NASA Technical Memorandum 106836 k__S 1:" 7c Closed-Form Static Analysis With Inertia Relief and Displacement-Dependent Loads Using a MSC/NASTRAN DMAP Alter Alan R. Bamett and Timothy W. Widrick Analex Corporation Brook Park, Ohio Damian R. Ludwiczak Lewis Research Center Cleveland, Ohio Prepared for the 1995 World Users' Conference sponsored by the MacNeal-Schwendler Corporation Los Angeles, California, May 8-12, 1995 (NASA-TM-106836) CLOSED-FCRM N95-19649 STATIC ANALYSIS WITH INERTIA RELIEF AND DISPLACEMENT-DEPENDENT LOADS USING A MSC/NASTRAN OMAP ALTER NationalAeronauticsand Unclas SpaceAdministration (NASA. Lewis Research Center) 16 p 83139 0039661 CLOSED-FORM STATIC ANALYSIS WITH INERTIA RELIEF AND DISPLACEMENT-DEPENDENT LOADS USING A MSC/NASTRAN DMAP ALTER Alan R. Barnett and Timothy W. Widrick Analex Corporation 3001 Aerospace Parkway Brook Park, Ohio 44142 Damian R. Ludwiczak National Aeronautics and Space Administration Lewis Research Center Cleveland, Ohio 44135 Abstract Solvingforthedisplacementsof free-freecoupledsystemsactedupon by staticloadsiscommonly performed throughouttheaerospaceindustry.Many times,theseproblemsaresolvedusingstaticanalysiswithinertiarelief. This solutiontechniqueallowsfora free-freestaticanalysisby balancingthe appliedloadswithinertialoads generatedbytheappliedloads.Forsome engineeringapplicationst,hedisplacementsofthefree-freceoupledsystem induce additional static loads. Hence, the applied loads are equal to the original loads plus displacemem-dependent loads. Solving for the final displacements of such systems is commonly performed using iterative solution techniques. Unfortunately, these techniques can be time-consuming and labor-intensive. Since the coupled system equations for free-free systems with displacement-dependent loads can be written in closed-form, it is advantageous to solve for the displacements in this manner. Implementing closed-form equations in static analysis with inertia relief is analogous to implementing transfer functions in dynamic analysis. Using aMSC/NASTRAN DMAP Alter, displacement-dependent loads have been included in static analysis with inertia relief. Such an Alter has been used successfully to efficiently solve a common aerospace problem typically solved using an iterative technique. CLOSED-FORM STATIC ANALYSIS WITH INERTIA RELIEF AND DISPLACEMENT-DEPENDENT LOADS USING A MSC/NASTRAN DMAP ALTER Alan R. Barnett and Timothy W. Widrick Analex Corporation 3001 Aerospace Parkway Brook Park, Ohio 44142 Damian R. Ludwiczak National Aeronautics and Space Administration Lewis Research Center Cleveland, Ohio 44135 Nomenclature Abbreviations Matrices Set Notation DOF Degree-of-freedom F Applied forces a a-set (assembled DOF) DMAP Direct Matrix Abstraction I Identity g g-set (structural grid DOF) Program K Stiffness 1 l-set (left over DOF) ELV Expendable Launch M Mass r r-set (reference DOF) Vehicle P Applied loads uI Ul-Set (subset of 1-set DOF) T Displacements transformation y y-set (subset of l-set DOF) u Displacements a a-set (r-set + y-set DOF) ti Accelerations Steady-state accelerations 12 Steady-state accelerations transformation p Loads transformation Mode shapes Introduction Solving for the displacements of free-free coupled systems acted upon by static loads is commonly perfomwA throughout the aerospace industry. Such analyses areperformed for ELV/spacecraft systems during assumed-static, or quasistatic, phases of flight. For these flight event analyses, it is assumed that only steady-state loads act on the system and system transient responses have dampened out. Many times, these problems are solved using static analysis with inertia relief. This solution technique allows for a free-free static analysis by balancing the applied loads with inertia loads. Static analysis with inertia relief is offered in MSC/NASTRAN via Solution 91 [1]. For some engineering applications, the displacements of a free-fIee coupled system induce additional static