Module 6 Spectrum Analysis

Module 6 Spectrum Analysis A. Define a spectrum analysis and its purpose. B. Understand the underlying concepts and terminology. C. Learn how to do a response spectrum analysis. D. Guidelines for spectrum analysis. E. Random Vibration Analysis January 30, 2001 Inventory #001447 6-2

Spectrum Analysis A. Definition & Purpose What is spectrum analysis? A technique to compute a structure’s response to transient excitations that contain many frequencies. Excitations could be from sources such as earthquakes, aircraft noise/ flight history, missile launches. A spectrum is a representation of a load’s time history in the frequency domain. This is also referred to as response spectrum. January 30, 2001 Inventory #001447 6-3

Spectrum Analysis … Definition & Purpose El Centro Earthquake ( 1940 ) Acceleration vs. time Acceleration spectrum (G vs. Hz) A structure subject to the El Centro earthquake can be analyzed using either a Transient analysis or spectrum analysis. January 30, 2001 Inventory #001447 6-4

Spectrum Analysis … Definition & Purpose Spectrum analysis follows a modal analysis. Computes the maximum response of the structure to a given spectrum at each natural frequency. This maximum response is computed as scale factor*mode shape. These maximum responses are then combined to give a total response of the structure. January 30, 2001 Inventory #001447 6-5

Spectrum Analysis … Definition & Purpose An alternative is to perform a transient analysis. Transient analysis is generally more time consuming, especially when a number of components and load conditions have to be considered. However, transient analysis is more accurate. In spectrum analysis the focus is to get the maximum response quickly, and some information is lost (phase). January 30, 2001 Inventory #001447 6-6

Spectrum Analysis … Definition & Purpose Used in the design of: Nuclear power plants (buildings and components) Airborne Electronic equipment (aircraft / missile) Spacecraft components Aircraft components Any structure or component that is subjected to seismic or other erratic loads Building frames and bridges January 30, 2001 Inventory #001447 6-7

Spectrum Analysis … Definition & Purpose ANSYS allows four types of spectrum analysis: Single-point response spectrum** A single response spectrum excites all specified points in the model. Multi-point response spectrum ** Different response spectra excite different points in the model. Dynamic design analysis method (DDAM) A specific type of spectrum defined by the U.S. Naval Research Laboratory to evaluate shock resistance of shipboard equipment. Power Spectral Density (PSD)** A probabilistic approach used in random vibration analysis. ** Covered in this seminar January 30, 2001 Inventory #001447 6-8

Spectrum Analysis B. Terminology & Concepts Topics covered: Definition of a spectrum How a response spectrum is used to calculate a structure’s response to the excitation Participation factor Mode coefficient Mode combination January 30, 2001 Inventory #001447 6-9

Spectrum Analysis - Terminology & Concepts Definition of spectrum What is a spectrum? A curve representing the maximum response of an idealized system to an excitation. The response may be acceleration, velocity, displacement, or force. Consider, for example, four single-DOF spring-mass systems mounted on a shaker table. Their frequencies are f1, f2, f3, and f4, with f1 < f2 < f3 < f4. 1 2 3 4 January 30, 2001 Inventory #001447 6-10

Spectrum Analysis - Terminology & Concepts … Definition of spectrum If the shaker table is excited at frequency f1 and the displacement response of the four systems is recorded, it will look as shown on the right. Now add a second excitation of frequency f3 and record the displacement response. Systems 1 and 3 will each reach their peak response. If now a general excitation containing several frequencies is applied and only the peak responses are recorded, we might get the curve shown. This curve is the spectrum, specifically a response spectrum. u f u f u f January 30, 2001 Inventory #001447 6-11

Spectrum Analysis - Terminology & Concepts … Definition of spectrum Thus a response spectrum is an envelope of the maximum responses of a number of single DOF systems to a given excitation. Input to a spectrum analysis consists of a response spectrum curve and a direction of excitation. January 30, 2001 Inventory #001447 6-12

