection of ‘‘fitter’’ designs; (3) evolutionary programming [11], which is a more heuristic approach to the problem but which has an increasing number of adherents. The author has applied all of these methods to structural problems and found that, for the current case, the G.A. works best. The G.A. used here is fairly typical of those discussed in the well-known book by Goldberg [10] but encompasses a number of new ideas that are particularly suited to engineering design problems [12, 13]. Such methods work by maintaining a pool or population of competing designs which are combined to find improved solutions. In their basic form, each member
Figure 7. Optimized design with limits of 225% on all 18 joints, 2000 evaluations over 10 generations.
447
Figure 8. Frequency response of optimized design with limits of 225% on all 18 joints, 2000 evaluations over 10 generations (band average = 0·32 × 10−9; initial frequency response shown dotted).
of the population is represented by a binary string that encodes the variables characterizing the design. The search progresses by manipulating the strings in the pool to provide new generations of designs, hopefully with on average better properties than their predecessors. The processes that are used to seek these improved designs are set up to mimic those of natural selection: hence the method’s name. The most commonly used operations are currently: (1) selection according to fitness, i.e., the most promising designs are given a bigger share of the next generation; (2) crossover, where portions of two good designs, chosen at random, are used to form a new design, i.e., two parents ‘‘breed’’ an ‘‘offspring’’; (3) inversion, whereby the genetic encoding of a design is modified so that subsequent crossover operations affect different aspects of the design; (4) mutation, where small but random changes are arbitrarily introduced into a design. In addition, the number of generations and their sizes must be chosen, as must a method for dealing with constraints (usually by application of a penalty function). The algorithm used here works with 16 bit binary encoding. It uses an elitist survival strategy which ensures that the best of each generation always enters the next generation and has optional niche forming to prevent a few moderately successful designs dominating and so preventing wide ranging searches. Two penalty functions are available to deal with constraints. The main parameters used to control the method may be summarized as follows: Ngen , the number of generations allowed (default 10); Npop , the population size or number of trials used per generation which is therefore inversely related to the number of generations given a fixed number of trials in total (default 100); P[best], the proportion of the population that survive to the next generation (default 0·8); P[cross], the proportion of the surviving population that are allowed to breed (default 0·8); P[invert], the proportion of this population that have their genetic material re-ordered (default 0·5); P[mutation], the proportion of the new generation’s genetic material that is randomly changed (default 0·005); a proportionality flag, which selects whether the new generation is biased in favour
448
. .
Figure 9. Convergence of the first three optimization runs as a function of generation number. q, Initial design; · · · · · · · · ·, 25% on all joints, 1000 evaluations over five generations; – – – –, 225% on six mid-span joints, 1000 evaluations over five generations; – · – · –, 225% on all joints, 2000 evaluations over 10 generations.
of the most successful members of the previous generation or alternatively if all P[best] survivors are propagated equally (default TRUE); the penalty function choice. When using the G.A. to explore large design spaces with many variables, it has also been found that the method must be prevented from being dominated by a few moderately good designs which prevent further innovation. A number of methods have been proposed to deal with this problem; that used here is based on MacQueen’s Adaptive KMEAN algorithm [14] which has recently been applied with some success to multi-peak problems [15]. This algorithm subdivides the population into clusters that have similar properties. The members of each cluster are then penalized according to how many members the cluster has and how far it lies from the cluster centre. It also, optionally, restricts the crossover process that forms the heart of the G.A., so that large successful clusters mix solely with themselves. This aids convergence of the method, since radical new ideas are prevented from contaminating such sub-pools. The version of the algorithm used here is controlled by the following: Dmin , minimum non-dimensional Euclidean distance between cluster centres, with clusters closer than this being collapsed (default 0·05); Dmax , the maximum non-dimensional Euclidean radius of a cluster, beyond which clusters subdivide (default 0·1); Nclust , the initial number of clusters into which a generation is divided (default 25); Nbreed , the minimum number of members in a cluster before exclusive inbreeding within the cluster takes place (default 5); a, the penalizing index for cluster members, which determines how severely members sharing an overcrowded niche will suffer, with large numbers giving greater penalty (default 0·5): i.e., the objective functions of members of a cluster of m solutions are scaled by
Figure 10. Optimized design with limits of 225% on all 18 joints, 4400 evaluations over 22 generations.
