S2-1 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation SECTION 2 NORMAL MODES ANALYSIS

S2-2 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation

S2-3 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation OVERVIEW n The previous section looked at an SDOF problem of a spring mass system. n This section looks at normal modes analysis of multi degree-of-freedom problems and how to define these problems using MSC.Patran and MSC.Nastran. n Steps to follow are u Create a 2 DOF equation of motion using engineering approach. u Summarize some important ideas about normal modes that emerge. u Define the same problem using a matrix approach. u Build the example using MSC.Patran and MSC.Nastran.

S2-4 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation n Consider the system with 2 masses and 3 spring stiffnesses as shown. n Use an engineering approach to solve the equations of motion. u First, create free body diagrams for the masses. u Equating the Inertia and Elastic terms l For 1st mass l For 2nd mass 2 DOF EQUATION OF MOTION USING AN ENGINEERING APPROACH k k k 2MM x1x1 x2x2 M kx 1 k( x 2 - x 1 ) kx2kx2

S2-5 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation 2 DOF EQUATION OF MOTION USING AN ENGINEERING APPROACH (Cont.) Assume the motion of x 1 and x 2 is harmonic so n The object is to solve for the frequency and amplitudes. u Now n Substituting these equations into the equations obtained from the free-body diagrams the following is found u For the 1st mass: l So u For the 2nd mass: l So This means they vibrate at the same frequency but have different amplitudes A 1 and A 2.

S2-6 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation 2 DOF EQUATION OF MOTION USING AN ENGINEERING APPROACH (Cont.) n Assemble these two equations in matrix form, the result in: 3 unknowns exist, 2 and the pair of amplitudes

S2-7 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation 2 DOF EQUATION OF MOTION USING AN ENGINEERING APPROACH (Cont.) Solve this by using the determinant of the above equation, letting 2 =. Two roots of the equation are found as 1 and 2. u These roots are called eigenvalues. So the two frequencies where the inertia and elastic terms balance are 1 and 2.

S2-8 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation 2 DOF EQUATION OF MOTION USING AN ENGINEERING APPROACH (Cont.) n The amplitudes are found by substituting the frequencies into the equations of motion. n In turns out that only the ratio of the amplitudes can be determined. n This is an important physical point in an analysis to obtain normal modes. The absolute value of the amplitudes cannot be determined, only the value of the amplitudes relative to another amplitude.

S2-9 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation 2 DOF EQUATION OF MOTION USING AN ENGINEERING APPROACH (Cont.) n Arbitrarily set A 2 = The relative amplitudes are expressed as normal modes or eigenvectors Mode 1 Mode

S2-10 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation SUMMARIZING SOME IMPORTANT IDEAS ABOUT NORMAL MODES THAT EMERGE n The motion of all displacements is assumed harmonic. n Resonance is found at a set of natural frequencies where the inertia terms balance the elastic terms. n The natural frequencies are calculated by an eigenvalue method n The relative amplitude, or normal mode, is found for each natural frequency.

S2-11 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation n Now consider the system in terms of a matrix solution. n The individual element stiffness matrices [K 1 ],[K 2 ], and [K 3 ] are SETTING THE SAME PROBLEM USING A MATRIX APPROACH

S2-12 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation SETTING THE SAME PROBLEM USING A MATRIX APPROACH (Cont.) n To derive the model stiffness matrix [K], assemble the individual element stiffness matrices [K 1 ],[K 2 ], and [K 3 ] n Constrain out DOFs 1 and 4 as they are set to 0.0 DOF: 1234 and Lump mass at DOFs

S2-13 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation SETTING THE SAME PROBLEM USING A MATRIX APPROACH (Cont.) n The equation of motion in matrix form is: n Substitute n And n Then n So This means a normal mode, { }, and corresponding frequency can be found, where the inertia and elastic terms balance. This means there is a normal mode, { }, which varies sinusoidal with frequency and time, t. The Eigenvalue problem

S2-14 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation SETTING THE SAME PROBLEM USING A MATRIX APPROACH (Cont.) n So at, the motion is defined by: u is in balance at this first resonant or natural frequency. n And at, the motion is defined by: u is also in balance at this second resonant or natural frequency

S2-15 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation SETTING THE SAME PROBLEM USING A MATRIX APPROACH (Cont.) n Let us add some values in and check out the numbers u Let k = 1000 units of force / length u Let m = 20 units of mass n Then u Notice the conversion of Frequency from radians/sec to cycles/sec (Hertz)

S2-16 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation SETTING THE SAME PROBLEM USING A MATRIX APPROACH (Cont.) n Load this model with a time dependent set of forces at DOF 2 and 3. This results in a net displacement response which is a combination of the response of the two calculated normal modes. n For an n DOF system u For this case The scaling factor i for each normal mode { i } is called the modal displacement. There will be further reference to this when loading is applied in later sections