Spectrum Analysis - Terminology & Concepts Approach Spectrum analysis follows a modal analysis in which natural frequencies and mode shapes have been computed. In doing a spectrum analysis you will encounter three new terms: Participation factor Mode coefficient Mode combination We will define these three terms along with the general outline of how a spectrum analysis is done. January 30, 2001 Inventory #001447 6-13

Spectrum Analysis - Terminology & Concepts … Approach - Participation factor For each mode of the structure, a participation factor gi is calculated in the excitation direction. The participation factor is a function of the mode shape and the direction of excitation. This is a measure of how much a mode will contribute to the deflections (and hence stresses) in the direction of excitation. January 30, 2001 Inventory #001447 6-14

For example, consider the cantilever beam shown. Spectrum Analysis - Terminology & Concepts … Approach - Participation factor For example, consider the cantilever beam shown. If an excitation is applied in Y direction, mode 1 will have the highest PF and mode 2 a lower PF. Mode 3 will have zero PF. If the excitation is in the X direction, then modes 1 and 2 will have zero PF, whereas mode 3 will have a high PF. January 30, 2001 Inventory #001447 6-15

The mode coefficient Ai for each mode is Ai = Sigi * Spectrum Analysis - Terminology & Concepts … Approach - Mode coefficient The mode coefficient is the “scale factor” used to multiply the mode shapes to get the maximum response. The mode coefficient Ai for each mode is Ai = Sigi * Si is the response spectrum value at frequency wi gi is the participation factor for mode i The maximum modal response is then computed as {U}i max = Ai {f}i *A different formula is used for acceleration, velocity and force spectra; see the ANSYS Theory Manual. January 30, 2001 Inventory #001447 6-16

Spectrum Analysis - Terminology & Concepts … Approach - Mode combination Once the maximum response at each mode is known for a given response spectrum, these need to be combined in some way to get the total response. The simplest combination is to add all the maximum modal responses. However, it is highly unlikely that all the maximum modal responses will occur at the same time. Several standard combination methods are published in the literature. Usually each industry’s regulating authority recommends or enforces a technique most suitable for that industry. January 30, 2001 Inventory #001447 6-17

Six different combination methods are available in the ANSYS program: Spectrum Analysis - Terminology & Concepts … Approach - Mode combination Six different combination methods are available in the ANSYS program: Complete Quadratic Combination (CQC) method Grouping Method (GRP) Double Sum method (DSUM) Square Root of the Sum of the Squares (SRSS) method Naval Research Laboratory (NRL) sum method (DDAM) Power Spectral Density method January 30, 2001 Inventory #001447 6-18

Spectrum Analysis … Terminology & Concepts We will discuss the procedure for a single-point response spectrum analysis. In the following discussion, we will use the term “response spectrum” to mean single-point response spectrum. To learn about multi-point response spectrum and DDAM, please refer to the ANSYS Structural Analysis Guide. January 30, 2001 Inventory #001447 6-19

C. Procedure Five main steps: Build the model Obtain the modal solution Switch to spectrum analysis type Define the response spectrum Solve and review results January 30, 2001 Inventory #001447 6-20

Response Spectrum Procedure … Obtain the Modal Solution Mode extraction: Only valid methods are Block Lanczos, subspace, or reduced. Block Lanczos strongly recommended Extract enough modes to cover the spectrum’s frequency content. Expand all modes. Only expanded modes can be used for the spectrum solution. Loads and BC’s: For a base excitation, be sure to constrain the appropriate DOFs. Files: The .mode file contains the eigenvectors and is needed for the spectrum solution. January 30, 2001 Inventory #001447 6-21

Response Spectrum Procedure Switch to Spectrum Analysis Type Build the model Obtain the modal solution Switch to spectrum analysis type Exit and re-enter Solution New analysis: Spectrum Analysis options: Discussed next Damping: Discussed next January 30, 2001 Inventory #001447 6-22

Response Spectrum Procedure … Switch to Spectrum Analysis Type Analysis options Type of spectrum: Single point Number of modes: If 0 or blank, all expanded modes are used for solution. January 30, 2001 Inventory #001447 6-23