449
Figure 11. Frequency response of optimized design with limits of 225% on all 18 joints, 4400 evaluations over 22 generations (band average = 0·13 × 10−9; initial frequency response shown dotted).
m min (a,1)[1 − (E/Dmax )a] + (E/Dmax )a, where E is the Euclidean distance of the member from its cluster centre (which is always less than Dmax ; moreover, when E = Dmax no penalty is applied). In addition, the implementation of the G.A. used here allows the solution of individual members of the population to be run in parallel if a multiple processor computer is available. 6. RESULTS
To begin the optimization process relatively tight limits of 25% were placed on the joint positions and a modest number of 1000 trials used, spread over five generations (giving 200 members per generation so as to ensure reasonable coverage over the rather large domain being investigated here, at the expense of final convergence). This run gave rise to the configuration shown in Figure 3 and the response illustrated in Figure 4. With this configuration, the total energy flow in the 100 Hz band investigated has been reduced from 0·33 × 10−6 to 0·21 × 10−7 (for unit forcing), i.e., a reduction of 23 dB, indicating how purely periodic structures can give rise to very significant noise transmission problems. Next, the limits on the joint positions were relaxed to 225% but only six of the mid-span joints were allowed to move, again using 1000 trials and five generations. This leads to the configuration of Figure 5 and the response shown in Figure 6. This gives slightly less isolation with the
Figure 12. Optimized design with limits of 225% on all 18 joints, 4500 evaluations over 15 generations.
450
. .
Figure 13. Frequency response of optimized design with limits of 225% on all 18 joints, 4500 evaluations over 15 generations (band average = 0·13 × 10−9; initial frequency response shown dotted).
total energy flow now being 0·31 × 10−7 , i.e., a reduction of 21 dB over the base design. As can be seen from Figure 5, a kind of notch has been driven into the structure and this clearly causes reflections of energy-carrying waves in the desired frequency ranges. After having gained some initial impressions, the G.A. was next applied with a 225% constraint on the joint positions applied to the whole structure, but now using 2000 trials over 10 generations, i.e., with an increased number of generations so as to improve convergence. This yields the geometry shown in Figure 7 together with the response shown in Figure 8. The reduction in the frequency averaged response is now from 0·33 × 10−6 to 0·32 × 10−9 , i.e., a change of 60 dB. Clearly, this has been achieved by rather radical geometrical distortions and such severe modifications to the structure might not be tolerable in practice.
Figure 14. Convergence of all five optimization runs as a function of generation number. q, Initial design; · · · · · ·, 25% on all joints, 1000 evaluations over five generations; – – – –, 225% on six mid-span joints, 1000 evauations over five generations; – · – · – · –, 225% on all joints, 2000 evaluations over 10 generations; - - - - -, 225% on all joints, 4400 evaluations over 22 generations; – · · – · · –, 225% on all joints, 4500 evaluations over 15 generations.
451
Figure 15. Complete frequency response of optimized design with limits of 225% on all 18 joints, 4500 evaluations over 15 generations (band average = 0·13 × 10−9 initial frequency response shown dotted).
At this stage it is useful to consider in more detail the convergence of the optimizer when producing the structures of Figures 3, 5 and 7. Figure 9 illustrates this convergence and shows how the best member of each generation performs. As can be seen, most of the improvement in each case is had from the first generation, implying that a simple randomization of the design is beneficial (as has been noted above, purely regular structures often suffer from rather poor noise performance over certain ranges of frequencies, as here). Moreover, the figure shows that none of the trajectories have stabilized with the number of generations allowed. Therefore, another two runs were attempted with around 4500 evaluations used in each. In the first of these the population size was kept at 200 and 22 generations were used while in the second 15 generations of 300 were tried. These resulted in the structures of Figures 10 and 12 together with the responses of Figures 11 and 13, respectively. The convergence traces of these two runs are shown in Figure 14 together with those from the previous runs (notice that the three traces for the overall optimization with 225% limits do not start at the same point because different seeds were used for the random number sequences used, it being a feature of G.A.s that different sequences of random numbers will give different results even if no other parameters are changed). Here the energy flows have been reduced to 0·13 × 10−9 in both cases, i.e., a reduction of 68 dB. In both cases the structures are rather smoother than that of Figure 7, with that of Figure 12 being particularly interesting. This latter design shows a strong overall pattern where, the depth of the composite cantilever is increased at both ends and reduced mid-span. Such a design is clearly workable and demonstrates the power of the technique being used. Figure 15 further illustrates this point, showing the full frequency response of this design compared to the base structure, and illustrating that gains over the 150–250 Hz band have not been achieved at the expense of significantly worsened behaviour at other frequencies. 7. CONCLUSIONS
This paper has shown that significant noise isolation characteristics can be introduced into a regular structure by modifying it in a controlled way. Moreover, the structures produced
452
. .
