dynamic-pdf教程awb120_dynamics_02_modal

1、Chapter 2: Modal AnalysisANSYS Mechanical DynamicsANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-1 July 2009Inventory #002666Modal AnalysisTraining ManualA. Define modal analysis and its purpose.B. Discuss associated concepts, terminology, and mode extraction methods.C. Learn how to

2、do a modal analysis in Workbench.D. Work on one or two modal analysis exercises.ANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-2July 2009Inventory #002666Description & PurposeTraining Manual A modal analysis is a technique used to determine the vibration characteristics of structures

3、: natural frequencies at what frequencies the structure would tend to naturally vibrate mode shapes in what shape the structure would tend to vibrate at each frequency mode participation factors the amount of mass that participates in a given direction for each mode Most fundamental of all the dynam

4、ic analysis types.ANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-4July 2009Inventory #002666Benefits of modal analysis Allows the design to avoid resonant vibrations or to vibrate at a specified frequency (speaker box, for example). Gives engineers an idea of how the design will resp

5、ond to different types of dynamic loads. Helps in calculating solution controls (time steps, etc.) for other dynamic analyses.Recommendation: Because a structures vibration characteristics determine how it responds to any type of dynamic load, it is generally recommended to perform a modal analysis

6、first before trying any other dynamic analysis.Description & PurposeTerminology A “mode” refers to the pair of one natural frequency and corresponding mode shape. A structure can have any number of modes, up to the number of DOF in the model.Training Manualmode 1 f1f1 = 109 Hzmode 2 f2f2 = 202 Hzmod

7、e 3 f3f3 = 249 HzANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-5July 2009Inventory #002666TheoryAssumptions & Restrictions The structure is linear (i.e. constant stiffness and mass).Training ManualANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-6July 2009Inventory #00

8、2666 There is no damping. Damped eigensolvers (MODOPT,DAMP or MODOPT,QRDAMP) may be accessed using Commands Objects, but will not be covered here. The structure has no time varying forces, displacements, pressures, or temperatures applied (free vibration).TheoryDevelopmentTraining Manual Start with

9、the linear general equation of motion:M u&+ Cu&+ Ku= F Assume free vibrations, and ignore damping:00M u&+ Cu&+ K u= FM u&+ K u= 0ANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-8July 2009Inventory #002666 Assume harmonic motion:u=u&=fiwifisin(wit +qi ) cos(wit+qi)iiiiu& =-w2fsin (w t

10、+q ) Substitute and simplifyM u&+ K u= 0iiiiiii-w2M f sin (w t +q )+ Kf sin (w t +q )= 0(- w2M + K )f = 0ii This equality is satisfied if fi = 0 (trivial, implies no vibration) or ifidet(K-w2 M)= 0 This is an eigenvalue problem which may be solved for up to n eigenvalues, wi2, and n eigenvectors, fi

11、, where n is the number of DOF.TheoryExtraction & NormalizationTraining Manual Note that the equationidet(K-w2 M)= 0has one more unknown than equations; therefore, an additional equation is needed to find a solution. The addition equation is provided by mode shape normalization. Mode shapes can be n

12、ormalized either to the mass matrixfT M f = 1iior to unity, where the largest component of the vector fi is set to 1. Workbench displays results normalized to the mass matrix. Because of this normalization, only the shape of the DOF solution has real meaning.ANSYS, Inc. Proprietary 2009 ANSYS, Inc.

13、All rights reserved.2-9July 2009Inventory #002666TheoryEigenvalues & EigenvectorsTraining Manual The square roots of the eigenvalues are wi, the structures natural circular frequencies (rad/s). Natural frequencies fi can then calculated as fi = wi/2p (cycles/s). It is the natural frequencies, fi in

14、Hz, that are input by the user and output by Workbench. The eigenvectors fi represent the mode shapes, i.e. the shape assumed by the structure when vibrating at frequency fi.mode 1 f1f1 = 109 Hzmode 2 f2f2 = 202 Hzmode 3 f3f3 = 249 HzANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-10J

15、uly 2009Inventory #002666TheoryEquation Solversi The equationdet(K-w2 M)= 0Training ManualANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-11July 2009Inventory #002666can be solved using one of two solvers available in Workbench Mechanical: Direct (Block Lanczos) To find many modes (ab

16、out 40+) of large models. Performs well when the model consists of shells or a combination of shells and solids. Uses the Lanczos algorithm where the Lanczos recursion is performed with a block of vectors. Uses the sparse matrix solver. Iterative (PCG Lanczos) To find few modes (up to about 100) of

17、very large models (500,000+ DOFs). Performs well when the lowest modes are sought for models that are dominated by well-shaped 3-D solid elements. Uses the Lanczos algorithm, combined with the PCG iterative solver. In most cases, the Program Controlled option selects the optimal solver automatically

18、.TheoryParticipation Factors (Solution Information)Training Manual The participation factors are calculated byg= fTM Diiwhere D is an assumed unit displacement spectrum in each of the global Cartesian directions and rotation about each of these axes. This measures the amount of mass moving in each d

19、irection for each mode. The “Ratio” is simply another list of participation factors, normalized to the largest. The concept of participation factors will be important in later chapters.ANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-13July 2009Inventory #002666 A high value in a direc

20、tion indicates that the mode will be excited by forces in that direction.mode 1mode 3mode 5TheoryEffective Mass (Solution Information) Also printed out is the effegct2ive mass.Training ManualANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-14July 2009Inventory #002666Meff ,i= T i= g 2f

21、T f = i,ifiMi1f iM f i Ideally, the sum of the effective masses in each direction should equal total mass of structure, but will depend on the number of modes extracted. The ratio of effective mass to total mass can be useful for determining whether or not a sufficient number of modes have been extr

22、acted.TheoryPrestress EffectsTraining Manual A prestressed modal analysis can be used to calculate the frequencies and mode shapes of a prestressed structure, such as a spinning turbine blade. The prestress influences the stiffness of the structure through the stress- stiffening matrix contribution.

