SECTION 12 COMMUNICATIONS TOWER S12-1 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation

SECTION 12 COMMUNICATIONS TOWER S12-2 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation

SECTION 12 COMMUNICATIONS TOWER S12-3 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Topics covered in this case study: u Normal modes analysis

SECTION 12 COMMUNICATIONS TOWER S12-4 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Problem Description u A communications tower is in the final design stage. You are asked to analyze the tower structure under dynamic loading including wind and seismic loading. u Before you can perform any dynamic analysis, you must first perform a normal modes analysis to determine the dynamic characteristics of the tower. n Analysis Objectives u Perform a normal modes analysis to determine the natural frequencies and mode shapes of the tower structure.

SECTION 12 COMMUNICATIONS TOWER S12-5 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Getting started on the case study u The tower structural members are made from steel open sections. These members are modeled by CBAR elements. u The communications equipment is mounted at the top of the tower. The equipment is fairly compact so it will be modeled as lumped masses attached to the top corners of the tower. u The model is shown on the next page.

SECTION 12 COMMUNICATIONS TOWER S12-6 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation Lumped mass element CBAR element

SECTION 12 COMMUNICATIONS TOWER S12-7 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Before performing a normal modes analysis on the tower structure, lets look at the theory behind the normal modes analysis: u Equation of motion for free vibrations u Mass Matrix Formulation u Solving the equations

SECTION 12 COMMUNICATIONS TOWER S12-8 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Governing Equations u Consider the undamped single-degree-of-freedom system shown below: u The equation of motion for free vibrations (i.e., without external load or damping) is where m = mass k = stiffness mx = -kx or mx + kx = 0..

SECTION 12 COMMUNICATIONS TOWER S12-9 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n For a multi-degree-of-freedom system, this equation becomes M x + K x = 0.. where[K] = the stiffness matrix of the structure (the same as in static analysis) [M] = the mass matrix of the structure (it represents the inertia properties of the structure)

SECTION 12 COMMUNICATIONS TOWER S12-10 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Formulating the mass matrix u The mass matrix represents the inertia properties of the structure. MSC.Nastran provides the user with two choices: l Lumped Mass Matrix (default) Contains only diagonal terms associated with translational degrees of freedom l Coupled Mass Matrix Also contains off-diagonal terms coupling translational degrees of freedom and rotational degrees of freedom. (Note: for a rod element, only translational DOFs are coupled.)

SECTION 12 COMMUNICATIONS TOWER S12-11 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Example of Mass Matrix Lumped Mass Matrix Coupled Mass Matrix Where = mass density and A = cross sectional area

SECTION 12 COMMUNICATIONS TOWER S12-12 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Lumped vs. Coupled Mass u Coupled mass is generally more accurate than lumped mass. u Lumped mass is preferred for computational speed in dynamic analysis. u User-selectable coupled mass matrix for elements l PARAM,COUPMASS,1 to select coupled mass matrices for all BAR, ROD, and PLATE elements that include bending stiffness l Default is lumped mass. u Elements that have either lumped or coupled mass: l CBAR, CBEAM, CONROD, CHEXA, CPENTA, CQUAD4, CQUAD8, CROD, CTETRA, CTRIA3, CTRIA6, CTRIAX6, CTUBE

SECTION 12 COMMUNICATIONS TOWER S12-13 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Lumped vs. Coupled Mass (cont.) u Elements that have lumped mass only l CONEAX, CSHEAR u Elements that have coupled mass only l CBEND, CHEX20, CTRAPRG, CTRIARG u Lumped mass contains only diagonal, translational components (no rotational ones). u Coupled mass contains off-diagonal translational components as well as rotations for CBAR (though no torsion), CBEAM, and CBEND elements.

SECTION 12 COMMUNICATIONS TOWER S12-14 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Solving the equation of motion for free vibration n Assume a harmonic solution of the form (Physically, this means that all the coordinates perform synchronous motions and the system configuration does not change its shape during motion only its amplitude.) (1) (2)

SECTION 12 COMMUNICATIONS TOWER S12-15 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Differentiating equation (2) twice we get n Substituting equations (2) and (3) into equation (1), we get which simplifies to n This is an eigenvalue problem. (3) (4)..