are not unworkable and do not suffer from degraded performance at frequencies away from those that have been optimized. It must be stressed that the improved vibration isolation characteristics arise from the constructive reflections of travelling waves caused by the discontinuities introduced. Part of the improvement is seen to be the result of moving away from perfect geometric periodicity. However, even more significant gains arise from the unusual geometries adopted when significant changes to the design are made possible. Therefore, a simple small scale randomization, although often beneficial when dealing with a structure that originally exhibits precise geometric periodicity, does not realize all the gains that can be made. It is, however, much harder to select a particular set of large scale modifications that control the noise performance in a particular way, while still maintaining a reasonable overall shape. Other studies [5] have shown that the modes of the optimized structures do not show any overall pattern when the band of frequencies being studied contain more than one or two modes. This happens because many complex interactions must be achieved simultaneously to gain the desired effects in such cases. Conversely, when only a single mode contributes to the behaviour being optimized (i.e., when the band is very narrow compared to the modal density) there is a tendency towards near periodic structures which show the well-known behaviour of localized mode shapes. In such cases the mode in question tends to show low amplitudes at the points of driving and response; moreover the modal frequency tends to lie at the edges of, or even outside, the band being considered. This in turn leads to the desired noise isolation characteristics. Conversely, because it is so hard to predict the frequency averaged behaviour of a structure over a band containing many modes, it proves impossible to give simple design rules to effect noise isolation for such problems. This leads to the need for optimization, and sophisticated optimization at that. Indeed, if this were not the case where would be no need to engage in studies such as that given here: the correct configurations would merely be used from the outset. This is where modern optimization techniques are so powerful. None the less, caution must still be exercized when using methods like the G.A. Although beguilingly simple to describe they are, in fact, quite difficult to operate to best effect. They are not unlike may other new and advanced techniques in this respect. Moreover, it should be noted that, even when using a four processor work-station, some of the runs discussed here take more than five days of computer time to carry out. Clearly, larger structures would be even more expensive to deal with. However, when considering the cost of satellite or aircraft design and the penalties associated with redundant weight, such calculations should be well worthwhile.
REFERENCES 1. D. J. M and N. S. B 1986 Journal of Sound and Vibration 111(2), 229–250. Free vibration of a thin cylindrical shell with discrete axial stiffeners. 2. D. J. M and N. S. B 1987 Journal of Sound and Vibration 115(3), 449–520. Free vibration of a thin cylindrical shell with periodic circumferential stiffeners. 3. R. H. L 1975 Statistical Energy Analysis of Dynamical Systems: Theory and Applications. Cambridge, MA: MIT Press. 4. W. G. P and A. J. K (editors) 1994 Statistical Energy Analysis – Philosophical Transactions (A) Theme Issue 346(1681). The Royal Society of London. 5. A. J. K and C. S. M 1993 Journal of Sound and Vibration 168(2), 253–284. Energy flow variability in a pair of coupled stochastic rods. 6. K. S and A. J. K 1995 Journal of Sound and Vibration 181, 801–838. A study of the vibrational energies of two coupled beams using finite element and Green function (receptance) methods. 7. K. S and A. J. K (to appear) Journal of Sound and Vibration. Energy flow predictions in a structure of rigidly joined beams using receptance theory.
453
8. R. E. D. B and D. C. J 1960 The Mechanics of Vibration. Cambridge: Cambridge University Press. 9. S. K, C. D. G, J., and M. P. V 1983 Science 220(4598), 671–680. Optimization by simulated annealing. 10. D. E. G 1989 Genetic Algorithms in Search, Optimization and Machine Learning. Cambridge, MA: Addison-Wesley. 11. D. B. F 1993 Cybernetics and Systems 24(1), 27–36. Applying evolutionary programming to selected traveling salesman problems. 12. A. J. K 1993 Proceedings of the International Conference on Artificial Neural Nets and Genetic Algorithms (editors R. F. Albrecht, C. R. Reeves and N. C. Steele); Innsbruck: Springer-Verlag, 536–543. Structural design for enhanced noise performance using Genetic Algorithm and other optimization techniques. 13. A. J. K 1995 Artificial Intelligence in Engineering, 9(2), 75–83. Genetic algorithm optimization of multi-peak problems: studies in convergence and robustness. 14. M. R. A 1975 Cluster Analysis for Applications. New York: Academic Press. 15. X. Y and N. G 1993 Proceedings of the International Conference on Artificial Neural Nets and Genetic Algorithms (editors R. F. Albrecht, C. R. Reeves and N. C. Steele); Innsbruck, Springer-Verlag, 450–457. A fast genetic algorithm with sharing scheme using cluster methods in multimodal function optimization.
