This analysis type (accessed with ANTYPE,MODAL) is used for natural frequency and mode shape determination. The equation of motion for an undamped system, expressed in matrix notation using the above assumptions is:

(15–46)

Note that , the structure stiffness matrix, may include prestress effects (PSTRES,ON). For a discussion of the damped eigensolver (MODOPT,DAMP or MODOPT,QRDAMP) see Eigenvalue and Eigenvector Extraction.

For a linear system, free vibrations will be harmonic of the form:

(15–47)

where:

= eigenvector representing the mode shape of the ith natural frequency

= ith natural circular frequency (radians per unit time)

= time

Thus, Equation 15–46 becomes:

(15–48)

This equality is satisfied if either or if the determinant of is zero. The first option is the trivial one and, therefore, is not of interest. Thus, the second one gives the solution:

(15–49)

This is an eigenvalue problem which may be solved for up to n values of and n eigenvectors which satisfy Equation 15–48 where n is the number of DOFs. The eigenvalue and eigenvector extraction techniques are discussed in Eigenvalue and Eigenvector Extraction.

Rather than outputting the natural circular frequencies , the natural frequencies () are output; where:

(15–50)

where:

= ith natural frequency (cycles per unit time)

If normalization of each eigenvector to the mass matrix is selected (MODOPT,,,,,,OFF):

(15–51)

If normalization of each eigenvector to 1.0 is selected (MODOPT,,,,,,ON), is normalized such that its largest component is 1.0 (unity).

A discussion of effective mass is given in Spectrum Analysis.