SECTION 12 COMMUNICATIONS TOWER S12-16 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n There are two possible solutions to the eigenvalue problem: 1. If, then the only possibility is which is the so-called trivial solution and is not interesting from a physical point of view. 2. If, then there is a nontrivial solution to the eigenvalue problem. The eigenvalue problem is reduced to or Where = 2 is called the eigenvalue (5) =

SECTION 12 COMMUNICATIONS TOWER S12-17 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation If the structure has N dynamic degrees of freedom (degrees of freedom with mass), then there are N number of s that are solution of the eigenvalue problem. These s ( 1, 2,..., N ) are the natural frequencies of the structure (also known as normal frequencies, characteristic frequencies, fundamental frequencies, or resonant frequencies). The eigenvector associated with the natural frequency j is called normal mode or mode shape. The normal mode corresponds to deflected shape patterns of the structure. n When a structure is vibrating, its shape at any given time is a linear combination of its normal modes.

SECTION 12 COMMUNICATIONS TOWER S12-18 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Example of normal modes Simply Supported Beam Mode 1 Mode 2 Mode 3 n The number of grid points (degrees of freedom) in the model must be adequate to describe the mode shapes.

SECTION 12 COMMUNICATIONS TOWER S12-19 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Reasons to compute natural frequencies and normal modes: u Assess the dynamic characteristics of the structure. For example, if rotating machinery is going to be installed on a certain structure, it might be necessary to see if the frequency of the rotating mass is close to one of the natural frequencies of the structure to avoid excessive vibrations. u Assess possible dynamic amplification of loads. Use natural frequencies and normal modes to guide subsequent dynamic analysis (transient response, response spectrum analysis), i.e., what should be the appropriate t for integrating the equation of motion in transient analysis? u Use natural frequencies and mode shapes for subsequent dynamic analysis, i.e., transient analysis of the structure using modal expansion. u Guide the experimental analysis of the structure, i.e., the location of accelerometers, etc.

SECTION 12 COMMUNICATIONS TOWER S12-20 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation The natural frequencies ( 1, 2,..., j ) are expressed in radians/seconds. They can also be expressed in hertz (cycles/seconds) using

SECTION 12 COMMUNICATIONS TOWER S12-21 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n If a structure is not totally constrained, i.e., if it admits a rigid body mode (stress-free mode) or a mechanism, at least one natural frequency will be zero. Example: The following unconstrained structure has a rigid body mode.

SECTION 12 COMMUNICATIONS TOWER S12-22 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Scaling of normal modes is arbitrary. For example, the following three mode shapes represent the same model of vibration.

SECTION 12 COMMUNICATIONS TOWER S12-23 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Determination of the natural frequencies, i.e., solution of is a difficult problem. The solution to this problem must be determined using a numerical approach.

SECTION 12 COMMUNICATIONS TOWER S12-24 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n MSC.Nastran provides the user with the following three types of methods for eigenvalue extraction: u Tracking Method 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. This approach is more convenient when few natural frequencies are to be determined. In general, SINV is more reliable than INV.

SECTION 12 COMMUNICATIONS TOWER S12-25 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation u Transformation Method l The original eigenvalue problem is transformed to the form l Next, 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 for small models when a large proportion of eigenvalues are needed.

SECTION 12 COMMUNICATIONS TOWER S12-26 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation u Lanczos Method l This is a combined tracking-transformation method and is the most modern method. l This method is most efficient for computing a few eigenvalues of large, sparse problems (most structural models fit into this category). l This is the recommended method for most structural problems. By default, Patran uses this method when setting up a Nastran input file.

SECTION 12 COMMUNICATIONS TOWER S12-27 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n In order to perform a normal modes analysis, following entries are required in the Nastran input data file: u Executive Control Section l SOL 103 u Case Control Section l METHOD = n where n is the ID number for the EIGR or EIGRL entry that is included in the bulk data section. Multiple subcases can be used to control output requests u Bulk Data Section l EIGRL entry – Lanczos method l EIGR entry – Other eigenvalue extraction methods l Mass properties are required

SECTION 12 COMMUNICATIONS TOWER S12-28 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n The EIGRL entry u Defines data needed to perform vibration or buckling analysis with the Lanczos Method EIGRLSIDV1V2NDMSGLVLMAXSETSHFSCL NORM EIGRL FieldContents SIDSet identification number (unique integer > 0) V1, V2Vibration analysis: Frequency range of interest Buckling analysis: Eigenvalue range of interest (V1 < V2, real). If all modes below a frequency are desired, set V2 to the desired frequency and leave V1 blank. It is not recommended to put 0.0 for V1 (It is more efficient to use a small negative number or leave it blank).