PASSIVE VIBRATION CONTROL VIA UNUSUAL GEOMETRIES: THE APPLICATION OF GENETIC ALGORITHM OPTIMIZATION TO STRUCTURAL DESIGN A. J. K Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, England (Received 28 April 1994, and in final form 8 August 1994) In the majority of aerospace structures, vibration transmission problems are dealt with by the application of heavy, viscoelastic damping materials. More recently, interest has focussed on using active vibration control methods to reduce noise transmission. This paper examines a third, and potentially much cheaper method: that of redesigning the load bearing structure so that it has intrinsic, passive noise filtration characteristics. It shows that very significant, broadband noise isolation characteristics (of around 60 dB over a 100 Hz band) can be achieved without compromising other aspects of the design. Here, the genetic algorithm (GA), which is one of a number of recently developed evolutionary computing methods, is employed to produce the desired designs. The problem is set up as one in multi-dimensional optimization where the geometric parameters of the design are the free variables and the band averaged noise transmission the objective function. The problem is then constrained by the need to maintain structural integrity. Set out in this way, even a simple structural problem has many tens of variables; a real structure would have many hundreds. Consequently, the optimization domain is very time consuming for traditional methods to deal with. This is where modern evolutionary techniques become so useful: their convergence rates are typically less rapidly worsened by increases in the number of variables than those of more traditional methods. Even so, they must be used with some care to gain the best results. 7 1995 Academic Press Limited
1. INTRODUCTION
Structure-borne noise and vibration control is an aspect of design where there are few reliable design techniques available and one that is relevant to almost all vehicles and many other lightweight engineering structures. Moreover, structures such as cars, aeroplanes, ships, suspension bridges, etc., all suffer from exposure to noise and vibration sources. These sources often excite unwanted structural resonances which can cause damage or the transport of vibrational energy to distant parts of the structure where they cannot be tolerated. For example, the engines in cars, ships and aircraft always vibrate to some degree and, despite isolation treatments, excite broadband motions of their mounting points. Since most structures are quite stiff and have inherently low damping characteristics the subsequent resonant motions may well be relatively large. This vibrational energy can then flow through the structure and cause significant motions in, say, cabin panels. These in turn radiate noise into the working environments of crew or passenger spaces. Vibrations of this kind may also effect sensitive equipment or lead to fatigue failures in light-weight alloy structures. The most common treatment for such problems is to coat the structural elements with heavy viscoelastic damping materials with consequent weight and cost penalties. 441 0022–460X/95/330441 + 13 $12.00/0
7 1995 Academic Press Limited
442
. .
Moreover, the effectiveness of such treatments diminishes with the vibration levels which makes continuously improving noise and vibration targets difficult to meet. Clearly, if the vibrational energy could be contained near to the points of excitation there would be a reduced need for damping treatments and, additionally, they could be concentrated in regions where they were most effective. This is precisely the aim of the vibration isolators used between most pieces of equipment and their supporting structure. However, it is difficult and expensive to isolate large pieces of structure in this way, although this is done in modern ship design where entire deckhouses are sometimes placed on resilient mountings. Of course, some pieces of structure, such as the wings on an aircraft, cannot be mounted in this way. The consequence of this problem is the desire for some kind of widely applicable, generic structural filter design capability that could be used to build desirable characteristics into a structure, retaining their ability to carry static loads while blocking higher frequency motions. Recently, much effort has been expended on so-called ‘‘active’’ vibration control measures for use in this area. These employ ‘‘anti-noise’’ to cancel out unwanted vibrations and so block noise propagation. However, they are inevitably expensive to install and maintain, and passive solutions would be preferable if they could be found. One passive approach that has been extensively studied, mainly for use in aircraft fuselages, makes use of the vibration characteristics exhibited by structures with geometric periodicity. Such structures possess so-called ‘‘stop’’ bands which are regions of the frequency domain where natural frequencies do not occur and where travelling waves are very rapidly attenuated by constructive reflections [1, 2]. These characteristics become increasingly complex as the nature of the periodicity becomes more complex and also as the structural type changes from one- through two- to three-dimensional. It has in practice, so far proved impossible to design real structures with the desired characteristics by using this behaviour. Moreover, such success as there has been, has been found to be extremely sensitive to the accuracy of manufacture and subsequent modifications, the desired stop bands being easily disrupted or shifted by the addition of extra structural features, changes in payload, etc. Clearly, there is no point in adopting such methods if they work only for a single cargo configuration or if they preclude even minor modifications through the life of the structure concerned. In short, to be of practical use, noise isolation design techniques must be widely applicable, yielding robust designs, even if this is achieved at the expense of ultimate theoretical performance. As part of a recent programme of work concerned with improving noise prediction methods, statistical energy analysis (S.E.A.) was applied to simplified aerospace structural models. S.E.A. is a method dating back to the early 1960s [3] which is the subject of much current research [4]. It is used to predict the flow of energy around vibrating structures and is seen by many as providing the best long term means for analyzing their noise performance. During the work being undertaken the design of anomalous structural configurations became of interest [5]. The primary reason for this was the desire to quantify the tails in the statistical descriptions of the responses of structures with random physical properties. It became clear that the most direct way of finding configurations where energy flowed well or badly was to employ optimization techniques to maximize or minimize the flows. This proved difficult when using classical methods because of the many variables involved and a G.A. approach was adopted instead. The flow of vibrational energy around a complex structure is dominated by the many resonances exhibited by such structures and also the large number of physical parameters needed to specify typical structural designs (in the previous case 40, many hundreds in a full car, ship or aircraft structural model), i.e., the G.A. is well suited to such problems. This approach, combined with S.E.A. methods, showed that dramatic reductions in energy flows between coupled structures could be achieved, and moreover such reductions could be made