Response Spectrum Procedure … Switch to Spectrum Analysis Type Damping Available forms of damping are: Beta (stiffness) damping Constant damping ratio. Can be material dependent but only if specified as a material property* in the modal step. Frequency dependent damping ratio (modal damping) Some form of damping must be specified for the CQC mode combination method. *Material property DAMP in this case is damping ratio, not beta damping. January 30, 2001 Inventory #001447 6-24

Response Spectrum Procedure Define the Response Spectrum Build the model Obtain the modal solution Switch to spectrum analysis type Define the response spectrum Settings: type of spectrum and excitation direction Table of spectral value versus frequency Mode combination method January 30, 2001 Inventory #001447 6-25

Response Spectrum Procedure … Define the Response Spectrum Settings: Type of spectrum Seismic or force (not PSD) Seismic spectra - automatically applied at the base Force spectrum - manually applied at desired nodes as a force Excitation direction (global Cartesian) Specified by a unit vector for seismic spectra: 1,0,0 means X; 0,1,0 means Y; 0,0,1 means Z. Implied by FX, FY, or FZ labels for force spectrum. January 30, 2001 Inventory #001447 6-26

Response Spectrum Procedure … Define the Response Spectrum Spectral value vs frequency table First define frequency table. Up to 20 points are allowed. Then define corresponding spectral values. Specify damping ratio only for multiple spectral curves. For a force spectrum, the spectral values can be scaled by the applied force value. January 30, 2001 Inventory #001447 6-27

Response Spectrum Procedure … Define the Response Spectrum Mode combination method Determines how the individual modal responses are combined. Five methods are available: CQC (Complete Quadratic Combination) GRP (Grouping) DSUM (Double Sum) SRSS (Square Root of Sum of Squares) NRLSUM (Naval Research Laboratory Sum) Which method you choose typically depends on company or government standards being followed. January 30, 2001 Inventory #001447 6-28

Response Spectrum Procedure … Define the Response Spectrum Mode combinations (continued) The significance threshold allows you to include only significant modes in the mode combination. It is the ratio of the mode coefficient of a mode to the maximum mode coefficient. Use a zero value to include all modes. Type of output allows calculation of different response quantities: displacement, velocity, or acceleration. January 30, 2001 Inventory #001447 6-29

Response Spectrum Procedure Solve and Review Results Build the model Obtain the modal solution Switch to spectrum analysis type Define the response spectrum Solve and review results Solve the current load step. Mode combination calculations are written as POST1 commands to the .mcom file. Review results: discussed next. January 30, 2001 Inventory #001447 6-30

Response Spectrum Procedure … Solve and Review Results Enter POST1 (general postprocessor). Perform mode combinations Commands to do this are written to .mcom file during solution. Read the file jobname.mcom using Utility Menu > File > Read Input from... Review deformed shape. Plot and list stresses and strains. January 30, 2001 Inventory #001447 6-31

Response Spectrum Analysis Procedure Build the model Obtain the modal solution Switch to spectrum analysis type Define the response spectrum Solve and review results January 30, 2001 Inventory #001447 6-32

D. Spectrum Analysis Guidelines Modal analysis Make sure you extract and expand enough modes in the modal analysis to cover the frequency range of interest. For example, if the spectrum extends from 1 to 1000 Hz, a rule of thumb is to extract and expand modes up to 1500 Hz. Block Lanczos extraction technique recommended Use Lagrange multiplier (accurate) method if large numbers of constraint equations are present. If you have material dependent damping ratio, this should be specified in the modal analysis. January 30, 2001 Inventory #001447 6-33

… Spectrum Analysis Guidelines Remember that no results file is written in a spectrum analysis. Instead the instructions for mode combination are written to jobname.mcom. Most combination methods involve squaring operations causing the component stresses to lose their signs. Hence deriving equivalent or principal stresses from these unsigned components will be non-conservative and incorrect. If equivalent or principal stresses and strains are of interest then you need to issue the command SUMTYPE,PRIN ( General Postprocessor > Load Case > Stress Option …) before reading in jobname.mcom. This causes direct operation on derived quantities leading to more conservative results. January 30, 2001 Inventory #001447 6-34

