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vector of measured control forces generated by m MR dampers; x g is ground acceleration; F is the
column vector of ones; and A is the matrix determined by the placement of MR dampers in the
structure. The displacement can be expressed as the linear combination
(0=*»,r = J.2,...,n (2)
where r\ r (r) is the r th modal displacement; fy r is the r th eigenvector; 0 is a eigenvector set; and 77 is a
modal displacement vector. In modal control, only a limited number of lower modes are controlled.
Hence, / controlled modes can be selected with I < n and the displacement may be partitioned into
controlled and uncontrolled parts as jr(r) = x c (r) + X R () , where x c and X R represent the controlled
and uncontrolled displacement vector, respectively. We refer to the uncontrolled modes as residual.
The eigenvectors are orthogonal and can be normalized so as to satisfy the orthononnality conditions.
Inserting Eq. (2) into Eq. (1), multiplying by$ r r and considering orthogonal condition between
eigenvectors, we obtain
(3)
where C are r modal damping ratios, ® r is a natural frequency. Then, Equation (3) can be rewritten in
state-space form such as
w c (t) = A c w c (t) + B c f(l) + E c x s , y c (t} = C c w c (t} (4)
where w c is a 2/-dimensional modal state vector by the controlled modes and
' °
-al -/ ''
/c I s B - - I E -
—
Er ~
~ ''
(5)
are the 21x21, 2Lxm matrixes and a 2/xl vector, respectively, and Ac is the diagonal matrix listing
2co r £ r ; ii 2 is the diagonal matrix listing fi),..., a$\ B c '=® T A\ and E c '=