443
to hold over significant frequency bandwidths while at the same time remaining insensitive to minor structural modifications. When the geometries that gave rise to these reductions were studied it was seen that they showd characteristic periodic fluctuations in properties (i.e., alternating light and heavy sections), although these were not so precise as those that could be adopted when using classical periodic structure theories. In short, although only at a very preliminary stage and applied to a very idealized model, this finding suggested that the possibility of designing the highly desirable filtering characteristics sought during the earlier programme of work by using periodic structures need not be abandoned. Instead, designs produced and analyzed in the normal way might be modified by using G.A. optimizers seeking to retain the desired structural characteristics while at the same time incorporating robust noise filtration features. Moreover, this approach should be applicable to a wide range of structural noise problems. This paper is a sequel to this earlier work and is concerned with the flow of energy around a network of 40 coupled beams, of the sort that may be found in satellite booms (and girder bridges, tower cranes, etc.). Here the vibration analysis is carried out by using matrix receptance methods based on the Green functions of the individual beam elements, which are set up to calculate the forces and velocities at the joints in the structures [6, 7]. This approach enables the energy flows around the structure to be calculated directly and efficiently and this, in turn, allows for the solution of the thousands of configurations that must be considered when dealing with designs consisting of many elements. 2. THE STRUCTURE
Figure 1 indicates the structure to be optimized. This consists of 40 Euler–Bernoulli beams all having the same properties per unit length. Here, EA is taken as 69·87 MN, EI as 12·86 kNm2 and the mass per unit length as 2·74 kg/m. the beams are all either 1 m or 1·414 m long and the joints at (0, 0) and (0, 1) are taken to be pinned to ground; all other joints are free to move. The structure has been chosen to be two-dimensional for simplicity. It is excited by a point transverse force halfway between (0, 0) and (1, 0) and, for this study, the aim is set as the minimization of the vibrational energy level in the right-hand end vertical beam (which in practice might carry an instrumentation package or similar). The damping of the structure is fixed so that the normal modes of the uncoupled beam elements all have a constant bandwidth of 20 s−1 . 3. THEORY
The forced vibrational performance of such a structure may be readily computed by using finite element methods but this is rather cumbersome when evaluating the response of a
Figure 1. Initial structural design.
444
. .
Figure 2. Initial frequency response (band average = 0·33 × 10−6 ).
structure to broadband forcing, especially when very many different configurations will need to be considered during optimization. Instead, the well-known characteristics of Euler–Bernoulli beams combined with receptance methods are used. Such methods have been studied for many years [8] and, as has already been mentioned, may be set up to solve directly for the energetic quantities of interest [6, 7]. Using this approach leads to the problem being defined in terms of the unknown displacements and forces at the joints. The displacements at the ends of each member meeting at a joint are equated and the joint coupling forces and moments summed to zero. The effects of external driving and the joint forces on any beam are specified in terms of the Green functions of the elements which, in turn, are based on the modes of the elements in isolation. This approach requires some care to ensure convergence of the Green functions but allows the selection of various models for the elements to be made with great ease. The references show that energy flows calculated in this way agree very well with those based on finite elements. Figure 2 shows how the vibrational energy level of the end beam varies with frequency for this base design. As can be seen from the figure, the energy level exhibits the many peaks and troughs characteristic of a complex, lightly damped structure. Moreover, there is a large response in the region 100–200 Hz. The calculation of a single frequency point in this plot requires the solution of some 260 complex simultaneous equations and this takes around 20 s on a Sun Sparc 10 workstation; i.e., the calculation of a frequency band averaged takes
Figure 3. Optimized design with limits of 25% on all 18 joints, 1000 evaluations over five generations.
445
Figure 4. Frequency response of optimized design with limits of 25% on all 18 joints, 1000 evaluations over five generations (band average = 0·21 × 10−7; initial frequency response shown dotted).
around seven minutes when using the 21 point rule adopted here. Even so, this is considerably faster than can be achieved by using a finite element approach with the same level of accuracy. 4. THE OPTIMIZATION PROBLEM
The aim of this study is set as the reduction of the frequency averaged response of the end beam in the range 150–250 Hz by the use of optimization: i.e., to trim down the upper half of the major response peak. In practice, such a requirement would reflect known sensitivities of the payload or known excitation frequencies. To meet design requirements the optimization is constrained to keep the end beam unchanged in length and position with respect to the fixed points. Further, to meet structural requirements, all of the joints within the structure must be kept within fixed distances from their original positions. This ensures that no beam is too long or short and also restricts the overall envelope of the structure. The free variables in the problem are thus set as the x and y co-ordinates of the 18 mid-span joints: i.e., 36 variables in all.
Figure 5. Optimized design with limits of 225% on all six mid-span joints, 1000 evaluations over five generations.
446
. .
