Now the order of the matrix to invert is equal to the number of co-ordinates j. This is advantageous because the size of j will be – in general – much smaller than that of i. The question is which matrix ijW to use. There is not an easy answer, but probably the most logical one is to use Aij H : 1 B A A A A C A A A A jj ji ij ji ii ii ij ji ij jj H HHHH HH HH H (16) 2.2.2 Using only the coordinates of the joint Using eq. (7) based on C jj H , one has: 1 C A A A B A jj jj jj jj jj jj H H H H H H (17) Solving eq. (17) with respect to B jj H , it follows that: 1 B A A C A jj jj jj jj jj jj H H H H I H (18) Based on the dynamic stiffness matrix of the structure, Ambrogio [1] obtains an alternative formulation (eq. (19) can be derived from eq. (18)): 1 -1 B C A C jj jj jj jj jj H I H H H (19) 2.2.3 Using coordinates i and j Back again to eq. (8), the third option to evaluate B jj H starts from: 1 C A A A B A ij ij ij jj jj jj + H H H H H H (20) or 1 A A B A A C ij jj jj jj ij ij + H H H H H H (21) Using once again arbitrary matrices, now ji W and jj W , with i greater or equal to j, one has 1 A A B A A C ji ij jj jj jj jj ji ij ij jj + WHH H HW WH HW (22) from which 1 -1 -1 A B A A C A jj jj ji ij ji ij ij jj jj jj + H H WH WH HWHW (23) Solving in order to B jj H , gives: -1 B A A C A A jj jj jj ji ij ij jj ji ij jj H HWWH HW WH H (24) As before, one shall make A ji ji W H and A jj jj W H : -1 B A A A A C A A A A jj jj jj ji ij ij jj ji ij jj H HHHH HH HH H (25) 386
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