The n second order modal equations (Equation 14–142) are transformed into 2n first order equations, where n is input as NMODE on the SPMWRITE command, using the following coordinate transformation:

(14–360)

The equation becomes:

(14–361)

is a (2n x 2n) state-space matrix defined by:

(14–362)

Where is the frequency of mode , is the effective modal damping of mode (see Modal Damping), and is the vector of input forces:

(14–363)

Where ninput is the number of scalar input forces derived from Inputs on the SPMWRITE command.

is a (2n x ninput) state-space matrix defined by:

(14–364)

With

(14–365)

Where is the matrix of eigenvectors and is a unit force matrix with size (ndof x ninput). It has 1 at the degrees of freedom where input forces are active and 0 elsewhere.

Now that the states have been expressed as a function of the input loads, the equation for the degrees of freedom observed (outputs ) is written as:

(14–366)

is a (3*noutput x 2*n) state-space matrix, where noutput is derived from Outputs on the SPMWRITE command, and is defined by:

(14–367)

with

(14–368)

is a unit displacement matrix with size (noutput x ndof). It has 1 on degrees of freedom where output is requested and 0 elsewhere.

is a (3*noutput x ninput) state-space matrix defined by:

(14–369)

and are included only if VelAccKey = ON on the SPMWRITE command, otherwise the last two rows of are not written and is zero so it is not written.