The use of negative structures in modelling cavities was discussed in previous symposia [1-3]. A formal proof that the use of negative structures to remove corresponding positive structural components in calculating natural frequencies has been established for discrete systems. This approach introduces extra spurious modes and methods of identifying and eliminating these modes have also been studied for the same. However, a corresponding proof for continuous systems, and its application poses some challenges. The focus of. this paper is on these challenges

Ilanko, Sinniah

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!SVCS 10, 1 Olh International Symposium on Vibrations of Continuous SystemsStanley Hotel, Estes Park, Colorado, USA, July 26-31, 2015 Theoretical Basis of assembling structures in the Rayleigh Ritz Method S. Ilanko"Abstract *The School of EngineeringThe University of WaikatoGate 1 Knighton Road Private Bag 3105 Hamilton 3240 New Zealand ilanko@waikato.ac.nz The use of negative structures in modelling cavities was discussed in previous symposia [1-3]. A formal proof that the use of negative structures to remove corresponding positive structural components in calculating natural frequencies has been established for discrete systems. This approach introduces extra spurious modes and methods of identifying and eliminating these modes have also been studied for the same. However, a corresponding proof for continuous systems, and its application poses some challenges. The focus of. this paper is on these challenges. To start with, let us consider a discrete n degree of freedom vibratory system A, which when connected to another m dof discrete system c+ subject to r constraints that connect A and c+ results in an n+m-r dof system B. See Figure I. System A System B I 7 L 7 7 7 7 7 System c+ + I 7 L 7 7 7 7 7 System B' Figure I 31 Here, the conne.ction constraints are such that each of the corresponding pairs of r degrees of freedom of A and c+ vibrate together. The constraint conditions may be written as dA9 - de!= == 0, for i == 1,2 .. r,' ., (1) Where dA9, dct are the common degrees of freedom in A and C+. ' . There is an implication that r < n, and r < m.Let [xA], lxs], [xcJ be the normalised modal matrices of A, B, C respectively. Then the ith modes of A, B, C are given by the ith columns of the corresponding matrices and are denoted by vectors {xA..J, CX8.;)fxc.d respectively. Let VA(£ ). vs( fs ), Ve ( fc) etc, be the potential energy of A, B, C+ expressed in terms of assumed displacement vector forms{{,.}, {/8}, {le} respectively. The displacement forms are obtained from a liner combination of sets of admissible vectors {cpA}, {«j,8}, {.Pd which have the same length as the corresponding number of dofs. l. e T _ �na :r. T _ �n+m-ra :r. T _�ma :r.• • JA,j - ,t.,; i'l'A,J,i• IB - ,l.,i i'l'IJ,J,i, JC - ,t.,; l'+'C,j,I (2a,b,c) Similarly the kinetic energy functions of A, B, c+ are tJ;i IA), t/Je(f8), 1/Jc(fc).The energy terms of a negative structure C- are simply negative of their positive counterparts.i.e. Vc-(1'·) = -Ve( fc); t/Jc-(fc) = -t/Jc(fc) (3a,b)For discrete systems, the exact frequencies may be readily obtained by minimising the Rayleighcoefficients in terms of an assumed form. For example, for A, an assumed form of displacement[A formed by the admissible functions [¢A], will give the natural frequencies and modes of Ausing the Rayleigh-Ritz Method (RRM), if the functions(ef>A} are independent, and the numberof admissible functions is equal to the number of degrees of freedom.In terms of the exact modes, the ith eigenvalue (the square of the natural frequency) of A is [4]w2. = v<xA(t)) (4a) A,1 ,J,(X,4(0)But from Rayleigh's Principle, the ith eigenvalue of A is also given by z . v(t A) I T ,, . 1 2 . 1WA; = mm-.- < JA,XAJ >, 10rJ = , . .  t -' 1/i(T ,t) • (4b) Now consider the combination of A and c+, without the connections (see Figure 1), which for convenience may be labelled B'. B' =Aue• (5) In B', it is possible for one of the constituting structures to vibrate in a natural mode while the other remains stationary. Therefore, the eigensolutions of B' are combination of the eigensolutions of A and c+. 