12 The Importance of Phase-Locking in Nonlinear Modal Interactions 125 ˇ| D!2 r1 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 8 9 r2 0 1 r2 0 1 r2 8 9 r2 .r C1/.r C3/ 4.1Cr/ r2 1 0 .r 1/.r 3/ 4.1 r/ .r 1/.r 3/ 4.1 r/ r2 1 0 .r C1/.r C3/ 4.1Cr/ 4r.r C1/ .3r C1/.r C1/ 0 1 r2 4r.r 1/ .3r 1/.r 1/ 4r.r 1/ .3r 1/.r 1/ 0 1 r2 4r.r C1/ .3r C1/.r C1/ 9r2 1 8r2 r2 1 0 r2 1 0 9r2 1 8r2 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : (12.11) The resonant equation of motion is written RuCƒuCNu .u/ D0; (12.12) where Nu is a vector of resonant nonlinear terms, defined using Nu .u/ DŒnu u up; um ; (12.13) where Œnu is a matrix of the coefficients of the resonant nonlinear terms. The nonlinear terms represented by Nq are defined as resonant if they correspond to an element in ˇthat contains a zero; hence such a term is also represented in Œnu . Conversely, if a term is non-resonant (i.e. it contributes to a harmonic) then the corresponding element inˇis non-zero, and hence the element in Œnu must be zero. This is expressed by the relationship defining element fi; `g of Œnu as Œnu i;` D ( Nq i;` if W ˇi;` D0; 0 if W ˇi;` ¤0: (12.14) Note that, whilst the harmonics are neglected here, they may be computed using the second-order normal form technique— see [13] for further details. It can be seen from Eq. (12.11) that the terms in ˇmay be separated into three categories: • Non-resonant, which are non-zero, regardless of the value of r, • Unconditionally-resonant, which are zero for all values of r, • Conditionally-resonant, which are only zero for specific values of r. Furthermore, from Eq. (12.11), it can be seen that, for this case, the conditionally-resonant terms become resonant for three different values of r, namely r D1=3, r D1 and r D3. Therefore, using Eqs. (12.11) and (12.14), the matrix of resonant coefficients, Œnu , may be found, from which Eq. (12.13) may be used to write
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