… Spectrum Analysis Guidelines During the spectrum analysis the effective mass for each mode as well as the sum of all the effective mass is printed out. For a lumped mass system the sum of the effective masses should approach the total mass of the structure as the number of modes used in the spectrum analysis is increased. The total effective mass is an indicator of whether enough modes are included in the spectrum analysis. January 30, 2001 Inventory #001447 6-35

E. Workshop - Response Spectrum Analysis In this workshop, you will determine the response of a workbench table to a response spectrum excitation. See your Dynamics Workshop supplement for details. (Response Spectrum Workshop - Workbench Table, Page W-40. ). January 30, 2001 Inventory #001447 6-36

F. Random Vibration Analysis Topics covered: Definition and purpose Overview of ANSYS capabilities ANSYS procedure January 30, 2001 Inventory #001447 6-37

Random Vibration Analysis Definition and Purpose What is random vibration analysis? A spectrum analysis technique based on probability and statistics. Meant for loads such as acceleration loads in a rocket launch that produce different time histories during every launch . Reference: Random vibrations in mechanical systems by Crandall & Mark January 30, 2001 Inventory #001447 6-38

Random Vibration Analysis … Definition and Purpose Transient analysis is not an option since the time history is not deterministic. Instead, using statistics the sample time histories are converted to Power Spectral Density function (PSD), a statistical representation of the load time history. Image from “Random Vibrations Theory and Practice” by Wirsching, Paez and Ortiz. January 30, 2001 Inventory #001447 6-39

Random Vibration Analysis … Definition and Purpose What is a PSD? A PSD records the mean square value of the excitation and response as a function of frequency. The area under a PSD curve is the variance of the response (square of the standard deviation). The units used in PSD is mean square/Hz (e.g. an acceleration PSD will have units of G2/Hz). The quantity represented by PSD may be displacement, velocity, acceleration, force, or pressure. January 30, 2001 Inventory #001447 6-40

Random Vibration Analysis … Definition and Purpose Typical applications include Aircraft electronic packaging Airframe parts under atmospheric loading Blast deflectors Laser guidance systems Stable optical platform for telescopes Seismic loading of large structures January 30, 2001 Inventory #001447 6-41

Random Vibration Analysis … Definition and Purpose Input: The structure’s natural frequencies and mode shapes The PSD curve (explained next) Output: 1s displacements and stresses that can be used for fatigue life prediction. Response PSD curves that show the frequency content of any output quantity ( RPSD ). Undocumented (FPAS and RISK ) life prediction capability. January 30, 2001 Inventory #001447 6-42

Random Vibration Analysis Overview of ANSYS Capabilities Loading: Base or nodal excitation Single-point excitation e.g. Single PSD excitation applied to all ground nodes Multi-point (i.e., multi-spectra) excitation Uncorrelated Partially correlated Fully correlated Partial correlation in terms of spatial coordinates Partial correlation in terms of a traveling wave January 30, 2001 Inventory #001447 6-43

Random Vibration Analysis … Overview of ANSYS Capabilities Solution: Relative or absolute 1s output Option for calculating 1s forces/stresses etc. Solution for complete structure i.e., results can be contoured. Output in form of 1s displacements, velocities or accelerations January 30, 2001 Inventory #001447 6-44

Random Vibration Analysis … Overview of ANSYS Capabilities Postprocessing: 1s results can be contoured like any other analysis. Response PSD can be computed for any result quantity ( e.g. stress or nodal force at a node of an element) or cross response spectra can be computed between any two quantities (RPSD). This enables the user to look at the frequency content of output. Covariance between any two quantities can be computed (CVAR). Undocumented commands RISK and FPAS allow user to compute equivalent stress / predict life. January 30, 2001 Inventory #001447 6-45

Random Vibrations Procedure Six main steps: Build the model Obtain the modal solution Switch to spectrum analysis type Define and apply the PSD excitation Solve Review results January 30, 2001 Inventory #001447 6-46

Random Vibrations Build the Model Same considerations as a modal analysis. Linear elements and materials only. Nonlinearities are ignored. Remember density! Also, if material-dependent damping is present, it must be defined in this step. See also Modeling Considerations in Module 1. January 30, 2001 Inventory #001447 6-47