S2-17 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE n Model the system using MSC.Patran and MSC.Nastran n Allocate arbitrary dimensions to the model as above and input the nodes and spring and mass elements directly into Patran. K K K DOF: 1234 MM

S2-18 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE Create the grids using Create/Node/Edit

S2-19 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE Create the Bar2 generic Patran Elements for the Springs using Create/Element/Edit

S2-20 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE Create the Point generic Patran Elements for the Masses using Create/Element/Edit

S2-21 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE Define the specific MSC.Nastran type of Spring via the Physical Property definition: Create/1D/Spring Input the name Input the properties (1000 force/length units) Select the bar2 elements

S2-22 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE n The MSC.Nastran.bdf file will contain the connectivity definition of the CELAS elements and the PELAS property definition. $ Elements and Element Properties for region : spring_stiff PELAS $ Pset: "spring_stiff" will be imported as: "pelas.1" CELAS CELAS CELAS ID PID End A Grid DOF End B Grid Stiffness PID

S2-23 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE Define the specific MSC.Nastran type of Mass via the Physical Property definition: Create/0D/Mass Input the name Input the properties 20 and 40 mass units Choose the Lumped option Select the point elements

S2-24 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE n The MSC.Nastran bdf file will contain the CONM2 elements, there is one for each mass point. $ Elements and Element Properties for region : m1 CONM $ Elements and Element Properties for region : m2 CONM IDGridMass

S2-25 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE GRID: fixedmake_1d Two constraint sets are created fixed: constrains the ends in all DOF make_1d: constrains all DOF except x at GRIDS 2 and 3

S2-26 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE The Analysis Type is selected as Normal Modes (SOL103) Solution Parameters are set: Mass Calculation is Lumped Wt. Mass Conversion is 1.0

S2-27 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE n The MSC.Nastran.bdf file will contain the solution sequence definition and implied parameters for weight mass conversion and choosing lumped mass. n The implications of the parameters will be discussed later.

S2-28 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE $ Normal Modes Analysis, Database SOL 103 TIME 600 $ Direct Text Input for Executive Control CEND SEALL = ALL SUPER = ALL TITLE = MSC.Nastran job created on 10-Jan-02 at 12:52:18 ECHO = NONE MAXLINES = BEGIN BULK PARAM,WTMASS,1.0 {do not appear as these are the defaults} PARAM,COUPMASS,-1 {do not appear as these are the defaults}...

S2-29 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE A Subcase is created and Parameters defined: MSC.Nastran will select the upper and lower bound on frequencies found using Lanczos. No of roots is 2 Use the Maximum method in Normalization

S2-30 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE n The MSC.Nastran.bdf file contains the Subcase Definition and the Lanczos data entry... SUBCASE 1 $ Subcase name : Default SUBTITLE=Default METHOD = 1 SPC = 2 VECTOR(SORT1,REAL)=ALL... EIGRL MAX... a b ed a.Eigenvalue Set definition b.Eigenvector Output Request c.EIGRL keyword indicates the Lanczos method will be used A description of the Lanczos method occurs later d.ID of Lanczos request e.Number of Roots = 2 f.Normalization Method (Maximum) cf

S2-31 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE n Once the analysis carried out, inspect the MSC.Nastran F06 File: R E A L E I G E N V A L U E S MODE EXTRACTION EIGENVALUE RADIANS CYCLES GENERALIZED GENERALIZED NO. ORDER MASS STIFFNESS E E E E E E E E E E+03 ab The MSC.Nastran.f06 file has an Eigenvalue Summary. Important pieces of information in this summary are: a.Values of the Eigenvalues in (radians/sec) 2. b.The Natural Frequencies in hz and Radians/Sec. b

S2-32 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE n The values agree with those calculated earlier. The meaning of the Generalized Mass and Stiffness will be discussed in the next few pages.

S2-33 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE n The Eigenvector results for the 2 modes are also shown in the F06 file. 0 SUBCASE 1 EIGENVALUE = E+01 CYCLES = E-01 R E A L E I G E N V E C T O R N O. 1 POINT ID. TYPE T1 T2 T3 R1 R2 R3 1 G G E G E G MSC.NASTRAN JOB CREATED ON 10-JAN-02 AT 12:52:18 JANUARY 10, 2002 MSC.NASTRAN 6/11/01 PAGE 8 DEFAULT 0 SUBCASE 1 EIGENVALUE = E+02 CYCLES = E+00 R E A L E I G E N V E C T O R N O. 2 POINT ID. TYPE T1 T2 T3 R1 R2 R3 1 G G E G E G Eigenvector 1 clearly matches the hand-calculation from before, but Eigenvector 2 does not. The reason is because of a difference in the way MSC.Nastran and the hand calculation normalize the results.

