講義文稿分析lndyn

1、Lesson content:Introduction Steady-State Dynamics Solution ProceduresExcitation and OutputSteady-State Dynamics UsageComparative ExampleWorkshop 4: Steady-State Dynamics (IA)Workshop 4: Steady-State Dynamics (KW)Lesson 7: Steady-State Harmonic Response2 hoursBoth interactive (IA) and keywords (KW) v

2、ersions of the workshop are provided. Complete only one.Introduction (1/11)STEADY-STATE DYNAMICS (SSD) analysis procedures provide solutions to the linear equations of motion when the loading is harmonic. Harmonic loading repeats in time with a period corresponding to a complete cycle of load. The c

3、yclic frequency F of the applied load is F = 1/t, and the radian frequency is w = 2p F = 2p /t.Harmonic loading has the form of a trigonometric function: P(t) = Pmagsin(w t + q) The phase angle q allows the loading to be described relative to any starting point for the cycle (time axis position)t1ax

4、ist2axisTime PmagtIntroduction (2/11)Steady-state dynamicsWhen a damped structure that is initially at rest is excited with a harmonic load, it has a transient response that disappears rather quickly and is rarely of much interest. Eventually the structure reaches a steady-state condition that is ch

5、aracterized by harmonic response with the same frequency as the applied harmonic load.The trigonometric form of the response (vector, tensor components) is described by a magnitude and a phase angle which positions the response relative to the starting point for the solution cycle (time t = 0). time

6、 axisTime UmagU(t) = Umagsin(w t + q)Introduction (3/11)Steady-state dynamics procedures.are frequency domain solutions that represent a single complete cycle of harmonic vide solutions for each excitation frequency that are independent of the other excitation frequencies (no initial con

7、ditions).excitation input may consist of two parts corresponding to a magnitude and a phase duce response output consisting of two parts corresponding to a magnitude and a phase angle. have twice the number of active solution variables compared to static or transient dynamic solutions. Two

8、unknowns are being solved for each degree of freedom (magnitude and phase angle). Introduction (4/11)Complex representation Magnitude and phase angle of excitation and/or output can also be expressed in terms of complex quantities consisting of real (Re) and imaginary (Im) partsFor a generic quantit

9、y the relationship is: = ARe+ iAIm complex value ARe = Amag cos(q) AIm = Amag sin(q) Excitation is input as complex (real and imaginary components)Response can be viewed in multiple formats (magnitude/phase or real/imaginary)Introduction (5/11)Complex planeReal AxisImaginary AxisAmagAReAImIntroducti

10、on (6/11)Complex plane (contd)The time variation of an excitation or output quantity during a cycle of response is equal to its projection on a rotating “Solution Axis.”Real Axis = Solution Axis at t = 0Imaginary AxisqAmagSolution Axis radian position (w t) at time tSolution Axis rotates at w radian

11、s/secwA at time t of the cycleIntroduction (7/11)Complex plane (contd) Example: Unit force due to an imbalance for a z-axis rotation.Fx = 1 + 0i, Fy = 0 + 1i (rotates about - z-axis)Fx = 1 + 0i, Fy = 0 - 1i (rotates about + z-axis)Real Axis = Solution Axis at t = 0Imaginary AxisSolution Axis radian

12、position (w t) at time tFx and Fy at time t of the cycle(each goes through a +/- cycle)They are 90 degrees out-of-phase.FxFywIntroduction (8/11)Equation of motion:Assume harmonic response and applied loads:Acceleration is 180 degrees out-of-phase with complex displacement:Velocity is 90 degrees out-

13、of-phase with complex displacement:Load vector is complex:Let = excitation frequencyLet * designate a complex quantityIntroduction (9/11)Harmonic equation of motion:Damping forces are 90 degrees out-of-phase with the elastic and inertial forces.Without damping the solution es unbounded when equals a

14、 system natural frequency wn.Introduction (10/11)For structural damping models, the damping forces are proportional to displacements, not velocity, for which the concept of a complex stiffness in steady-state dynamics is useful. Thus,where KL is a loss modulus and KS is the storage modulus. The moti

15、on equation in terms of a structural damping constant S iswhere KL = S KS.Introduction (11/11)The behavior of lightly-damped systems near resonant frequencies.can be described equally well in terms of effective viscous and structural damping models.The effective structural damping constant is twice

16、the effective viscous damping ratio (fraction of critical damping):The half-power bandwidth of a resonant response can be used to estimate the effective damping.Half-power points (F1, F2) are those where the response is 0.707 of the resonant peak. A measure of the sharpness of a resonant condition i

17、s referred to as the Quality Factor (Q). Steady-State Dynamics Solution Procedures (1/6)Three types of steady-state dynamics analysis procedures are available in Abaqus:Direct solutionSolves the complete set of model DOFs at each excitation frequency.Subspace projectionProjects the equation of motio

18、n onto a subspace of mode shapes and solves for the generalized displacements (GU)Traditional and SIM architectures.Modal superpositionApproximates the system response by a superposition of modal responses. The reduced set of equations is totally uncoupled (Traditional Architecture) . SIM Architectu