2 • Y(f,i)+V(fc)I ;, ,, . . w8,; = mm ·•·c- , c. > < 18,,Xg, ,· > , 1or J = 1,2 .. i - 1' '#' fA1+lfi Jc ' ButB = Auc•1 eq.(1) (7) (6) 2 • V(JA)+v<Jc)I ;, ,, · · d ( ) )w8,; = mm I/JCf1t)+lJl<fc> < 18,,Xs,,J >, 1orJ = 1,2 . . l -1 an eq. 1 (8 Here, fe,denote the union of admissible forms for A and c+ and, x8,,1 denotes the jth mode from the union of the sets of modes of A and c+. Since all possible degrees of freedom have been included in the chosen set of admissible forms and the constraint conditions (1) are enforced, eq. (8) gives the exact natural frequencies ofB. Thus the natural frequencies of B may be obtained by using the admissible displacement forms for A and c+ in the RRM. Now let us consider systems A, B and C as discretised models of continuous systems A, 11!, and IC with numbers of dofs that are of interest (let us refer to these as "significant degrees of freedom") being n, m and n+m-r respectively. It is noted here that in order to get accurate modal 32 values for these displacements, additional degrees of freedom may have to be introduced in the models, such that the required natural frequencies and modes of all these systems are correct to the desired degree of accuracy. Therefore, for all practical purposes, the discretised models with the significant degrees of freedom can be used instead of the actual system. The significant dofs correspond to displacements (translations or rotations) at specified locations. The question then arises as to how to relate this to a typical Rayleigh-Ritz analysis of a continuous systems. Let the actual number of degrees of freedom needed to obtain results with any desired level of convergence be nn and nc for systems Band C, where ni>> n+m-r and nc>:> m. Then a linear combination of admissible functions <pA.i for i=l,2 .. nn and <fJc.J forj=l,2 .. nc willyield independent coordinate values for all of the required number of independent degrees of freedom. This proves that the application of the Rayleigh Ritz Method to an assembly of structures based on the admissible functions for the component structures with the application of relevant connection constraints will yield the natural frequencies and modes of the assembled structure. The proof above is applicable for any Rayleigh-Ritz model of continuous systems consisting of positive structures. If this argument is valid for combining positive and negative structures, we could then also infer that the natural frequencies of A could be obtained by combining the modes of B and C- in a Rayleigh-Ritz procedure subject to continuity constraints at the interface/boundary. This would prove to be useful because the natural frequencies and modes of negative structures are identical to those of their positive counterparts because both kinetic energy function and potential energy for the negative structures are equal and opposite to their positive counterparts as given in eq. (3). However, it is not clear whether the Rayleigh's theorem of separation is applicable for negative structures. Currently work is in progress to find a proof that is applicable for assembling negative and positive structures. References 1 Ilanko, S (2009) Embedding Negative Structures to Model Holes and Cut-Outs, Proceedings of the 7th International Symposium on Vibrations of Continuous Systems, pp 30-33, July 2009, Zakopane, Poland. 2 Ilanko, S., Mochida, Y., Hagedorn, P., A. Wagner, A., and Kennedy, D. (2013) Modelling voids using embedded negative structures - the story of the spurious modes, Proceedings of the 9th International Symposium on Vibrations of Continuous Systems, pp 34-36, July 2013, Courmayeur, Italy. 3 Kennedy, D., Ilanko, S., and Mochida, Y., (2013), Understanding spurious modes of structures with voids using dynamic stiflness method, Proceedings of the 9th International Symposium on Vibrations of Continuous Systems, pp 37-39, July 2013, Courmayeur, Italy. 4 Gould, S.H., Variational methods for eigenvalue problems, Third Edition (1995), Dover Publications Inc, New York. Acknowledgement The author wishes to acknowledge the financial support from the Marsden Fund (http://www.royal society .org.nz/programmes/funds/Jnarsden/). 33 

Theoretical Basis of assembling structures in the Rayleigh Ritz Method

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