S2-34 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE n The difference in the normalization methods between the hand calculation and MSC.Nastran is interesting: u In the hand calc. displacement was normalized at Grid 3 to quite arbitrarily u The MSC.Nastran method selected in this run was Maximum. This means the maximum value in the e-vector list is set to for each e-vector, so u This emphasizes all that is known about the e-vectors in a Normal Modes analysis is their relative values – the shape is known, but not the amplitude. u There is another commonly used normalization method called Mass normalization that will be discussed.

S2-35 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE n Mass normalization is a useful way of normalizing an Eigenvector because it can be thought of as a universal standard. Scale { } so that for each mode n If there is comparison between modes of different analyses, or even to test data, then it becomes meaningful to compare the mass normalized eigenvectors as

S2-36 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE The term { } T [M]{ } is called the Generalized Mass. u It is clearly orthogonal if you can equate it to an Identity matrix in Mass Normalization. l In general, orthogonality is defined as Generalized Stiffness, { } T [K]{ }, can be defined in a similar way

S2-37 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE n Reviewing the Generalized Stiffness and Mass terms from the current example, it is obvious that the Eigenvector is not mass normalized in this analysis: R E A L E I G E N V A L U E S MODE EXTRACTION EIGENVALUE RADIANS CYCLES GENERALIZED GENERALIZED NO. ORDER MASS STIFFNESS E E E E E E E E E E+03

S2-38 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE n It is difficult to visualize the mode shapes by looking at printed e-vectors, so Patran and its deformation plots and animation methods will be used to help understand the behavior. n The analysis just carried out will create an XDB file containing binary results data. Attach that to the Patran database, so the results can be viewed.

S2-39 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE The results file mass2_k3_ex.xdb is attached to the database

S2-40 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE Select Results: Create/Deformation and select the first Modal result. The deformation we want to plot is Eigenvector: Translational Do not hit Apply yet because we want to set up some plotting options. Select the Plot Options Icon See next slide for Plot Options

S2-41 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE Using the Plot Options form we can: Increase the line width of the spring elements Switch off the Undeformed plot option, to make things clearer Switch off Maximum Label

S2-42 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE 1. The size of nodes can be increased to help visualization. Select the Node Size Increase Icon from the Main Toolbar. This icon is magnified for clarification. 2. Notice the PCL call in the History Window 3. Copy this to the Command line by clicking on it 4. Modify it to read node_size(30) and hit Enter 5. Now select the Animation Options

S2-43 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation 1. In Animation Options increase the Number of Frames to 20 to get a smooth animation 2. Check the Animate box and hit Apply The first Mode will be animated Repeat for the second Mode The Tutor will play the mpeg files if available CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE 1 2

S2-44 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation n Please now carry out Workshop 2 in the Workshop Section to allow you to set up this model and visualize the results. n The workshop will take you through step by step if you are unfamiliar with MSC.Nastran or MSC.Patran. n If you have some experience, then try to set up the analysis without referring to the step by step guide. n Please feel free to ask your tutor for help. WORKSHOP 2 NORMAL MODES ANALYSIS OF A 2 DOF STRUCTURE

S2-45 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation EXTENDING TO MULTI DOF PROBLEMS n Consider normal modes of multi degree-of-freedom (MDOF) problems like the beam shown below. u The beam model here contains 54 DOFs. n The linear dynamic equation for an MDOF system Built in at both ends

S2-46 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation THEORETICAL APPROACH FOR MDOF n Consider an MDOF system w/o damping and no force n Assume the following harmonic solution u Physically, this means that all the coordinates perform synchronous motions. The system configuration does not charge its shape during motion, only its amplitude. n From this equation

S2-47 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation THEORETICAL APPROACH FOR MDOF (Cont.) n Substituting the latter two equations into the first equation the following is obtained After factoring-out e i t, the remaining term is seen to be equal to zero u This equation is the foundation for the eigenvalue problem. Solve for { } as follows

S2-48 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation n There are two cases for the eigenvalue problem u If det the solution is trivial u When det the solution is non-trivial, and a physically meaningful solution can be obtained n The eigenvalue problem is then obtaining a solution to for all possible values of, i, where k = k 2 Once the eigenvalues; 1, 2, …; are obtained, the corresponding eigenvectors; { 1 }, { 2 }, …; are obtained u There are numerous methods for obtaining the eigenvalues and eigenvectors for a model THEORETICAL APPROACH FOR MDOF (Cont.)