loads. Hence, the applied loads are equal to the original loads plus displacement-dependent loads. Such is the case when analyzing ELV/spacecraft systems acted upon by quasistatic aerodynamic loads [2]. Quasistatic aerodynamic loads are generated as the system flies through the atmosphere at an angle-of-attack. The system static deformations cause local changes in the angle-of-attack which result in additional aerodynamic loads. The system will reach a state of 2 static equilibrium under the static-aeroelastic loading. Solving for the final displacements of such systems is commonly performed using iterative solution techniques. Unfortunately, these techniques can be time-consuming and labor-intensive. Since the free-free coupled system equations with displacement-dependent loads can be written in closed-form, it is advantageous to solve for the final displacements in this manner. The objective of this work was to develop a closed-form methodology for including displacement-dependent loads during static analysis with inertia relief using MSCfNASTRAN. Implementing closed-form equations in static analysis with inertia relief is analogous to implementing transfer functions in dynamic analysis in that induced load terms are added to the system stiffness resulting in a nonsymmea'ic ma_ix. A MSC/NASTRAN DMAP Alter has been developed for including displacement-dependent loads during static analysis with inertia relief. The Alter has been used successfully to efficiently solve a quasistatic ELV/spacecraft aerodynamic loads problem once solved using an iterative solution technique. Closed-form static analysis with displacement-dependent loads is illustrated in the next section. A simple example problem is solved to demonstrate the basic principles behind the development of the new Alter. In a subsequent section, the underlying theory of closed-form static analysis with inertia relief and displacement-dependent loads is described. Implementation of the theory within a MSCdNASTRAN DMAP Alter is then explained. Lastly, a quasistatic ELV/spacecraft aerodynamic loads problem is solved to demonstrate the accuracy of using the new Alter versus using an iterative solution technique. Closed-form Static Analysis with Displacement-dependent Loads As previously stated, the objective of this work was to develop a methodology using MSC/NASTRAN, whereby, a static analysis with displacement-dependent loads, typically solved using iterative solution techniques, could be solved in closed-form. To demonstrate the basic principles of closed-form static analysis with displacement- dependent loads, consider the three DOF system with applied forces shown in Figure 1. Applied forces fl and f2 are assumed constant. Applied force f3 is assumed to be a function of displacements uIand u2, or f3 ----C(Ul -- U2) (1) where c is a constant. The static equations describing the system are ikl+k 2 -k2 0 1 fl (2) 0 -k 3 k3+k 4] [U3J or (3) [K]{u} = {F} When not using a closed-form solution technique, the solution for the displacements {u} must be found using an iterative solution technique because the applied force f3 shown in Eq. (2) is not known. The system static equations rewritten for an iterative solution are "kl+l_ -k 2 0 ' (4) +11- ok k2+k3 -k3 -k3 k3+k4 ÷lj or [K]{ui+l}= {Fi} (5) where i signifies the iteration number (i = 0, 1, 2.... ). To begin solving for {u i+l } using an iterative solution technique, an initial value for applied force f3 is assumed. Let f3be equal to zero when i--0; hence, I! °ltfit (6) k2+k3 -k 3 ,_ = -k3 k3+k4 _ J A first set of system displacements {U1} is solved forusing Eq. (6). Once {u1}is solved for, the applied force f31 is calculated, the applied forces {Fi}arc updatod, and system displacements {U2} ale solved for. This procedure continues until the difference in system displacements between two successive iterations satisfies a convergence criterion, or (7) II{ui+l } - {Ui}ll < e (where e << 1) At this point, the