Random Vibrations Obtain the Modal Solution Build the model Obtain the modal solution Same procedure as a normal modal analysis. A few differences, discussed next. January 30, 2001 Inventory #001447 6-48

Random Vibrations … Obtain the Modal Solution Mode extraction: Only valid methods are Block Lanczos, subspace, or reduced. Block Lanczos strongly recommended Extract enough modes to cover the spectrum’s frequency content. Expand all modes. Only expanded modes can be used for the spectrum solution. January 30, 2001 Inventory #001447 6-49

Random Vibrations … Obtain the Modal Solution Loads and BC’s: For a base excitation, be sure to constrain the appropriate DOFs. For a pressure PSD, apply the pressures on desired surfaces in this step. Files: The .mode file contains the eigenvectors and is needed for the spectrum solution. January 30, 2001 Inventory #001447 6-50

Random Vibrations Switch to Spectrum Analysis Type Build the model Obtain the modal solution Switch to spectrum analysis type Exit and re-enter Solution New analysis: Spectrum Analysis options: Discussed next Damping: Discussed next January 30, 2001 Inventory #001447 6-51

Random Vibrations … Switch to Spectrum Analysis Type Analysis options Type of spectrum: PSD Number of modes: If 0 or blank, all expanded modes are used for solution. Element calculations: can be ON only if they were ON in the modal step. January 30, 2001 Inventory #001447 6-52

Random Vibrations … Switch to Spectrum Analysis Type Damping All four forms are available. Alpha (mass) damping Beta (stiffness) damping Constant damping ratio Frequency dependent damping ratio (modal damping) If no damping is specified, ANSYS uses a 1% constant damping ratio as default. January 30, 2001 Inventory #001447 6-53

Random Vibrations Define and Apply the PSD Excitation Build the model Obtain the modal solution Switch to spectrum analysis type Define and apply the PSD excitation Specify PSD settings Define PSD versus frequency table Apply excitation at desired nodes January 30, 2001 Inventory #001447 6-54

Random Vibrations … Define and Apply the PSD Excitation PSD settings Spectrum type (units) Acceleration (normal units or g2/Hz) Velocity Displacement Force Pressure Table number defaults to 1. Used for multiple PSD curves. January 30, 2001 Inventory #001447 6-55

Random Vibrations … Define and Apply the PSD Excitation PSD versus frequency table Specify table number (usually 1). Then enter frequency and PSD value pairs. January 30, 2001 Inventory #001447 6-56

Random Vibrations … Define and Apply the PSD Excitation PSD versus frequency table (continued) Graph the PSD table to verify the input. January 30, 2001 Inventory #001447 6-57

Random Vibrations … Define and Apply the PSD Excitation Procedure depends on the type of PSD. Acceleration, velocity, or displacement PSD: These are base excitations and can be applied only at previously constrained nodes. Apply as a constraint in UX, UY, or UZ (excitation direction) with a value of 1.0. Pick nodes... January 30, 2001 Inventory #001447 6-58

Random Vibrations … Define and Apply the PSD Excitation Apply the PSD (cont'd.) Force PSD Nodal excitation Apply as a force in FX, FY, or FZ (excitation direction) with a value of 1.0 (or desired scale factor). Pressure PSD Requires pressure to be applied in the modal step. Use the load vector (calculated during modal solution) to apply the pressure PSD excitation. Set value to 1.0 or desired scale factor. January 30, 2001 Inventory #001447 6-59

Random Vibrations Solve Build the model Obtain the modal solution Switch to spectrum analysis type Define and apply the PSD excitation Solve Activate PSD mode combination method Specify items to be calculated* Calculate participation factors* Initiate PSD solution* *Discussed next January 30, 2001 Inventory #001447 6-60

Random Vibrations … Solve Items to be calculated: Default is to calculate the displacement solution (including stresses and strains) relative to base excitation. Velocity and acceleration solutions are also available, relative to base or absolute. January 30, 2001 Inventory #001447 6-61