S2-49 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation If the structure has n degrees-of-freedom with associated mass, there will be n eigenvalues/natural frequencies, k. The natural frequencies are sometimes called characteristic frequencies or resonant frequencies. An eigenvector, { k } (there is an associated natural frequency, k ), is frequently called a normal mode. A normal mode corresponds to a deflected shape (pattern) of a structure. n When a structure is vibrating linearly its shape, at any given time, is a linear combination of its normal modes. THEORETICAL APPROACH FOR MDOF (Cont.)

S2-50 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation n Example THEORETICAL APPROACH FOR MDOF (Cont.)

S2-51 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation FACTS REGARDING NORMAL MODES n When [K] and [M] are symmetric and real (this is the case for all the standard structural finite elements), the following orthogonality properties hold u and u also l The units of the natural frequencies are radians/second. The natural frequencies can also be expressed in Hertz (cycles/seconds), using

S2-52 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation n For practical considerations, modes should be normalized by a chosen convention. In MSC.Nastran there are three methods of normalization, except when using Lanczos. u The generalized masses are each normalized to be 1.0 l This is the default method u The value of the largest a-set component for each mode is normalized to 1.0 u The value of a specific component for each mode is normalized to 1.0 (not recommended) n For the Lanczos method, normalization is done using a unit value of generalized mass, or a unit value of the largest a-set component. FACTS REGARDING NORMAL MODES (Cont.)

S2-53 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation EXTENDING TO MULTI DOF PROBLEMS (Cont.) n When a normal modes analysis is run, results are obtained as shown in the table on the next page. Built in at both ends

S2-54 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation EXTENDING TO MULTI DOF PROBLEMS (Cont.) n MSC.Nastran.F06 excerpt showing Eigenvalue table R E A L E I G E N V A L U E S MODE EXTRACTION EIGENVALUE RADIANS CYCLES GENERALIZED GENERALIZED NO. ORDER MASS STIFFNESS E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+07 a.Note the generalized mass is 1.0, this is a result of Mass Normalization Method being chosen, as explained earlier. a

S2-55 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation EXTENDING TO MULTI DOF PROBLEMS (Cont.) n It is important to be able to describe all the modes. Patran is used to identify and characterize the modes as described in the table. u The modes increase in complexity for a given type of mode. This leads us to the use of a subset of modes to describe the physical behavior. ModeF (Hz)Description st bend xz plane nd bend xz plane st bend xy plane rd bend xz plane nd bend xy plane th bend xz plane th bend xz plane rd bend xy plane th bend xz plane th bend xz plane

S2-56 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation EXTENDING TO MULTI DOF PROBLEMS (Cont.) n Modes in XZ plane Mode 1 = Hz Mode 2 = Hz Mode 4 = HzMode 6 =

S2-57 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation n Modes in XZ plane (cont.) EXTENDING TO MULTI DOF PROBLEMS (Cont.) Mode 7 = 595 HzMode 9 = 809 Hz Mode 10 = 1027 Hz

S2-58 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation n Modes in XY plane EXTENDING TO MULTI DOF PROBLEMS (Cont.) Mode 3 = HzMode 5 = Hz Mode 8 = Hz

S2-59 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation EXTENDING TO MULTI DOF PROBLEMS (Cont.) n Remember, the contribution of each mode is the product of the mode and its modal coordinate u So u For the 2 DOF case u For the beam example, it may be possible to represent the response to loading in the XZ plane only using the first two modes. The assumption is that the higher modes do not contribute significantly to the solution. This is a significant advantage of modal methods, the response of the beam, {x(t)}, has 54 physical degrees of freedom. But the response can be represented by 2 Modal DOF, i n EXTREME CARE must be taken when assuming which modes contribute and this will be discussed more in later sections. u As a taster consider the following over page

S2-60 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation EXTENDING TO MULTI DOF PROBLEMS (Cont.) n Possible Problems with 2 modes only: u If loading in the XY plane exists, it simply will not be able to be represented and will therefore be ignored u If the XZ loading excites at multiple inputs as shown, then the first two modes may not represent the response, and the mode number 4 (the third bending mode in the x-z plane) may be needed

S2-61 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation EIGENVALUE EXTRACTION METHOD n Now consider the method of solving the Eigenvalue problem in MSC.NASTRAN. n 3 types of methods for eigenvalue extraction are available: u Tracking Methods l Eigenvalues (or natural frequencies) are determined one at a time using an iterative technique. Two variations of the inverse power method are provided, INV and SINV. In general, SINV is more reliable than INV. u Transformation Methods The original eigenvalue problem([K] - [M]) { } = 0 is transformed to the form[A] { } = { } where [A] = [M] -1 [K] l Then the matrix A is transformed into a tridiagonal matrix using either the Givens technique or the Householder technique. Finally, all the eigenvalues are extracted at once using the QR algorithm. Two variations of the Givens technique and two variations of the Householder technique are provided: GIV, MGIV, HOU, and MHOU. These methods are more efficient when a large proportion of eigenvalues are needed.

