Sub-structuring of mechanical systems based on the path concept

Transcription

1 Sub-structuring of mechanical systems based on the path concept Francesc Xavier MAGRANS 1 ; Jordi POBLET-PUIG 2 ; Antonio RODRÍGUEZ-FERRAN 3 1 Ingeniería para el Control del Ruido S.L., Spain 1,2,3 Universitat Politècnica de Catalunya, Spain ABSTRACT Real mechanical systems are usually complex. Their modelling with discretisation techniques involve a large number of degrees of freedom/unknowns. This leads to high computational costs especially at high frequencies. An alternative, the statistical methods, are sometimes limited by strong behavioural requirements. So, sub-structuring can be considered very often as a good alternative. The present research studies the coupling between subsystems from the point of view of transmission path analysis. The interest is focused on the signal transmission rather than the energy distribution. A subsystem identification method is proposed. It is based on the expression of the solution in terms of the powers of the transfer matrix. This is related with the description of high-order paths which are more affected by damping and system properties. Consequently, the identification of subsystems can be more easily done. The method provides a quantification of the degree of coupling between subsystems and of the error caused by the detachment of a system part from the global system in the modelling phase. The output of the presented technique can be a sub-system definition valid for the SEA method or relevant information to be used in the design of measurement procedures. Keywords: vibroacoustic, path, SEA, subsystems 1. INTRODUCTION The subdivision of a mechanical system into subsystems according to their vibroacoustic response is a required task in several modelling methods or in order to better understand the behaviour of the mechanical system. A clear example is Statistical Energy Analysis (SEA) that requires, as preliminary step, the definition of subsystems which satisfy several physical properties (high modal density, equipartition of energy between modes, equal probability of mode excitation, weak coupling between subsystems). Or, not related with modelling and simulations, it can be interesting to know which parts of a big system (train wagon, building, airplane) can be tested in the laboratory isolated from the other parts and the results will still be meaningful. The splitting of the domain is sometimes performed by hand, mainly based on experience and intuition. However, some more systematic methods have also been proposed. Some of them simply follow some material or geometrical criteria [1, 2], are based on energy models [3 5], regard the frequency response function of a system [6], or study the shape of some eigenfunctions of the system [7]. In these last two methods [6, 7], cluster algorithms are used. Cluster analysis is a general purpose and very powerful tool that groups sets of objects in a big population according to their similitude, the group of objects, the parameters that characterise them, the comparison criteria and how to measure their similitude (distance). In this ongoing research, we plan to apply cluster analysis in order to automatically define the subsystems in a vibroacoustic problem. The core of the method is based on the transfer matrices and how they (and their powers) are representative of transmission paths inside the system. Some brief background concepts and the core idea of the method are explained in Section 2 and some illustrative numerical examples are presented in Section fxmagrans@icrsl.com 2 jordi.poblet@upc.edu 3 antonio.rodriguez-ferran@upc.edu

2 2. METHODOLOGY 2.1 Overview of path analysis and transfer matrices The problem statement as well as a brief summary of the transfer matrix concept is exposed here. The main goal is to see that a solution of a linear system can be expressed by means of the superposition of multiple paths. This has been recently proven in [8]. Assume that the mechanical system is properly described by the linear system of equations: Ax = b x i, b i, a ij C (1) where the system matrix A can be split into diagonal D, strictly lower triangular L and upper triangular U matrices A = D + L + U (2) The linear system in (1) can be then rewritten as x = D 1 b + Tx (3) where T = D 1 (L + U) (4) is the transfer matrix. This matrix is the transpose of the Direct Transfer Matrix defined in [9] with zeros in the diagonal. It was clearly demonstrated in [10, 11] that a solution of a mechanical problem can be described by means of the Neumann series of the transfer matrix T (the powers of T are a representation of paths of different order in the mechanical system). The series has strict convergence conditions, which in practise mean that the solution description through transmission paths and Neumann series 1 is not always possible. From Eq. (3), the solution of system (1) may be expressed as x = (I T) 1 D 1 b (5) This modified form of the linear system has the advantage that the inverse of the matrix (I T) can be expressed as the limit of a matrix series as lim m S j,m = (I T) 1 (6) where S j,m is the partial sum of order j (j modified parameters) and m + 1 terms and can be expressed as S j,m = m j k=0 T k + m k=m j+1 γ m k T k (7) Contrary to the Neumann series mentioned above, the series in Eq. (7) is unconditionally convergent if coefficients γ m k are properly chosen. In fact, Eq. (7) can be seen as the Neumann series with m + 1 terms where the last j have been modified. Eqs. (6) and (7) as well as expressions for the coefficients γ m k are the main results in [8]. They allow the generalisation of the path superposition idea to any linear mechanical system (and not only to those with convergent Neumann series). So, a theoretical procedure based on the path concept has now a solid basis. In the present research, this idea is applied to the formulation of an algorithm for automatic sub-structuring of mechanical systems. 1 A Neumann series has the form k=0 Tk