Figure 6. Frequency response of optimized design with limits of 225% on six mid-span joints, 1000 evaluations over five generations (band average = 0·31 × 10−7; initial frequency response shown dotted).
5. THE OPTIMIZER
Optimization problems of the sort posed here are characterized by having many variables, highly non-linear relationships between the variables and the objective function, and an objective function that has many peaks and troughs. Moreover, as has already been noted, any one configuration is time consuming to evaluate. In short they are difficult to deal with. The search for methods that can cope with such problems has led to the subject of evolutionary computation. Techniques in this field are characterized by a stochastic approach to the search for improved solutions, guided by some kind of evolutionary control strategy. There are three main methods in use today: (1) simulated annealing [9], where the control strategy is based on an understanding of the kinetics of solidifying crystals; (2) genetic algorithms [10], where the methods of Darwinian evolution are applied to the selection of ‘‘fitter’’ designs; (3) evolutionary programming [11], which is a more heuristic approach to the problem but which has an increasing number of adherents. The author has applied all of these methods to structural problems and found that, for the current case, the G.A. works best. The G.A. used here is fairly typical of those discussed in the well-known book by Goldberg [10] but encompasses a number of new ideas that are particularly suited to engineering design problems [12, 13]. Such methods work by maintaining a pool or population of competing designs which are combined to find improved solutions. In their basic form, each member
Figure 7. Optimized design with limits of 225% on all 18 joints, 2000 evaluations over 10 generations.
447
Figure 8. Frequency response of optimized design with limits of 225% on all 18 joints, 2000 evaluations over 10 generations (band average = 0·32 × 10−9; initial frequency response shown dotted).
of the population is represented by a binary string that encodes the variables characterizing the design. The search progresses by manipulating the strings in the pool to provide new generations of designs, hopefully with on average better properties than their predecessors. The processes that are used to seek these improved designs are set up to mimic those of natural selection: hence the method’s name. The most commonly used operations are currently: (1) selection according to fitness, i.e., the most promising designs are given a bigger share of the next generation; (2) crossover, where portions of two good designs, chosen at random, are used to form a new design, i.e., two parents ‘‘breed’’ an ‘‘offspring’’; (3) inversion, whereby the genetic encoding of a design is modified so that subsequent crossover operations affect different aspects of the design; (4) mutation, where small but random changes are arbitrarily introduced into a design. In addition, the number of generations and their sizes must be chosen, as must a method for dealing with constraints (usually by application of a penalty function). The algorithm used here works with 16 bit binary encoding. It uses an elitist survival strategy which ensures that the best of each generation always enters the next generation and has optional niche forming to prevent a few moderately successful designs dominating and so preventing wide ranging searches. Two penalty functions are available to deal with constraints. The main parameters used to control the method may be summarized as follows: Ngen , the number of generations allowed (default 10); Npop , the population size or number of trials used per generation which is therefore inversely related to the number of generations given a fixed number of trials in total (default 100); P[best], the proportion of the population that survive to the next generation (default 0·8); P[cross], the proportion of the surviving population that are allowed to breed (default 0·8); P[invert], the proportion of this population that have their genetic material re-ordered (default 0·5); P[mutation], the proportion of the new generation’s genetic material that is randomly changed (default 0·005); a proportionality flag, which selects whether the new generation is biased in favour
448
. .
Figure 9. Convergence of the first three optimization runs as a function of generation number. q, Initial design; · · · · · · · · ·, 25% on all joints, 1000 evaluations over five generations; – – – –, 225% on six mid-span joints, 1000 evaluations over five generations; – · – · –, 225% on all joints, 2000 evaluations over 10 generations.
of the most successful members of the previous generation or alternatively if all P[best] survivors are propagated equally (default TRUE); the penalty function choice. When using the G.A. to explore large design spaces with many variables, it has also been found that the method must be prevented from being dominated by a few moderately good designs which prevent further innovation. A number of methods have been proposed to deal with this problem; that used here is based on MacQueen’s Adaptive KMEAN algorithm [14] which has recently been applied with some success to multi-peak problems [15]. This algorithm subdivides the population into clusters that have similar properties. The members of each cluster are then penalized according to how many members the cluster has and how far it lies from the cluster centre. It also, optionally, restricts the crossover process that forms the heart of the G.A., so that large successful clusters mix solely with themselves. This aids convergence of the method, since radical new ideas are prevented from contaminating such sub-pools. The version of the algorithm used here is controlled by the following: Dmin , minimum non-dimensional Euclidean distance between cluster centres, with clusters closer than this being collapsed (default 0·05); Dmax , the maximum non-dimensional Euclidean radius of a cluster, beyond which clusters subdivide (default 0·1); Nclust , the initial number of clusters into which a generation is divided (default 25); Nbreed , the minimum number of members in a cluster before exclusive inbreeding within the cluster takes place (default 5); a, the penalizing index for cluster members, which determines how severely members sharing an overcrowded niche will suffer, with large numbers giving greater penalty (default 0·5): i.e., the objective functions of members of a cluster of m solutions are scaled by
Figure 10. Optimized design with limits of 225% on all 18 joints, 4400 evaluations over 22 generations.