S2-62 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation EIGENVALUE EXTRACTION METHOD (Cont.) u Lanczos Method (recommended method) l This method is a combined tracking-transformation method l Features are: n A trial eigenvalue (called a shift point) is assumed, and an attempt is made to extract all of the eigenvalues close to this value. n It is called a block method because it extracts several eigenvectors within a frequency block close to the trial eigenvalue. n A Sturm sequence check is made at each of the shift points to determine the number of eigenvalues below that shift point. n This information is used to determine when all of the eigenvalues of interest have been found.

S2-63 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation STURM SEQUENCE THEORY n Tracking method example: Sturm Sequence Theory Choose. Factor ([K - i M]) into [L][D][L T ] The number of negative terms on the factor diagonal is the number of eigenvalues below.

S2-64 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation LANCZOS METHOD n Real Eigenvalue Extraction Data, Lanczos Method u Defines data needed to perform real eigenvalue (vibration of buckling) analysis with the Lanczos method. u Example EIGRLSIDV1V2NDMSGLVLMAXSETSHFSCLNORM EIGRL This example requests 10 Eigenvalues calculated between 0.1 and 3.2 Hz using the Lanczos method.

S2-65 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation LANCZOS METHOD (Cont.) n Field Contents: u SID: Set identification number. (Unique Integer > 0) u V1, V2: The V1 field defines the lower frequency bound; the V2 field defines the upper frequency field. (Real or blank, ) l For vibration analysis: frequency range of interest. l For buckling analysis: eigenvalue range of interest. ND Number of eigenvalues and eigenvectors desired. (Integer > 0 or blank) u MSGLVL: Diagnostic level. (0 Integer 4; Default = 0) u MAXSET: Number of vectors in block or set. (1 < Integer < 15; Default = 7) u SHFSCL: Estimate of the first flexible mode natural frequency. (Real or blank) u NORM: Method for normalizing eigenvectors (Character: "MASS" or "MAX") u MASS: Normalize to unit value of the generalized mass. l Not available for buckling analysis. (Default for normal modes analysis.) u MAX: Normalize to unit value of the largest displacement in the analysis set. l Displacements not in the analysis set may be larger than unity. (Default for buckling analysis.)

S2-66 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation LANCZOS METHOD (Cont.) n Tips when using Lanczos method: u If User Fatal Message 5299 is reported, note that this is often caused by a massless mechanism, such as a grid point floating in space or a BAR not connected to the rest of the structure in torsion. The DOF should be eliminated or mass added as appropriate. u Another potential error, which may or may not indicate a modeling problem, may occur when a shift point is chosen too close to an actual eigenvalue. l So, if UFM 5299 occurs and massless mechanisms are not present in the model, try adjusting the frequency range on the EIGRL entry. This adjustment forces the Lanczos method to use different shift points that may result in a better numerically conditioned solution. u If rigid body modes are not present or are not needed, it is sometimes helpful to increase V1 to a small positive number. u If the Lanczos method cannot find all of the roots after modifying V1 and V2 several times, this generally indicates a modeling problem.

S2-67 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 2 – NORMAL MODES ANALYSIS OF A SATELLITE

S2-68 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 2 – NORMAL MODES ANALYSIS OF A SATELLITE Import a MSC.Nastran.bdf file

S2-69 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 2 – NORMAL MODES ANALYSIS OF A SATELLITE Select the Solution Type and enter the correct Weight-Mass conversion value for the Normal Modes Analysis.

S2-70 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 2 – NORMAL MODES ANALYSIS OF A SATELLITE n This model uses a value of as the weight mass conversion parameter - what does this mean? u MSC.Nastran requires consistent units. u Some systems of Units (including the US system) define density as being a Weight Per Unit Volume (eg. lbs/in 3 ). l This is not a consistent unit if used with loads of lbf and dimensions of inches. u The weight mass conversion parameter converts weight mass units to mass units by scaling by the appropriate units of acceleration due to gravity. n So for our model defined in lbs and inches, g = in/s 2 u PARAM,WTMASS, converts the mass of the structure to the correct units of (lbf/in/s^2). u Some industries also mix SI units for convenience, so density may be given in N/m 3 instead of the correct term Kg/m 3 l In this case PARAM,WTMASS,0.102 will scale by g = 9.81m/s^2

S2-71 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 2 – NORMAL MODES ANALYSIS OF A SATELLITE n Another common method used in the industry is to have length measured in mm for convenience, but still want to apply forces in N. n When converting a non-standard system of units the golden rule is to apply Newtons Law of Motion and then dimensional equivalence u Force = Mass * Acceleration l N = (mass units) * mm/s 2 l Kg *m/s 2 = (mass units) *mm/s 2 l Kg *m/s 2 = (Kg*10 3 ) * mm/s 2 n So mass is in Kg*10 3 or Metric Tonnes and density is then Tonnes/mm 3 and these units should be used in the model u In this case if units of Kg mass and Kg/mm 3 density are used in the model PARAM,WTMASS,0.001 will scale the mass and density units correctly.