3 2.2 Main idea of the method INTER-NOISE 2016 A subsystem is a set of nodes with similar behaviour (in particular, of the paths passing through them). The concept of node here is general in the sense that the transfer matrix T concept is very flexible. It can be defined from a linear system of equations as in Eq. (4) but also in other frameworks of analysis [10, 11]. Thus T can establish relationships between degrees of freedom in a discretisation by means of a numerical technique such as Finite Elements (FEM), or between points of excitation and measure in an experiment in the laboratory. The relationship between these nodes is already described in the system matrix A or the transfer matrix T. However, it is not easy to identify the subsystems there. The main hypothesis is that the identification of subsystems can be more easily done in the powers of the transfer matrix, T k. It is sustained on the fact that this matrix contains the information of the relationship between nodes by means of k-order paths. If a path remains inside a strongly connected zone (i.e. a subsystem), it is not attenuated. So, the coefficients of matrix T k that relate nodes in the same subsystems will be large. On the contrary, if a k-order path passes through weakly connected subsystems, it will be attenuated. Consequently, the coefficients of matrix T k that relate nodes in different subsystems are penalised and have a smaller numerical value. Based on this idea, the proposed strategy in order to identify the subsystems is: 1. Define the transfer matrix of a system. It may have as many degrees of freedom as possible (for example, all the degrees of freedom in a FEM model), but it can also be optimised and include only some nodes selected by means of a smart strategy (i.e. distribute them based on some hypothesis on the subsystems). 2. Compute the powers of the transfer matrix T. 3. Apply a cluster algorithm that can identify the nodes with similar measure of their behaviour/relationship. In this case, the value of the matrix coefficient. 3. RESULTS Some examples are shown. They illustrate the idea exposed in Section 2.2 by means of mechanical problems solved in the frequency domain (steady harmonic). 3.1 Exemple 1: Two weakly coupled plates The first example is composed of two steel plates with different sizes and weakly linked through five springs, see Fig. 1. The geometric and mechanical properties of the steel plates are listed in Table 1 and the stiffness of the springs and their positions are summarized in Table 2. The results are for a problem frequency of 100 Hz. The solution is obtained with finite differences, using a regular grid in both plates. Figure 1. Two plates and 5 springs as described in Table 2.Calculus at 100 Hz

4 Table 1: Geometric and mechanical properties of the two steel plates (ρ v is the volumetric density, E is the Young module, ν is the Poisson s ratio, h is the plate thickness and L x, L y the plate dimensions). Material ρ v (kg/m 3 ) ν E (Pa) L x (m) L y (m) h(cm) Steel E Steel E Table 2. Stiffness K and position of each spring that connects the steel plates. Spring K (N/m) x (m) y (m) E E E E E Fig. 2 shows the transfer matrix of the system T k, with k = 60. The grey scale illustrates the module of each coefficient. The matrix plotted in this way, clearly presents a block structure. The coefficients of the diagonal blocks are representative of paths that start and end in the same plate. On the contrary, the coefficients in the out-of-diagonal or coupling blocks represent paths that begin and end in different plates. In this case the ordering of the unknowns helps to see the block structure. However, for an arbitrary mechanical system the ordering will not be always coincident to the possible subsystems. To identify the subsystems in this arbitrary numbering is the main reason why cluster analysis is required. Fig. 3 shows the dendograms obtained as the outputs of the cluster analysis. This is the way how the subsystems can be automatically identified even if random numbering of the unknowns is used. Each branch corresponds to one grouping and the ordinate shows a measure of the distance. The lower part of the dendogram, corresponding to the very small groups is not shown in the plot because it is meaningless for our purposes. As done in [7], it is considered that the more adequate subdivision is the one which requires the largest distance jump in the dendogram. Understanding that the distance jump is the distance required to pass from one grouping to another. The dendograms in Fig. 3 are shown for several powers of the matrix T k, k = 1, 31 and 61. For k = 1 no clear grouping can be identified. However, as k increases we can see how the option of two groups (corresponding to the two plates) can be clearly identified (see the largest branches). The clarity of grouping is increased with the power of the transfer matrix. We can conclude that the method allowed us to identify the subsystems in agreement with the concept that a subsystem is a set of nodes with similar behaviour

5 Figure 2: Transfer matrix raised to the power 60. Two plates and 5 springs as described in Table 2 Figure 3: Results of clustering for various powers of T matrix. Two plates of different sizes connected by means of 5 springs