449
Figure 11. Frequency response of optimized design with limits of 225% on all 18 joints, 4400 evaluations over 22 generations (band average = 0·13 × 10−9; initial frequency response shown dotted).
m min (a,1)[1 − (E/Dmax )a] + (E/Dmax )a, where E is the Euclidean distance of the member from its cluster centre (which is always less than Dmax ; moreover, when E = Dmax no penalty is applied). In addition, the implementation of the G.A. used here allows the solution of individual members of the population to be run in parallel if a multiple processor computer is available. 6. RESULTS
To begin the optimization process relatively tight limits of 25% were placed on the joint positions and a modest number of 1000 trials used, spread over five generations (giving 200 members per generation so as to ensure reasonable coverage over the rather large domain being investigated here, at the expense of final convergence). This run gave rise to the configuration shown in Figure 3 and the response illustrated in Figure 4. With this configuration, the total energy flow in the 100 Hz band investigated has been reduced from 0·33 × 10−6 to 0·21 × 10−7 (for unit forcing), i.e., a reduction of 23 dB, indicating how purely periodic structures can give rise to very significant noise transmission problems. Next, the limits on the joint positions were relaxed to 225% but only six of the mid-span joints were allowed to move, again using 1000 trials and five generations. This leads to the configuration of Figure 5 and the response shown in Figure 6. This gives slightly less isolation with the
Figure 12. Optimized design with limits of 225% on all 18 joints, 4500 evaluations over 15 generations.
450
. .
Figure 13. Frequency response of optimized design with limits of 225% on all 18 joints, 4500 evaluations over 15 generations (band average = 0·13 × 10−9; initial frequency response shown dotted).
total energy flow now being 0·31 × 10−7 , i.e., a reduction of 21 dB over the base design. As can be seen from Figure 5, a kind of notch has been driven into the structure and this clearly causes reflections of energy-carrying waves in the desired frequency ranges. After having gained some initial impressions, the G.A. was next applied with a 225% constraint on the joint positions applied to the whole structure, but now using 2000 trials over 10 generations, i.e., with an increased number of generations so as to improve convergence. This yields the geometry shown in Figure 7 together with the response shown in Figure 8. The reduction in the frequency averaged response is now from 0·33 × 10−6 to 0·32 × 10−9 , i.e., a change of 60 dB. Clearly, this has been achieved by rather radical geometrical distortions and such severe modifications to the structure might not be tolerable in practice.
Figure 14. Convergence of all five optimization runs as a function of generation number. q, Initial design; · · · · · ·, 25% on all joints, 1000 evaluations over five generations; – – – –, 225% on six mid-span joints, 1000 evauations over five generations; – · – · – · –, 225% on all joints, 2000 evaluations over 10 generations; - - - - -, 225% on all joints, 4400 evaluations over 22 generations; – · · – · · –, 225% on all joints, 4500 evaluations over 15 generations.
451
Figure 15. Complete frequency response of optimized design with limits of 225% on all 18 joints, 4500 evaluations over 15 generations (band average = 0·13 × 10−9 initial frequency response shown dotted).
At this stage it is useful to consider in more detail the convergence of the optimizer when producing the structures of Figures 3, 5 and 7. Figure 9 illustrates this convergence and shows how the best member of each generation performs. As can be seen, most of the improvement in each case is had from the first generation, implying that a simple randomization of the design is beneficial (as has been noted above, purely regular structures often suffer from rather poor noise performance over certain ranges of frequencies, as here). Moreover, the figure shows that none of the trajectories have stabilized with the number of generations allowed. Therefore, another two runs were attempted with around 4500 evaluations used in each. In the first of these the population size was kept at 200 and 22 generations were used while in the second 15 generations of 300 were tried. These resulted in the structures of Figures 10 and 12 together with the responses of Figures 11 and 13, respectively. The convergence traces of these two runs are shown in Figure 14 together with those from the previous runs (notice that the three traces for the overall optimization with 225% limits do not start at the same point because different seeds were used for the random number sequences used, it being a feature of G.A.s that different sequences of random numbers will give different results even if no other parameters are changed). Here the energy flows have been reduced to 0·13 × 10−9 in both cases, i.e., a reduction of 68 dB. In both cases the structures are rather smoother than that of Figure 7, with that of Figure 12 being particularly interesting. This latter design shows a strong overall pattern where, the depth of the composite cantilever is increased at both ends and reduced mid-span. Such a design is clearly workable and demonstrates the power of the technique being used. Figure 15 further illustrates this point, showing the full frequency response of this design compared to the base structure, and illustrating that gains over the 150–250 Hz band have not been achieved at the expense of significantly worsened behaviour at other frequencies. 7. CONCLUSIONS
This paper has shown that significant noise isolation characteristics can be introduced into a regular structure by modifying it in a controlled way. Moreover, the structures produced
452
. .