19、re can produce coupling via a projected modal damping matrix. Steady-State Dynamics Solution Procedures (2/6)Direct solutionAbaqus solves for the frequency response of a structure to harmonic loads by directly solving the original full set of coupled equations. The eigenmodes are not used.This proce

20、dure is accurate for:heavily damped systems,systems with large discrete dashpots/dampers,systems with frequency-dependent behavior,systems with viscoelastic materials, andwave propagation modeling (acoustics).It is rather expensive since a complex solution must be obtained for the full system of equ

21、ations at each excitation (driving) frequency. Steady-State Dynamics Solution Procedures (3/6)Subspace projectionThis procedure computes the systems response to harmonic excitation by projecting the dynamic equilibrium equations onto a subspace of user-selected eigenmodes.The basic idea behind subsp

22、ace projection involves calculating the response of a system by projecting the systems frequency-dependent damping and stiffness matrices onto a number of the systems undamped eigenmodes. The *SELECT EIGENMODES option can be used to select which eigenmodes will be used.If *SELECT EIGENMODES is not u

23、sed, then all of the extracted eigenmodes will be used in the analysis.The *SELECT EIGENMODES option is not support by Abaqus/CAE (use the Keywords Editor).Steady-State Dynamics Solution Procedures (4/6)Subspace projection (contd)A frequency extraction step must precede the SSD step.Provides a cost-

24、effective way to include frequency-dependent stiffness/damping behavior.Allows for modeling of multiple forms of material damping, including viscoelastic behavior.Allows for a nonsymmetric stiffness.Compared to the direct solution method:Less accurateCan be significantly cheaperEffective speed-up in

25、creases with model size.Mode-based SSD with the SIM architecture may provide significantly better performance if material properties are not frequency dependentSteady-State Dynamics Solution Procedures (5/6)Modal superpositionAbaqus solves for the frequency response of a structure by superimposing t

26、he response of a set of selected eigenmodes. A frequency extraction step must precede the SSD step.The modal responses are assumed to be uncoupled (Traditional Architecture).The *SELECT EIGENMODES option can be used to select which eigenmodes will be used.The harmonic response is calculated based on

27、 eigenmodes, generalized masses, eigenfrequencies, and modal damping.The method may not be suitable for heavily damped systems where the eigenfrequencies and eigenmodes do not accurately represent the response.Steady-State Dynamics Solution Procedures (6/6)Modal superposition (contd)Solutions are ob

28、tained at different frequencies within a user-specified range of excitation frequencies.By default, the frequency data points will be concentrated near the eigenfrequencies, since the response varies most rapidly there.If damping is absent, the response will be unbounded if the forcing frequency is

29、equal to an eigenfrequency of the structure. To get quantitatively accurate results, especially near natural frequencies, accurate specification of the damping properties is essential.Excitation and Output (1/8)LoadsConcentrated forces or distributed pressure and body forces.The loading is harmonic;

30、 amplitude curves for frequency dependence.Loads are defined as real (in-phase) and imaginary (out-of-phase) loads.*Cload, realload, 1, 10.*Cload, imaginaryload, 2, 50.Excitation and Output (2/8)Boundary conditions Both the real and imaginary parts of a degree a freedom are either restrained or unre

31、strained simultaneously.Physically impossible to have one part restrained and the other unrestrained.Direct steady-state dynamics:Boundary conditions may be applied to any of the active degrees of freedom.Modal steady-state dynamics (both mode- and subspace-based):It is not possible to prescribe non

32、-zero displacements and rotations directly.The motion of the nodes must be defined using the *BASE MOTION option (Lecture 4).Excitation and Output (3/8)Printed outputPrinted output in the data (.dat) file is available as either a real and imaginary complex data pair or in terms of magnitude and phas

33、e angle. Printed output is available for direct solution response variables:Scalars (acoustic pressures).Vector components (displacements, reaction forces).Tensors (stress/strain components).Printed output is not available for derived variables such as stress invariants (Mises, Max Principal, etc.)T

34、hese are not necessarily harmonic, though they are periodic. Excitation and Output (4/8)Output database fileDirect solution response variables are stored as real and imaginary:Scalars (acoustic pressures).Vector components (displacements, reaction forces).Tensors (stress/strain components).History a

35、nd field output can be viewed in a number of ways:MagnitudePhase angleReal componentImaginary componentSolution at a user-specified phase angle at a given point in time during a response cyclea response cycle is 360 degreesExcitation and Output (5/8)Field data can be animated on displaced shapes ove

36、r a complete response cycle of a driving frequency (AnimateHarmonicFull cycle).Derived field output can be computed and plotted over a complete animation cycle for a given excitation frequency. Mises StressTresca StressMaximum Principal StressMaximum Principal StrainEtc.Base motion response can be v

37、iewed with respect to either relative or total displacements (U or TU, respectively).Excitation and Output (6/8)Time history output specific to mode-based analysis results:BM = base motion (displacement, velocity, acceleration)GU = modal generalized displacements (modal solution DOF)KE = modal kinet