solution hasconverged, and the system displacements are known. The displacements for the system shown in Figure 1 can also be solved for in closed-forna. Let a fourth DOF, u4, be definedasthedifference betweensystem displacements U1and u2,or U4 = Ul _ U2 (8) The displacement-dependent applied force f3can then be written as f3= cu4 (9) Ifthedefinitioofnu4isincludewdithinthesystemstiffnemsastrix[K]andthevariablaeppliedforcef3ismoved totheleft-hand-soifdtehesystemstatiecquationst,hesystemequationbsecome kl+k 2 -k2 0 0 u1 fl -k2 k2+k3 -k3 0 u2 f2 (10) 0 -k3 k3+k4 -c u3 0 -1 1 0 1 u4 0 Adding the fourth linearequation tothe system and moving the variable applied force term to the left-hand-side is the procedure for rewriting the iterative solution shown byEq. (4) as a closed-form solution. Equation (10) defines 4 aclosed-form static problem with displacement-dependent loads. The system displacements {u} can be solved for immediately. Before describing the general implementation of closed-form static analysis with inertia relief and displacement- dependent loads within MSC/NASTRAN Solution 91, itis beneficial to look at the example system stiffness matrix shown in Eq. (10). Note that the matrix is nonsymmetric. Adding the fourth DOF to the problem is analogous to adding extra points in MSC/NASTRAN dynamic analysis in that the once symmetric matrices become nonsymmetric. Partitioning [K] of Eq. (10) into original physical DOF and added DOF partitions, [KI1] [Kt2]] (II) [K] = [[K2'] [K22]J where kt+k -k2 0 --k3 (12) [KII]= -k2 k2+k 3 k3+k4 0 -k3 ., (13) [K21] = [-1 1 O] (14) and [K22] = {I] 05) The physical stiffness partition, [KI1], is equal to the original system stiffness matrix shown in Eq. (2). The two lower partitions, [K21] and [K22], contain elements of the linear equation defining the added DOF displacement shown by Eq. (8). The upper-right partition, [KI2], defines the variable force applied to the physical DOF due to a unit displacement of the added DOF shown by Eq. (9). In summary, static equations with displacement-dependent loads solved using an iterative solution technique can be solved in closed-form by: 1. Generating additional DOF and writing linear equations defining their displacements as functions of displacements of the physical DOF and including these equations within the system static equations. 2. Redefining the displacement-dependent load relationships as functions of the additional DOF displacements and including these relationships within the system static equations. These basic principles were used when developing a MSC/NASTRAN DMAP Alter to Solution 91 for performing closed-form static analysis with inertia relief and displacement-dependent loads. Closed-form Static Analysis with Inertia Relief and Displacement-dependent Loads Including displacement-dependent loads during static analysis with inertia relief was implemented within MSC/NASTRAN Solution 91 using a DMAP Alter. The underlying theory upon which the Alter is based stems from the basic concepts presented in the previous section. Special considerations were made for including inertia relief effects during development of the Alter. The goal of the development effort was to generate closed-form static equations of the form shown by Eq. (10) to efficiently perform static analyses with inertia relief and displacement- dependentloads. Before developing closed-form static equations with inertia relief and displacement-dependent loads, it is beneficial to review the basic theory of static analysis with inertia relief. The equations-of-motion for all DOF of aflee-free coupled system under a steady-state loading condition are (16) [Mgg]{iig} + [Kgg] {ug} = {]'g} After accounting for DOF, defined via multi-point and single-point constraints, the system equations are reduced from g-set size to a-set size [1]; hence, [Maa]{iia}+ [Kaa]{Ua}= {Pa} (17) To solve for the displacements of the free-free system represented by Eq. (17), an