Random Vibrations … Solve Calculate participation factors: Must be done for each PSD table defined. Specify base or nodal excitation. Initiate PSD solution: Results are written to the .rst file. January 30, 2001 Inventory #001447 6-62

Random Vibrations Review Results Build the model Obtain the modal solution Switch to spectrum analysis type Define and apply the PSD excitation Solve Review results Plot and list 1s quantities (POST1) Generate a response PSD (POST26) Calculate covariance between two quantities (POST26) Life prediction January 30, 2001 Inventory #001447 6-63

Random Vibrations- Review Results Review 1-Sigma Stresses Random vibration results are 1s quantities: 1s displacements, 1s stresses, etc. All quantities assume a Gaussian (normal) distribution with zero mean. For example, a maximum displacement of Umax = 0.15 indicates a 68% probability (1s) that Umax will be 0.15 or less. It also indicates: a 95% probability (2s) that Umax will be 0.15x2 = 0.3 or less. a 98% probability (3s) that Umax will be 0.15x3 = 0.45 or less. Gaussian (normal) Distribution 1s 2s 3s January 30, 2001 Inventory #001447 6-64

Random Vibrations- Review Results … Review 1-Sigma Stresses To review 1s displacements & stresses: Enter POST1 (General Postproc). Read results from load step 3, which is where 1s results are stored on the results file. Note: 1s velocities and 1s accelerations, if requested, are stored in load steps 4 and 5, respectively always. Then plot and list the desired quantities. January 30, 2001 Inventory #001447 6-65

Random Vibrations- Review Results Review 1-Sigma Stresses January 30, 2001 Inventory #001447 6-66

Random Vibrations- Review Results Review 1-Sigma Stresses 1s results are typically used for: Fatigue calculations In PSD analyses, the average frequency of excitation (number of cycles/second) is given by 1s velocity / 1s displacement. Using normal distribution the stress level is at 1s 68% of the time, at 2s 27% of the time (95-68), and at 3s 3% of the time (98-95). Knowing the above two quantities, fatigue life can be predicted using usual S-N diagram procedures. January 30, 2001 Inventory #001447 6-67

Random Vibrations- Review Results Response PSD Gives engineers an idea of how a response quantity (stress, for example) varies with frequency. Results file contains 1s values, which is the square root of the area under the PSD curve. POST26, the time-history postprocessor, is used to calculate response PSD. January 30, 2001 Inventory #001447 6-68

Random Vibrations- Review Results … Response PSD To calculate response PSD 1. Enter POST26 and first store the frequency vector. You can use 1 to 10 additional data points on either side of a natural frequency for a smoother frequency curve. Default is 5. Variable 1 is automatically assigned to the frequency vector. January 30, 2001 Inventory #001447 6-69

Random Vibrations- Review Results … Response PSD 2. Identify results quantities for which response PSD is to be calculated. TimeHist Postpro > Define Variables... Can be any nodal or element result item. Choose category, then pick node... January 30, 2001 Inventory #001447 6-70

Random Vibrations- Review Results … Response PSD 3. Calculate and plot the response PSD. TimeHist Postpro > Calc Resp PSD... TimeHist Postpro > Graph Variables… January 30, 2001 Inventory #001447 6-71

Random Vibrations- Review Results Covariance Covariance represents the correlation between two quantities. Can be calculated between any two response quantities; for example, stress at two different points in the model. POST26, the time-history postprocessor, is used to calculate covariance. January 30, 2001 Inventory #001447 6-72

Random Vibrations- Review Results … Covariance To calculate covariance: 1. Reset or exit and re-enter POST26. 2. Identify the two response quantities for which covariance is to be calculated. January 30, 2001 Inventory #001447 6-73

Random Vibrations- Review Results … Covariance 3. Calculate and retrieve the covariance. TimeHist Postpro > Calc Covariance... Use *GET to retrieve the covariance: *GET,COVAR,VARI,#,EXTREM,CVAR -or- Utility Menu > Parameters > Get Scalar Data... January 30, 2001 Inventory #001447 6-74