38、ic energy (modal quantity equivalent SDOF)SNE = modal strain energy (modal quantity equivalent SDOF)Generating direct modal data is very efficient (very low cost). Generating model-based output is potentially much more expensive than the actual solution of the reduced modal equations.The basic comme

39、nts contained in Lecture 5 (Modal Dynamics) hold for the modal superposition and subspace projection solutions.Excitation and Output (7/8)Steady-state dynamics analysis using multiple load casesAn efficient alternative to multi-step SSD analysis when each step defines a distinct set of loads and bou

40、ndary conditions.Multiple load cases are defined in a single step rather than using multiple stepsCan be used in the following steady-state dynamic proceduresDirect analysisSIM-based modal analysisExcitation and Output (8/8)A SSD example: Chassis-bracket mobility analysisNumber of variables: 534,000

41、Number of equations: 483,000Number of load cases: 60Steady-state dynamics, Direct(10 frequency points)Output: U (output database)CPU time (sec)60 steps (projected based on 1 step)60 load casesSolver1290 60 = 77,4001990 (39 faster)Total1965 60 = 117,60011,600 (10 faster)Steady-State Dynamics Usage (1

42、/10)Steady-state dynamics: Pump assembly examplemotor and pump shaft mounted on a box girder structureoverhanging impeller is a rigid massmotor is rigid masshull mounted pedestals bolts modeled as CIRC beamscantilevered instrument panelmotorpump shaftflex couplinginstrument panelpedestalpedestalbox

43、girderimpellerSteady-State Dynamics Usage (2/10)Model characteristicsMaterials have density and elastic properties.The pedestal bases are fixed.Loading:A rotating force at the motor mass center due to an imbalance.The rotating force is applied using an amplitude curve.The motor rotational axis is al

44、igned with the global 1-axis. The rotating force has global 2- and 3-axis components. The 2-axis force is defined as imaginary.The 3-axis force is defined as real.Steady-State Dynamics Usage (3/10)Model characteristics (contd) Modal superposition solution.Frequency sweep in a range of 5 250 Hz.Eigen

45、value extraction step: All eigenmodes between 5 and 500 HzGeneral rule of thumb: maximum frequency extraction range should be twice the maximum frequency sweep range Store displacements, reaction forces and stresses. Mass matrix mode shape normalization.Set the fraction of critical damping (modal da

46、mping) to 0.03 (first 25 modes).Steady-State Dynamics Usage (4/10)Step 1: Frequency Extraction:Steady-State Dynamics Usage (5/10)Step 2: Steady-State Dynamics, ModalFrequency rangeDetermines the scale used for output.Distribution of frequency pointsNumber of points between each pair of eigenfrequenc

47、ies (includes endpoints):Steady-State Dynamics Usage (6/10)Step 2: Steady-State Dynamics, Modal*Step, name=Step-2, perturbationROTATIONAL UNIT FORCE DUE TO MOTOR IMBALANCE*Steady State Dynamics, frequency scale=LINEAR5., 250., 10, 1.*Modal Damping1, 25, 0.03Steady-State Dynamics Usage (7/10)Step 2:

48、Steady-State Dynamics, Modal (Alternate Input)Use logarithmic scale for output.Number of points over the entire frequency range*Step, name=Step-2, perturbationROTATIONAL UNIT FORCE DUE TO MOTOR IMBALANCE*Steady State Dynamics, interval=RANGE5., 250., 50, 1.*Modal Damping1, 25, 0.03Steady-State Dynam

49、ics Usage (8/10)Step 2: Steady-State Dynamics, Modal *Cload, real, amplitude=UNBALANCEDMOTOR-CG, 3, 1.*Cload, imaginary, amplitude=UNBALANCEDMOTOR-CG, 2, 1.*AMPLITUDE, name=UNBALANCED 0, 0f1, mew1f2, mew2An amplitude curve is used to specify the magnitude of the unbalance (F=mew)Steady-State Dynamic

50、s Usage (9/10)Animation of a complete time cycle at 1820 rpm (30.3Hz).Steady-State Dynamics Usage (10/10)Force transmissionForce transmissibility response functions obtained from history outputComparative Example (1/9)Steady-state dynamic analysis of a cantilever beamModal superposition, subspace pr

51、ojection, and direct solutions.Loading: 1N harmonic transverse force at the free end. Modal superposition damping:Modal damping (0.03 fraction of critical)Rayleigh b-damping (0.03 fraction of critical at first resonance)Structural (0.06 structural damping constant)Direct and subspace projection:Rayl

52、eigh b-damping (0.03 fraction of critical at first resonance).Structural damping (0.06 structural damping constant).Width = 0.0250 metersMaterial = AluminumLength = 0.2500 metersDensity = 2691 Kg/m3 Depth = 0.0065 metersModulus = 6.9e10 PaComparative Example (2/9)Cantilever beam natural frequencies

53、and mode shapesMode 1 - 85.0 HzMode 2 - 530.4 HzMode 3 - 1474.9 HzComparative Example (3/9)Steady-state dynamics Modal (modal damping = 0.03)Modes respond as SDOF systemsDamping limits the resonant responseStiffness controls below resonanceMass controls above resonanceModal damping value was verified (fraction of critical all mod