The n second order modal equations (Equation 14–122) are transformed into 2n first order equations, where n is input as NMODE on the SPMWRITE command, using the following coordinate transformation:

(14–316)

The equation becomes:

(14–317)

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

(14–318)

Where ω_j is the frequency of mode j, ξ_j is the effective modal damping of mode j (see Modal Damping), and {F} is the vector of input forces:

(14–319)

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

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

(14–320)

With

(14–321)

Where [Φ] is the matrix of eigenvectors and [F_u] 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 {z} have been expressed as a function of the input loads, the equation for the degrees of freedom observed (outputs {w}) is written as:

(14–322)

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

(14–323)

with

(14–324)

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

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

(14–325)

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