The modal solution obtained using the complex eigensolvers (UNSYM, DAMP, QRDAMP) and the solution from a harmonic analysis are complex. It can be written as
 | (17–153) |
where:
| R = the complex degree of freedom solution (a nodal displacement Ux, a reaction force Fy, etc.). |
| RR = the real part of the solution R. |
| RI = the imaginary part of the solution R. |
The same complex solution may also be expressed as:
 | (17–154) |
where:
| Rmax = the degree of freedom amplitude. |
| φ = the degree of freedom phase shift. |
The phase shift of the solution is different at each degree of freedom so that the total amplitude at a node is not the square root of the sum of squares of the degrees of freedom amplitudes (Rmax). More generally, total amplitudes (SUM), phases and other derived results (principal strains/stresses, equivalent strain/stress,… for example) at one node do not vary harmonically as degree of freedom solutions do.
The relationship between RR, RI, Rmax and φ is defined as follows:
 | (17–155) |
 | (17–156) |
RR = Rmaxcosφ
RI = Rmaxsinφ
In POST1, use KIMG in the SET command to specify which results are to be stored: the real parts, the imaginary parts, the amplitudes or the phases.
In POST26, use PRCPLX and PLCPLX to define the output form of the complex variables.
The complete complex solution is harmonic. It is defined as:
 | (17–157) |
where:
| Ω = the excitation frequency in a harmonic analysis, or the natural damped frequency in a complex modal analysis. |
In the equations of motion for harmonic and complex modal analyses, the complex notations are used for ease of use but the time dependant solution at one degree of freedom is real.
The solution of the harmonic analysis is defined as:
 | (17–158) |
The ANHARM command issued after the harmonic
analysis is based on Equation 17–158. The HRCPLXand LCOPER (with Oper2 = CPXMAX) commands
are based on Equation 17–158 for both modal and
harmonic analyses.
The complete solution of the complex modal analysis is defined as:
 | (17–159) |
Where σ is the real part of the complex frequency. See Complex Eigensolutions for more details about complex eigensolutions.
The ANHARM command issued after a complex modal analysis is based on Equation 17–159 except if NPERIOD is set to -1. In this case, the exponential term (decay or growth of the oscillation) is ignored and the equation reduces to Equation 17–158.
In the LCOPER command with Oper2 = CPXMAX, a
loop on the phase (Ωt), is performed to calculate the maximum (that may occur at any phase
at each location) of the degree of freedom results, displacements and stresses, the principal
stresses (S1, S2, and
S3), the stress intensity (SINT), the
equivalent stress (SEQV), the equivalent strain, and the principal strains.