23、ANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-16July 2009Inventory #002666 In free vibration with prestress analyses, two solutions are required. A linear static analysis is initially performed:Ku= F s Based on the stress state s from the static analysis, a stress stiffness matrix S

24、 is calculated (see Theory Reference for details):s S The free vibration with pre-stress analysis is then solved, including the S term:(K +S-w2M)f = 0ii Note that the prestress only affects the stiffness of the system. i.e. the static prestress will not be added to the modal stressRemarks & Comments

25、Contact RegionsTraining Manual Contact regions are available in modal analysis; however, since this is a purely linear analysis, contact behavior will differ for the nonlinear contact types, as shown below:Contact TypeStatic AnalysisLinear Dynamic AnalysisInitially TouchingInside Pinball RegionOutsi

26、de Pinball RegionBondedBondedBondedBondedFreeNo SeparationNo SeparationNo SeparationNo SeparationFreeRoughRoughBondedFreeFreeFrictionlessFrictionlessNo SeparationFreeFreeFrictionalFrictionalm = 0, No Separationm 0, BondedFreeFree Contact behavior will reduce to its linear counterparts. It is general

27、ly recommended, however, not to use a nonlinear contact type in a linear-dynamic analysisANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-17July 2009Inventory #002666Remarks & CommentsUnconstrained SystemsTraining Manual An unconstrained system is one that has no constraints or support

28、s and can move as a rigid body in at least one direction. Rigid-body motion can be considered to be a mode of oscillation with zero frequency. In practice, these modes may not have a frequency of exactly zero.“rigid-body” or“zero” modes Note that a well-connected system can have at most six rigid-bo

29、dy modes. Obtaining more than six rigid-body modes may indicate that assemblies are not well connected.ANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-18July 2009Inventory #002666Remarks & CommentsSymmetry Boundary ConditionsTraining Manual Symmetry BCs will only produce symmetrically

30、 shaped modes, so some modes can be missed. It may be necessary to apply several different symmetry conditions to find all modes. The full model below results in the frequencies listed in the tabular view. A quarter-symmetry model will require three sets of symmetry boundary conditions to find all m

31、odes (see next slide).ANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-19July 2009Inventory #002666Remarks & CommentsSymmetry Boundary ConditionsTraining ManualSymm-Asym BCFull ModelSymmetry BCetcAnti-Symmetry BCANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-20 July 200

32、9Inventory #002666Procedure: ModalANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-21 July 2009Inventory #002666ModalProcedureTraining Manual Drop a Modal (ANSYS) system into the project schematic.ANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-23July 2009Inventory #0026

33、66 Create new geometry, or link to existing geometry. Edit the Model cell to bring up the Mechanical application. ModalPreprocessingTraining Manual Verify materials, connections, and mesh settings. This was covered in Workbench Mechanical Intro.ANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights res

34、erved.2-25July 2009Inventory #002666 Add supports to the model. Displacement constrains must have a magnitude of zero.ModalSolution SettingsTraining Manual Choose the number of modes to extract. If needed, upper and lower bounds on frequency may be specified to extract the modes within a specified r

35、ange.ANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-27July 2009Inventory #002666 If the Program-Controlled solver selection is not appropriate, the solver type can be changed to either Direct or Iterative. Stress and strain results may be turned on under Output Controls.ModalPostproc

36、essingTraining Manual Total-deformation results may be quickly inserted by highlighting multiple rows in the tabular view or histogram view.ANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-29July 2009Inventory #002666 If stress/strain were requested, these results may also be access fr

37、om the Solution Toolbar.Procedure: Prestressed ModalANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-30July 2009Inventory #002666Prestressed ModalProcedureTraining Manual The procedure to do a prestressed modal analysis is essentially the same as a regular modal analysis, except that y

38、ou first need to prestress the structure by doing a static analysis. The static analysis results in a stressed structure, which is used as the initial condition for the modal analysis.ANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-34July 2009Inventory #002666 Drop a Static Structural

39、 (ANSYS) system into the project schematic. Drop a Modal (ANSYS) system onto the Solution cell of the Modal system. Note the circular-ended connector, indicating a data transfer from the Static to the Modal analysis. Create new geometry, or link to existing geometry. Edit the Model cell to bring up

40、the Mechanical application.Prestressed ModalPreprocessingTraining Manual In the Static Structural system, insert the loads and supports that will cause the prestressed-state to occur.ANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-35July 2009Inventory #002666Prestressed ModalPostproce

41、ssingTraining Manual Review the static results before proceeding.ANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.2-36July 2009Inventory #002666Prestressed ModalPreprocessing Workbench will automatically setup the data transfer between the systems.Training Manual To verify the data transfer, one can ensure that Future Analysis is set to Prestressed analysis in t