S2-72 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 2 – NORMAL MODES ANALYSIS OF A SATELLITE n Notice that in the case of the satellite, the base of the structure is constrained. The assumption is that it is rigidly built in to the launcher. n A free-free analysis could have been carried out where the structure is not connected to the launcher.

S2-73 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 2 – NORMAL MODES ANALYSIS OF A SATELLITE n For every DOF in which a structure is not totally constrained, it admits a rigid body mode (stress-free mode) or a mechanism. u There should be six rigid body modes. n The natural frequency of each rigid body mode would be zero. u You will try this in the next workshop. A later section is devoted to discussion of rigid body modes

S2-74 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 2 – NORMAL MODES ANALYSIS OF A SATELLITE Select the imported subcase that contained the correct boundary conditions for this Normal Modes Analysis. Submit the model to MSC.Nastran for analysis.

S2-75 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 2 – NORMAL MODES ANALYSIS OF A SATELLITE n The F06 file contains the Natural Frequencies or Eigenvalues. The results in terms of Cycles/sec or Hz are highlighted n The Generalized Mass is calculated by: { } T [M] { } n In this case the Mass normalization method for Eigenvectors was used, so the term reverted to: { } T [M] { } = 1 n The Generalized Stiffness is: { } T [K] { }

S2-76 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 2 – NORMAL MODES ANALYSIS OF A SATELLITE Use the Quick Plot option to see the Mode Shape and Frequency of each mode graphically. It can be useful to use fringes to see the displaced shape, particularly in high order panel modes.

S2-77 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 2 – NORMAL MODES ANALYSIS OF A SATELLITE The Mode Shape and Frequencies for the 10 Normal Modes requested are shown.

S2-78 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 2 – NORMAL MODES ANALYSIS OF A SATELLITE n It is very important to be able to characterize the Modes that are found, particularly when comparing one analysis with other, or against test. u To state I get 32 Hz for mode 4, against test of 36 Hz is meaningless unless it is confirmed that we are comparing the same physical modes u To help with this it is recommended that the modes are labeled with some simple description to help identification.

S2-79 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 2 – NORMAL MODES ANALYSIS OF A SATELLITE n So in the case of the satellite: ModeF (Hz)Description st bend main body st bend main body Orthogonal to mode st Torsion main body and all panels pant 1 st order, same sense +ve st order Panel pant st order Panel pant st order Panel pant, orthogonal to mode st order Panel pant st order Panel pant, orthogonal to mode nd order Panel pant nd order Panel pant, orthogonal to mode 9

S2-80 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CASE STUDY 2 – NORMAL MODES ANALYSIS OF A SATELLITE n In this case the first three modes include motion of the main body, the remainder are local panel modes. n It is very likely that the main response will be dominated by the first three modes. n Notice that many of the modes are described as orthogonal, and have identical or nearly identical frequencies. u This means that repeated roots have been found in the eigenvalue analysis, due to symmetry of the structural response. The eigenvectors are orthogonal to each other, meaning each is unique mathematically.

S2-81 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation REASONS TO CALCULATE NORMAL MODES n Now consider some reasons to compute natural frequencies and normal modes of structures u To assess the dynamic characteristics of a structure. l For example, if a structure is going to be subject to rotational or cyclic loading input, to avoid excessive vibrations, it might be necessary to see if the frequency of the input is close to one of the natural frequencies of the structure. n Passing blade frequencies of a helicopter n Rotational speed of an automobile wheel n Rotational speed of a lathe n Vortex shedding or flutter of bridge and deck structures u Assess the possible dynamic amplification of the loads. l If a structure is loaded near a natural frequency with an input that matches that frequency then the dynamic amplification can be significant for a lightly damped structure, perhaps being an order of magnitude higher than an equivalent static loading n Dynamic response of aircraft structure due to landing loads can exceed static loading n Dynamic response of Tacoma Narrows bridge, runaway loading

