A Truly Hybrid Approach to Substructuring Problems Using Mixed Assembly and Implicit Solving Strategies This reasoning has been illustrated using figures 7 and 8. Figure 7 is a simple three DoFmass - spring systemand in figure 8 three different plots are shown where the solid lines represent the driving point FRFs of: • x1 withbothx2 andx3 constrained, • x2 with x1 unconstrained andx3 constrained, • x3 withbothx1 andx2 unconstrained. The theoremstated above can also be easily verified fromfigure 8. The method devised by Wittrick and Williams uses the number of negative eigenvalues of the dynamic stiffness matrix at a certain frequency ωj to determine the number of natural frequencies of the dynamic systemwhich can be found below this particular frequency. In their work, the number of negative eigenvalues of the matrix is called its sign count s(Z(ωj). In order to compute the sign count of the matrix, we let Zr be the leading principle minor of order r of Z(ωj), i.e. the determinant of the r×r matrix formed by the first r rows and columns of Z(ωj) and the zeroth order minor Z0 is set to +1. Physically, one can understand this as fully constraining the systemand releasing one DoF at a time and computing the determinant of these constrained systems. The resulting sequence of determinants of a growing systemis called a Sturmsequence. Now it is known that the number of changes of sign between consecutive members of the Sturmsequence: {Z0,Z1,Z2,...,Zn} is equal to the number of natural frequencies of the systembelow ωj). So, using this sequence one can determine at which frequencies the sign count changes and thus find the natural frequencies of the system. This procedure is visualized in figure 8, where the development of the thee leading principle minors is plotted. It can be shown that the sign count of the dynamic stiffness matrix Z(ω) is equal to that of its inverse, the receptance matrix Y(ω) [29]. In order to show that this method can |Y11(ω)| FRF 0 Z1(ω) |Y22(ω)| 0 Z2(ω) Leading principle minor Frequency [ω] |Y33(ω)| 0 Z3(ω) ( U L .d U L .1 U L .n U Xw.D)d ( U L .d U L .1 U L .n U L Xw.D)( ( U L .d U L .1 U .n U L Xw.D)1 ( U L .d U .1 U .n U L Xw.D)1 ( U L .d U .1 U L .n U Xw.D)n Fig. 8: Computing the sign count also be used to compute the natural frequencies of the mixed assembled system(15), the dynamic systemof figure 7 is used again as an example. The first subsystemA, consisting of m1,m2,k1 andk2 is still in the formof a dynamic stiffness matrix: ZA(ω)=⎡ ⎣ ⎡ ⎣ k1 −k1 0 −k1 k1 +k2 −k2 0 −k2 k2 ⎤ ⎦− ω2⎡ ⎣ m1 0 0 0 m2 0 0 0 0 ⎤ ⎦ ⎤ ⎦ (17) SubsystemB, consisting of m3 andk3 is now in the formof a receptance matrix: 339
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