SECTION 12 COMMUNICATIONS TOWER S12-29 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n The EIGRL entry (cont.) FieldContents NDNumber of roots desired (integer > 0 or blank) MSGLVLDiagnostic level (integer 0 through 3 or blank) MAXSETNumber of vectors in block (integer 1 through 15 or blank) SHFSCLEstimate of the first flexible mode natural frequency (real or blank) NORMMethod for normalizing eigenvectors, either "MASS" or "MAX" MASSNormalize to unit value of the generalized mass (default) MAXNormalize to unit value of the largest component in the analysis set

SECTION 12 COMMUNICATIONS TOWER S12-30 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Mass properties are required for normal analysis. There are several ways to enter mass: u Structural Mass – Density field on MATi entries (mass/volume) u Non-Structural Mass – NSM field on element property entries (mass per unit length or area) MAT1MIDEGNURHOATREF GE MAT11010.E E PSHELLPIDMID1TMID212I/T 3 MID3T S /TNSM PSHELL

SECTION 12 COMMUNICATIONS TOWER S12-31 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation u Concentrated Mass – Mass property fields on concentrated element entries CONMi and CMASSi CONM2EIDGCIDMX1X2X3 I11I21I22I31I32 I33 COMN u Non-structural mass can also be defined on non- structural mass entries: NSM, NSM1, NSML, NSML1, and NSMADD.

SECTION 12 COMMUNICATIONS TOWER S12-32 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Mass Units u MSC.Nastran does not know units. It is up to you to use a consistent set of units. u For the steel tower structure in this case study, the SI system of units is used. l Units of Force = Newton Units of Length = meter Units of time = second Units of mass = kg Mass Density = 7,861 kg/m 3 l Concentrated Mass m = 1000 kg each

SECTION 12 COMMUNICATIONS TOWER S12-33 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Mass Units (cont.) u Another common system of units is the English in-lb f -sec system: l Units of Force = lb f Units of Length = in Units of time = second Units of mass = lb f sec 2 /in Mass Density = 7.36 x lb f sec 2 /in 4 l Concentrated Mass m = lb f sec 2 /in each l The consistent mass unit of lb f sec 2 /in must be used in this system of units. If your problem definition is based on the weight unit of lb f, use the following equation to convert lb f to the correct mass units: W = M x g M = W x 1/g = W x 1/386.1 = W x lb f x = lb f sec 2 /in lb f /in 3 x = 7.36 x lb f sec 2 /in 4

SECTION 12 COMMUNICATIONS TOWER S12-34 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Mass Units (cont.) u Alternatively, the user of the English system of units can use lb f as the mass unit in the model and use the WTMASS parameter to convert lb f to the correct mass unit. Mass Density = lb f /in 3 l Concentrated Mass m = 2,205 lb f each l Use PARAM, WTMASS, in the model. This is a multiplier for the mass matrix ( = 1/386.1 = 1/g).

SECTION 12 COMMUNICATIONS TOWER S12-35 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Lets now continue with the case study. n We want to determine the first 5 modes for the tower structure.

SECTION 12 COMMUNICATIONS TOWER S12-36 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation Set up the normal modes analysis

SECTION 12 COMMUNICATIONS TOWER S12-37 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation Click solution type and select normal modes analysis. Click solution parameters and enter a node ID for the Wt. Generator. The mass properties of the model will be computed about this node. Enter 0 to select the origin of the basic coordinate system.

SECTION 12 COMMUNICATIONS TOWER S12-38 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation Next click subcases and select the Default subcase Click subcase parameters and select the Lanczos method. Enter 5 in the number of desired roots box

SECTION 12 COMMUNICATIONS TOWER S12-39 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation Run the analysis, read the results into Patran, and animate the mode shapes one at a time.

SECTION 12 COMMUNICATIONS TOWER S12-40 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Examine the.f06 file

SECTION 12 COMMUNICATIONS TOWER S12-41 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Examine the.f06 file (cont.)

SECTION 12 COMMUNICATIONS TOWER S12-42 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation n Normal modes analysis summary: u Total mass of structure is 85,732 kg u Mode No.Frequency (Hz)Description Primary Bending Primary Bending Torsion Secondary Bending Secondary Bending

SECTION 12 COMMUNICATIONS TOWER S12-43 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation EXERCISE Perform Workshop 13 Normal Modes of a rectangular plate in your exercise workbook.

SECTION 12 COMMUNICATIONS TOWER S12-44 NAS120, Section 12, May 2006 Copyright 2006 MSC.Software Corporation