S2-82 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation REASONS TO CALCULATE NORMAL MODES (Cont.) n Use the Modal Data (natural frequencies and mode shapes) in a subsequent dynamic analysis u Later you will see that there is a class of transient and frequency response analysis methods that use modal techniques, using Modal data. n Assess requirements of subsequent dynamic analysis u For Transient response, calculate time steps based on the highest frequency of interest u For Frequency response, calculate the range of frequencies of interest n Guide the experimental analysis of structures u Identify optimum location of accelerometers, etc. u Avoid overstressing of components n Evaluate the effect of design changes u A normal modes analysis will give a clear indication of frequency shifts, changes in mode shapes to allow an early judgment on effect of design changes to be made

S2-83 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation WORKSHOP 13 MODEL ANALYSIS OF A CAR CHASIS n Objectives: u Importing the car model from a MSC.Nastran.bdf file into MSC.Patran and carry out a normal modes analysis. u Identify rigid body modes and describe the elastic modes. u Then carry out Workshop 13b, which sets the lowest frequency in the eigenvalue extraction to 0.1 Hz. n Note: u The workshop will take you through step by step if you are unfamiliar with MSC.Nastran or MSC.Patran. u If you have some experience, then try to set up the analysis without referring to the step by step guide. u Please feel free to ask your tutor for help.

S2-84 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation HOW ACCURATE IS THE NORMAL MODES ANALYSIS? n The following section and workshops explore the key ingredients: u mesh density u element type u mass distribution u detail of constraints u detail of joints

S2-85 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation MESH DENSITY n Mesh Density u The mesh must be fine enough to permit a representation of the of the highest mode considered l In the case of the beam, it was assumed that the second order mode was sufficient. The mesh is adequate for this. l However, if the higher order mode shown is required, then the mesh is inadequate.

S2-86 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation WORKSHOP 1a to 1c NORMAL MODES ANALYSIS WITH VARIOUS MESH SIZE n Objectives: u Building a simple plate model in MSC.Patran u Perform a normal modes analysis of the structure. u Workshop 1b and Workshop 1c vary mesh densities to investigate their effect on the results. u Identify and describe the elastic modes. n Note: u The workshop will take you through step by step if you are unfamiliar with MSC.Nastran or MSC.Patran. u If you have some experience, then try to set up the analysis without referring to the step by step guide. u Please feel free to ask your tutor for help.

S2-87 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation WORKSHOP 1a to 1c NORMAL MODES ANALYSIS WITH VARIOUS MESH SIZE n Comparison of results from Workshops 1a to 1c: u The mesh density of the plate structure was varied to check out the influence on the Normal Modes results. l These results show that the low frequency, first order modes are captured reasonable well in the coarser models, but as the higher modes are met then the frequency mode order drifts. u This illustrates how essential it is to describe each mode to be able to compare it.

S2-88 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation WORKSHOP 1a to 1c NORMAL MODES ANALYSIS WITH VARIOUS MESH SIZE n Workshop 1a – 1c results. ModefMesh 1a (10 x 4) Description ModefMesh 1b (2 x 1) Description ModefMesh 1c (50 x 20) Description st order Bending st Order Bending st order Bending st Order Torsion st Order Torsion st Order Torsion nd Order Bending nd Order Bending nd Order Bending nd Order Torsion nd Order Torsion nd Order Torsion rd Order Bending st Order Shear rd Order Bending st Order Shear nd Order Shear st Order Shear rd Order Torsion st Order Extension rd Order Torsion th Order Bending nd Order Shear II th Order Bending st Order Axial Bending nd Order Extension st Order Axial Bending nd Order Axial Bending nd Order Shear III th Order Torsion

S2-89 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation ELEMENT TYPE n Element Type u The type of element chosen is very important in dynamic analysis, in that it can control the stiffness representation and to a lesser extent the mass distribution of the structure. u Examples of poor choices are: l Using TET4 elements to model solid structures. If they are used in relatively thin regions that have plate or shell the results can be very poor. TET10 or preferably HEXA are a better choice. l If RBE2 is used instead of an RBE3 on a flexible structure such as a satellite platform then it may over stiffen the structure and influence the frequencies badly. u In the next workshop a classic structure is analyzed using different elements.

S2-90 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation WORKSHOP 15a – 15e TUNING FORK n Objectives: u Building a Tuning Fork in MSC.Patran u Perform a normal modes analysis of the structure. u Vary element types (tet10, tet4, beam) and mesh densities to investigate their effect on the results. u Identify and describe the elastic modes. n Note: u The workshop will take you through step by step if you are unfamiliar with Nastran or Patran. u If you have some experience, then try to set up the analysis without referring to the step by step guide. u Please feel free to ask your tutor for help.

