[MUii [CU ii [K U ii [Qpii

Combining equations 4.10 and 4.12 and substituting them into equation 4.8 without the force term, gives:

[G]{p }i+1+[C 0]{p }i+1+[K0 ]{p}i+1+p[Q]T [M ]-1([M ]+y At[C ]+£ At2[K ]){U }i+1 =0

The modally decomposed system is represented by a single degree of freedom equation. The single degree of freedom equivalent of equations 4.17 and 4.18 will be obtained by substituting the mass, damping and stiffness values m, c and k instead of [M], [C] and [K] in equation 4.17 and g, c andk instead of [G], [C ] and [K ] in equation 4.18. The coupling matrix [Q] would be represented by scalar quantity q. The characteristic equation of the coupled field can be written by substituting equations 4.15 and 4.16 into equations 4.17 and 4.18 as follows:

where:

16mg 8mc' 8gc

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