which replaces the three inversions by only one, restricted to the number of connection co-ordinates j, thereby limiting possible numerical problems. 2.2 FRF uncoupling Let us suppose that our goal is the dynamic characterization of a joint; joints can be considered as complex components, often difficult to analyze and model. A possible solution relies on obtaining their dynamic behavior by an inverse coupling procedure, i.e., an uncoupling procedure, having eq. (7) as a starting point. If our joint is defined as sub-structure B, one has co-ordinates i and j, whereas co-ordinates k (internal to B) do not play a role here. Therefore, eq. (7) is re-written as: 1 C C A A A ii ij ii ij ij A B A A jj jj ji jj C C A A A ji jj ji jj jj ª º ª º ª º ª º « » « » « » ¬ ¼ « » « » « » ¬ ¼ ¬ ¼ ¬ ¼ H H H H H H H H H H H H H H (8) From eq. (8) it is clear that there are three possibilities for the evaluation of B jj H : using Cii H , Cij H or C jj H . 2.2.1 Without the use of co-ordinates j Sometimes it may be difficult to undertake measurements on co-ordinates j, close to the joint, so ideally one should use measurements on the complete structure C involving only co-ordinates i. From (8), one has: 1 C A A A B A ii ii ij jj jj ji + H H H H H H (9) Rearranging (9), 1 A A B A A C ij jj jj ji ii ii + H H H H H H (10) Pre-multiplying (10) by 1 Aij H and post-multiplying by 1 Aji H leads to: 1 1 1 A B A A C A jj jj ij ii ii ji + H H H H H H (11) Note that this operation is mathematically possible only when i equals j; otherwise, if one of the (pseudo) inversions is possible, the other one is not, and vice-versa. Solving for B jj H , it follows that 1 B A A C A A jj ji ii ii ij jj H H H H H H (12) Generalizing to the case where i might be different from j (in fact, i j t ), one pre-multiplies equation (10) by an arbitrary matrix ji W and post-multiplies it by ijW : 1 A A B A A C ji ij jj jj ji ij ji ii ii ij + WHH H HWWH HW (13) Rearranging eq. (13), one can write, 1 1 1 A B A A C A jj jj ji ij ji ii ii ij ji ij + H H WH WH HWHW (14) This is only possible if i j t , not the other way around (which certainly is not a common case). From (14) one has: 1 B A A C A A jj ji ij ji ii ii ij ji ij jj H HWWH HW WH H (15) 385
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