S2-91 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation WORKSHOP 15a – 15e TUNING FORK (Cont.) n Comparison of results from Workshops 15a to 15e u The primary interest is the first elastic mode of the tuning fork. The theoretical value is 440 Hz, and is known as the A above middle C musical note. u As an aside, it is interesting to note that the vertical translation of the stem is what excites an instrument or another object that the tuning fork is placed against. WorkshopFrequency 1 st Elastic (Hz) % error (fr. theo. value) Element Type Mesh Density* # of Elements 15a Tet b Tet c Tet d Tet e Bar * Mesh Density is the global edge length

S2-92 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation WORKSHOP 15a – 15e TUNING FORK (Cont.) n Comparison of results from Workshops 15a to 15e (cont.): u The results clearly show the superiority of the TET10 Solid Elements over the TET4 Elements. In general, TET4 elements are NOT recommended in dynamic analysis. u Increasing the mesh density of the TET4 elements does not yield a significant change and it is likely that the element type is converging to an incorrect solution. This again underscores the need to avoid TET4s. u Increasing the mesh density improves the results significantly in the TET10 models, and this type of sensitivity is typical of what should be carried out in an analysis. u It is interesting to see that the BAR elements perform very well, because the quality of the idealization is high. Improvement may be possible by increasing the number of BARs around the curved regions.

S2-93 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation MASS DISTRIBUTION n Mass Distribution u A poor stiffness representation can influence a structure badly and a poor mass representation can also have the same effect. u The mass values may be wrong due to user error. The values can be checked in MSC.Patran and also the MSC.Nastran.f06 file (the next workshop shows how to do this) u There are two forms of mass representation in MSC.Nastran – lumped and coupled. Differences may occur in the analysis depending on which is selected. This is discussed in Section 3.

S2-94 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation WORKSHOP 14a MODEL ANALYSIS OF A TOWER n Objectives: u Build a Tower in MSC.Patran. u Perform a normal modes analysis of the structure. u Check the mass of the model. u Identify and describe the elastic modes. n Note: u The workshop will take you through step by step if you are unfamiliar with MSC.Nastran or MSC.Patran. u If you have some experience, then try to set up the analysis without referring to the step by step guide. u Please feel free to ask your tutor for help.

S2-95 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation DETAIL OF JOINTS n Detail of Joints: u Is the joint flexibility correct? l For example a corner of a formed sheet structure will have an internal radius which increases its torsional stiffness. It may be important in this case to include the torsional stiffness via ROD element. u If bolts are used to connect components together then the bolt stiffness may play an important role in dynamic analysis. CQUAD4 CROD

S2-96 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation DETAIL OF CONSTRAINTS n Detail of Constraints u When idealizing a structure assumptions are always made about the connection to an adjacent structure or to ground. l Hence if a panel is surrounded on all sides by reinforcing structure, is it represented as fully built in, simply supported, or is it necessary to model an equivalent edge stiffness using CELAS or CBUSH elements? u A particularly difficult case is where the connectivity is ill-defined, such as the push-fit and snap connectors of a typical car dashboard assembly. u Remember there is no such thing in nature as an infinitely stiff connection or structure. u In the workshop that follows, the mode shapes of the tower are significantly changed by the fact that the connection to ground is not rigid. A major redesign of this structure is needed.

S2-97 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation WORKSHOP 14b MODEL ANALYSIS OF A TOWER WITH SOFT GROUND CONNECTION n Objectives: u Using the model from Workshop 14a: l Account for the soil-base interaction using CBUSH elements. n Note: u The workshop will take you through step by step if you are unfamiliar with MSC.Nastran or MSC.Patran. u If you have some experience, then try to set up the analysis without referring to the step by step guide. u Please feel free to ask your tutor for help. Soil Stiffness modeled with CBUSH Elements Tower Leg RBE2

S2-98 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation HAND CALCULATIONS n Hand Calculations u Manual checking of the frequencies in an analysis to make sure answers are in the right ballpark can involve: l Using simple analogies of the structure to match standard solutions in Roark or Blevins l Applying a 1g load in relevant directions and using the resultant displacement at the cg. to calculate an equivalent SDOF frequency. l Using idealization techniques to create simple FE models to verify important modes of a complex model. u Remember the frequency is proportional to (k/m) 1/2, therefore consider whether stiffness or mass dominates errors and that frequency can be relatively insensitive to errors in both.

S2-99 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation CHECK LIST FOR NORMAL MODES PRIOR TO DOING FURTHER ANALYSIS n RBEs - are they as expected n Is the frequency domain adequate (this will be discussed more in the section on modal effective mass) n Are the modes clearly identified n Is Mesh Density adequate n Is the Element Type appropriate n Is the Mass distribution correct n Is coupled vs. lumped mass important n Are the internal joints modeled correctly n Are the constraints modeled correctly n Do the results compare with hand calculations, previous experience or test

S2-100 NAS122, Section 2, August 2005 Copyright 2005 MSC.Software Corporation