6 3.2 Example 2: four plates linked by means of springs The second example deals with four plates that have the same size and are also linked by five springs each, see Fig. 4. The four plates are like the second plate in Table 1 (material and geometric properties). The spring stiffness and positions are the same as in Section 3.1, except for the stiffness of the springs linking the second and third levels (in the middle) which is K = 10 1 EN/m (they are clearly stiffer than the others). Figure 4. Four plates and 5 springs linking each couple, the frequency is 100 Hz The results of the methodology for this example are shown in Fig. 5 (transfer matrix) and Fig. 6 (dendograms). Four subsystems are identified (the four plates). This can be seen both in the visual inspection of the coefficients (grey scale) of the transfer matrix or by counting the largest branches in the dendograms for advanced power of the transfer matrix. The four blocks in the diagonal correspond to each plate due to the used ordering of the unknowns numbering. The coupling blocks (out-of-diagonal) present a different darkness (proportional to the module of the matrix coefficient) depending on the degree of coupling between blocks (plates). Consequently, in the central part of the four times four block matrix, the coupling blocks between the second and third plates appear almost with the same colour than the diagonal blocks. This denotes an stronger coupling between these plates that are linked with the stiffer springs. They could almost be considered as a single block. On the contrary, the coupling matrices between plates 1-2 and 3-4 is clearly different (higher contrast between blocks). Finally, blocks 1 and 4, are clearly uncoupled because their out-of-diagonal matrices are the darkest ones (with smaller coefficients). Again, the identification of the branches and groupings in the dendograms of Fig. 6 is more clearly done for higher values of the power k. 51 Figure 5. Transfer matrix raised to the power 51. Four plates and 5 springs

7 Figure 6: Results of clustering for various powers of T matrix. Four plates attached each couple via 5 springs

8 3.3 Example 3: four strongly coupled plates The same system as in Section 3.2 is considered but now all the spring stiffness are K = 10 4 EN/m. Consequently, strong coupling between all the plates is expected. Again, the matrix for a frequency of 100 Hz is computed. The grey scale plot of the transfer matrix in Fig. 7 shows the whole matrix as a single block. Moreover, no clear trend can be seen in the dendograms of Fig. 8. The cluster analysis is unable to identify the plates as isolated subsystems because in fact, this mechanical system with stiff springs behaves as a unity. It has no sense to define subsystems in that case and this is what we learn from the cluster analysis. Figure 7. Transfer matrix raised to the power 51.Four plates and 5 stiff springs. Figure 8: Results of clustering for various powers of T matrix. Four plates attached each couple via 5 stiff springs

9 4. CONCLUSIONS AND OUTLOOK INTER-NOISE 2016 A methodology for the automatic identification of subsystems in a mechanical system is being proposed. This is based in the powers of the transfer matrix, that represent high-order flanking paths. It has been shown by means of some numerical examples that these matrices tend to have a more proper structure in order to identify the subsystems. Cluster analysis is used in order to generalise the technique to more complex mechanical systems (where the variables could be not properly ordered). This manuscript includes the main idea of the automatic definition of subsystems and some numerical examples. However, some other tasks are required before that proposal could be considered a consolidated method. Among others: Find an estimation of the minimum power k required in order to make possible the proper subsystem identification in T k. Currently we are working on a probabilistic analysis related with the paths definition that could provide information on this aspect. Define strategies in order to optimise the number of nodes in the transfer matrix and how they are distributed all around the mechanical system. Once the subsystems are defined, a quantification of the degree of coupling between them, would be a very useful tool in order to take modelling decisions (i.e. if they can be used as SEA subsystems or they can be tested isolated from the other parts). Apply the technique to more complex systems ACKNOWLEDGEMENTS LaCàN research group is grateful for the sponsorship/funding received from Generalitat de Catalunya (Grant number 2014-SGR-1471). REFERENCES [1] X. Chen, D. Wang, and Z. Ma. Simulation on a car interior aerodynamic noise control based on statistical energy analysis. Chinese Journal of Mechanical Engineering, 25(5): , [2] J. Forssén, S. Tober, Corakci.A.C., A. Frid, and K. Wolfgang. Modelling the interior sound field of a railway vehicle using statistical energy analysis. Applied Acoustics, 73(4): , [3] M. Kassem, C. Soize, and L. Gagliardini. Structural partitioning of complex structures in the mediumfrequency range. An application to an automotive vehicle. J. Sound Vibr., 330(5): , [4] L. Kovalevsky and R.S. Langley. Automatic recognition of the components of a hybrid FE-SEA model. In Proceedings of the Acoustics 2012 Nantes Conference, Nantes, France, April 2012, [5] L. Gagliardini, L. Houillon, G. Borello, and L. Petrinelli. Virtual SEA FEA-based modeling of midfrequency structure-borne noise. Sound and Vibration, 39(1):22 28, [6] N. Totaro and J.L. Guyader. SEA substructuring using cluster analysis: The MIR index. J. Sound Vibr., 290(1-2): , [7] C. Díaz-Cereceda, J. Poblet-Puig, and A. Rodríguez-Ferran. Automatic subsystem identification in statistical energy analysis. Mechanical Systems and Signal Processing, 54: , [8] F.X. Magrans, J. Poblet-Puig, and A. Rodríguez-Ferran. The solution of linear mechanical systems in terms of path superposition. Submitted, [9] F. X. Magrans. Method of measuring transmission paths. J. Sound Vibr., 74(3): , [10] F. X. Magrans. Direct transference applied to the study of room acoustics. J. Sound Vibr., 96(1):13 21,

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