S16-1 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation SECTION 16 COMPLEX MODAL ANALYSIS

S16-2 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation

S16-3 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation COMPLEX MODAL ANALYSIS n Complex modal analysis is used to compute the damped modes of structural models, or to assess the stability of systems modeled with transfer functions, e.g. rotating systems. n Mass and stiffness matrices may be unsymmetric and they may have complex terms.

S16-4 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation COMPLEX MODAL ANALYSIS (Cont.) n Create an equation for complex modes. The equation of motion of a system, with viscous damping, without any forcing is given as follows. Assume the following relationship between the displacement vector {u(t)} and modal vector { } n Substitute this equation into the other one

S16-5 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation COMPLEX MODAL ANALYSIS (Cont.) n This equation represents the complex eigenvalue problem n Numerous steps are done in MSC.Nastran to develop a corresponding eigenvalue equation. This equation is the one that is solved. The roots of the corresponding characteristic equation, i, can be real, purely imaginary, or complex. n Also, the corresponding eigenvectors can be complex

S16-6 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation COMPLEX MODAL ANALYSIS (Cont.) n The viscous damping matrix, [B], is the same as the one that is used for frequency response analysis n The direct method (SOL 107) solves the eigenvalue equation using d-set (physical plus extra DOFs) mass, damping, and stiffness matrices n The modal method (SOL 110) solves the eigenvalue equation using h-set (modal plus extra DOFs) mass, damping, and stiffness matrices n There are four methods of performing a complex modal analysis. They are u INV – inverse power u HESS – Upper Hessenberg u CLAN – Complex Lanczos

S16-7 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation COMPLEX MODAL ANALYSIS (Cont.) n The HESS method u Related to the GIV method u Requires nonsingular [M] u May be very time consuming to use for large problems. Therefore, the modal method is recommended, except for small problems where either the direct or modal method can be used. u Uses canonical equation forms l [B] = [0] l [B] not null

S16-8 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation COMPLEX MODAL ANALYSIS (Cont.) n The HESS method (continued) u Uses canonical equation forms l [B] not null (continued)

S16-9 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation COMPLEX MODAL ANALYSIS (Cont.) n The CLAN method u Similar to real Lanczos u Hybrid of a transformation and tracking method n The INV method u Related to real INV method u [M] can be singular u This method can be used for large problems. However, this method can be time consuming. u The user must select the search region in the complex plane u Less reliable that the HESS method

S16-10 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation COMPLEX MODAL ANALYSIS (Cont.) n A complex eigenvalue problem is specified as follows

S16-11 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation COMPLEX MODAL ANALYSIS (Cont.) n Executive section u Direct complex eigenvalue extraction, SOL 107 u Modal complex eigenvalue extraction, SOL 110

S16-12 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation COMPLEX MODAL ANALYSIS (Cont.) n Case Control section

S16-13 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation COMPLEX MODAL ANALYSIS (Cont.) n Bulk Data section

S16-14 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation COMPLEX MODAL ANALYSIS (Cont.) n Bulk Data section (continued)

S16-15 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation COMPLEX MODAL ANALYSIS (Cont.) n Bulk Data section (continued)

S16-16 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation COMPLEX MODAL ANALYSIS (Cont.) n Bulk Data section (continued)

S16-17 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation COMPLEX MODAL ANALYSIS (Cont.) n Bulk Data section (continued)

S16-18 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation COMPLEX MODAL ANALYSIS (Cont.) n Bulk Data section (continued)

S16-19 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation COMPLEX MODAL ANALYSIS (Cont.) n Bulk Data section (continued)

S16-20 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation COMPLEX MODAL ANALYSIS (Cont.) n Bulk Data section (continued)

S16-21 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation COMPLEX MODAL ANALYSIS (Cont.) n Bulk Data section (continued)

S16-22 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation COMPLEX MODAL ANALYSIS (Cont.) n Bulk Data section (continued)

S16-23 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation COMPLEX MODAL ANALYSIS (Cont.) n Bulk Data section (continued)

S16-24 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY: COMPLEX MODAL ANALYSIS n Calculate the complex frequencies and modes of the pile driver model Exciter Pile Ground M2M2 1.5 lbf*sec 2 /inch M3M3 3.0 lbf*sec 2 /inch K1K lbf/inch K2K lbf/inch C1C lbf*sec/inch K1K1 K2K2 C1C1 M2M2 M3M

S16-25 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY: COMPLEX MODAL ANALYSIS (Cont.) n Pile driver model u File name is pile_driver.bdf

S16-26 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY: COMPLEX MODAL ANALYSIS (Cont.) n Pile driver model (continued)

S16-27 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY: COMPLEX MODAL ANALYSIS (Cont.) n Eigenvalues, complex

S16-28 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY: COMPLEX MODAL ANALYSIS (Cont.) n Eigenvectors, complex

S16-29 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY: COMPLEX MODAL ANALYSIS (Cont.) n Eigenvectors, complex (continued)

S16-30 NAS122, Section 16, August 2005 Copyright 2005 MSC.Software Corporation