Random Vibrations- Review Results RISK & equivalent stress In PSD analysis only component stresses are valid (the mode combinations work only on component stresses). Since the component stresses are 1s statistical quantities, equivalent stress and principal stresses cannot be computed in the usual way. The RISK command can calculate equivalent stress by Monte Carlo simulation. RISK can also be used for life prediction. January 30, 2001 Inventory #001447 6-75

Random Vibrations- Review Results … RISK & equivalent stress Sample uses of RISK command CASE 1 INPUT: Location of interest and Design strength (or mean and standard deviation of strength) OUTPUT: Safety margin, Probability of failure, PDF, CDF CASE 2 INPUT: Location of interest and Acceptable probability of failure OUTPUT: Required design strength, PDF, CDF PDF - probability density function CDF- Cumulative probability density function January 30, 2001 Inventory #001447 6-76

Random Vibrations- Review Results … RISK & equivalent stress RISK analysis sample output RISK COMMAND WAS ISSUED FOR NODE 50 OF ELEMENT 28 SIX ESOL COMMANDS WITH VARIABLE NUMBERS 3 THRU 8 ARE CREATED THE COVARIANCE MATRIX OF CARTESIAN STRESS RESPONSES IS COMPUTED BELOW: SX SY SZ SXY SXZ SYZ 0.1665E+07 0.2156E+07 0.0000E+00 -0.5234E+05 0.0000E+00 0.0000E+00 0.2156E+07 0.2791E+07 0.0000E+00 -0.6776E+05 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -0.5234E+05 -0.6776E+05 0.0000E+00 1645. 0.0000E+00 0.0000E+00 A DETERMINISTIC DESIGN STRENGTH = 4687.5 WAS SPECIFIED STATISTICS BASED ON THE SAFETY MARGIN OF VON MISES STRESS WILL BE COMPUTED MEAN OF THE SAFETY MARGIN = 3452.3 STANDARD DEVIATION = 900.41 COEFFICIENT OF SKEWNESS = -1.0480 COEFFICIENT OF KURTOSIS = 3.9813 COMPUTED SAFETY INDEX = 3.8342 COMPUTED PROBABILITY OF FAILURE = 0.20000E-02 COEFFICIENT OF VARIATION FOR COMPUTED PROBABILITY OF FAILURE = 0.70640E-01 BASED ON 95% CONFIDENCE THE PROBABILITY OF FAILURE IS LESS THAN = 0.22324E-02 January 30, 2001 Inventory #001447 6-77

Random Vibrations- Review Results First Passage Failure When will it fail? What is the probability of failure? a X(t) t t January 30, 2001 Inventory #001447 6-78

Random Vibrations- Review Results First Passage Failure The FPAS command can be used to estimate first passage failure FPAS Works with displacement and component stresses Sample uses of FPAS command: CASE 1 INPUT: location , max. allowable value, desired probability of failure OUTPUT: Statistical average frequency, life in seconds CASE 2 INPUT: Location, maximum allowable value, time to failure OUTPUT: Statistical average frequency, probability of failure January 30, 2001 Inventory #001447 6-79

Random Vibrations- Review Results First Passage Failure Typical output for First Passage Failure fpas,ttfa,4,3,1,6.0,0.001 COMPUTED STATISTICAL AVERAGE FREQUENCY IS 285.36 THE EXPECTED NUMBER OF POSITIVE CROSSING OF THRESHOLD VALUE 1.0906 PER UNIT TIME IS 0.69170E-06 BASED ON FAILURE PROBABILITY OF 0.10000E-02 THE TIME TO FAILURE IS 1445.7 TIME TO FAILURE IS 0.144571E+04 January 30, 2001 Inventory #001447 6-80

Random Vibrations Procedure Build the model Obtain the modal solution Switch to spectrum analysis type Define and apply the PSD excitation Solve Review results January 30, 2001 Inventory #001447 6-81

G. Workshop – Random Vibration (PSD) In this workshop, you will determine the displacements and stresses in a model airplane wing due to an acceleration PSD. See your Dynamics Workshop supplement for details (Random Vibration Workshop - Model Airplane Wing , Page W-43 ). January 30, 2001 Inventory #001447 6-82