are not unworkable and do not suffer from degraded performance at frequencies away from those that have been optimized. It must be stressed that the improved vibration isolation characteristics arise from the constructive reflections of travelling waves caused by the discontinuities introduced. Part of the improvement is seen to be the result of moving away from perfect geometric periodicity. However, even more significant gains arise from the unusual geometries adopted when significant changes to the design are made possible. Therefore, a simple small scale randomization, although often beneficial when dealing with a structure that originally exhibits precise geometric periodicity, does not realize all the gains that can be made. It is, however, much harder to select a particular set of large scale modifications that control the noise performance in a particular way, while still maintaining a reasonable overall shape. Other studies [5] have shown that the modes of the optimized structures do not show any overall pattern when the band of frequencies being studied contain more than one or two modes. This happens because many complex interactions must be achieved simultaneously to gain the desired effects in such cases. Conversely, when only a single mode contributes to the behaviour being optimized (i.e., when the band is very narrow compared to the modal density) there is a tendency towards near periodic structures which show the well-known behaviour of localized mode shapes. In such cases the mode in question tends to show low amplitudes at the points of driving and response; moreover the modal frequency tends to lie at the edges of, or even outside, the band being considered. This in turn leads to the desired noise isolation characteristics. Conversely, because it is so hard to predict the frequency averaged behaviour of a structure over a band containing many modes, it proves impossible to give simple design rules to effect noise isolation for such problems. This leads to the need for optimization, and sophisticated optimization at that. Indeed, if this were not the case where would be no need to engage in studies such as that given here: the correct configurations would merely be used from the outset. This is where modern optimization techniques are so powerful. None the less, caution must still be exercized when using methods like the G.A. Although beguilingly simple to describe they are, in fact, quite difficult to operate to best effect. They are not unlike may other new and advanced techniques in this respect. Moreover, it should be noted that, even when using a four processor work-station, some of the runs discussed here take more than five days of computer time to carry out. Clearly, larger structures would be even more expensive to deal with. However, when considering the cost of satellite or aircraft design and the penalties associated with redundant weight, such calculations should be well worthwhile.
REFERENCES 1. D. J. M and N. S. B 1986 Journal of Sound and Vibration 111(2), 229–250. Free vibration of a thin cylindrical shell with discrete axial stiffeners. 2. D. J. M and N. S. B 1987 Journal of Sound and Vibration 115(3), 449–520. Free vibration of a thin cylindrical shell with periodic circumferential stiffeners. 3. R. H. L 1975 Statistical Energy Analysis of Dynamical Systems: Theory and Applications. Cambridge, MA: MIT Press. 4. W. G. P and A. J. K (editors) 1994 Statistical Energy Analysis – Philosophical Transactions (A) Theme Issue 346(1681). The Royal Society of London. 5. A. J. K and C. S. M 1993 Journal of Sound and Vibration 168(2), 253–284. Energy flow variability in a pair of coupled stochastic rods. 6. K. S and A. J. K 1995 Journal of Sound and Vibration 181, 801–838. A study of the vibrational energies of two coupled beams using finite element and Green function (receptance) methods. 7. K. S and A. J. K (to appear) Journal of Sound and Vibration. Energy flow predictions in a structure of rigidly joined beams using receptance theory.
453
8. R. E. D. B and D. C. J 1960 The Mechanics of Vibration. Cambridge: Cambridge University Press. 9. S. K, C. D. G, J., and M. P. V 1983 Science 220(4598), 671–680. Optimization by simulated annealing. 10. D. E. G 1989 Genetic Algorithms in Search, Optimization and Machine Learning. Cambridge, MA: Addison-Wesley. 11. D. B. F 1993 Cybernetics and Systems 24(1), 27–36. Applying evolutionary programming to selected traveling salesman problems. 12. A. J. K 1993 Proceedings of the International Conference on Artificial Neural Nets and Genetic Algorithms (editors R. F. Albrecht, C. R. Reeves and N. C. Steele); Innsbruck: Springer-Verlag, 536–543. Structural design for enhanced noise performance using Genetic Algorithm and other optimization techniques. 13. A. J. K 1995 Artificial Intelligence in Engineering, 9(2), 75–83. Genetic algorithm optimization of multi-peak problems: studies in convergence and robustness. 14. M. R. A 1975 Cluster Analysis for Applications. New York: Academic Press. 15. X. Y and N. G 1993 Proceedings of the International Conference on Artificial Neural Nets and Genetic Algorithms (editors R. F. Albrecht, C. R. Reeves and N. C. Steele); Innsbruck, Springer-Verlag, 450–457. A fast genetic algorithm with sharing scheme using cluster methods in multimodal function optimization.