inertia relief solution technique can be used because the original system stiffness matrix is singular [3]. The a-set DOF are the union of statically- determinate reference DOF (r-set) and the complement of the r-set DOF (l-set). Writing Eq. (17) in partitioned form, (18) [Mlr] [MII]J [{iil) J + L[Kir] [KIi] jL{BI }J ----[{i)l} J Under the steady-state loading condition, the system deforms elastically and accelerates as arigid-body. Using a rigid-body transformation [4], the a-set DOF steady-state accelerations are written as (19) {fir} l " [In] ] {i/l}j = _[Ku]_Z[KIr] {fir} = [Offi]{Ur} Using Eq. (19), the system equations shown by Eq. (18) are rewritten as (20) [Krr] [Kll]JL{ul} L{_.) t-- L_'] [K_] PremultiplyingEq. (20)by [Oar]T, {Or}= [Oar]T{Pa} (21) = [_]T{i'a} Solving for the r-set DOF steady-state accelerations {fir} from Eq. (21), (22) {fir}= [Mrr]-I[Oaf]T{Pa} 6 Because the free-free system has rigid-body modes, displacements {Ur}can be set arbitrarily. Letting {Ur}={Or}, the lower partition of Eq. (20) becomes (23) [KII]{Ui} - {PI} - [Mlal[Oar]{iir} where [Mla] is the lower partition of the mass matrix of Eq. (18). Defining the right-hand-side of Eq. (23) as {PI}, Eq. (23) simplifies to [Kll]{Ul} = {PI} (24) Equation (24) is the inertia relief solution for the l-set DOF displacements shown originally in Eq. (18). Now let the static equations with inertia relief be expanded to include displacement-dependent loads. Beginning with the a-set DOF equations, (25) [Maa]{iia} + [Kaal{Ua} = {Pa} + [Paallua} where [P_a] is a matrix for transforming the a-set DOF displacements into loads acting on the system. Rewriting Eq. (25) according to r-set and 1-set DOF partitions, r--1I<,'Il'!'l (26) [Mir] [Mil]J[{iil} + l[Klr] [Ku]J[{ul} J = I{P,} j + [_)ll]J[{ul} J To facilitate further development, let the l-set DOF be the union of user-defined DOF (Ul-Set) and the complement of the Ul-set DOF (y-set). The Ul-set DOF have no mass or stiffness and are added to the system DOF for applying the displacement-dependent loads. These DOF are analogous to extra points used to define transfer functions in MSC/NASTRAN dynamic analysis. The y-set DOF are those of the original system model. Let the displacement- dependent loads be defined solely by the ul-set DOF displacements, or (27) [_)aa ] {U a} --'_ [PaUl] {nil I} This is accomplished by defining {uul } via a set of linear equations which describe the system displacement dependencies, or I ur}] [[Tuf]f [.1.,]ul[yTutut][l{Uy}I= {0ul} (28) Taking into account the y-set and ui-set partitions of the l-set DOF, Eq. (26) is rewritten as Mo..01of.l.r.EUKrE.oo. .l1/.o. It0. 0' o o. .ll.0. L[O.,[ro].,.][o.,.,]Jt(°.,_L!m,.][%] m,=J,][_,,JJ lo_,j L[%]to.] [%,]Jli_,_J (29) As before, the system deforms elastically and accelerates as arigid-body under the steady-state loading condition. Using arigid-body transformation [4], the a-set DOF steady-state accelerations are written as (30) = [O_r]{iir} I{iio} ] I{iir}l [ [Irr] 1 r[Oor] 1 [{OnI} = l{iiy} I = I-[Kyy]-I [Kyr]_{iir} = [[0uff] J{iir} [_o.,Lljto.,.j] where the o-set DOF are the union of the r-set and y-set DOF (complement of Ul-Set in a-set). Using Eq. (30), the system equations shown by Eq. (29) are rewritten as [Krr] [Kry] [0m,] I ur}] {Pr}1 '_/y÷' [Ky_] [Kyy] [0yu_] {Uy}I = Frur,] [T.,y]LTu,u,]{%}J iO-l}j Premultiplying Eq. (31) by [_ffi]T, {Or}= [OffilT{_, } + [O.r]Z[_au,]iUuz} _ [Ofr]'r[Maa][_ffi]{Ur} (32) = [_ar]T{i})a+ [_ar]T[[)a{uU,u],}- [Mrr]{iir} Solving for the r-set DOF steady-state accelerations {iJr} from Eq. (32), {fir}= [Mrr1-I[(_ar]T{Pa } + [Mrr1-I[_]T[f)au,]{Uu,} (33) - {_]r}+ [J'Lrull{}Uul In Eq. (33), {1'Jr} are the steady-state accelerations due to the directly applied loads, and [gml] is a matrix for transforming system displacements into additional steady-state accelerations. Because the free-free system has rigid-body modes, {Ur} can be set arbitrarily. Letting {ur}={0r}, the lower partition of Eq. (31) becomes