Dynamics 70 M6 Spectrum

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Spectrum Analysis

Module 6


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Spectrum Analysis

A. Def ini t ion & Purpose

What is spectrum analysis?

A technique to compute a structures

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 loads

time history in the frequency domain.

This is also referred to as response

spectrum.


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Acceleration vs. time Acceleration spectrum (G vs. Hz)

Spectrum Analysis

Definition & Purpose

El Centro Earthquake ( 1940 )

A structure subject to the El Centro earthquake can be analyzed

using either a Transient analysis or spectrum analysis.


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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.

Spectrum Analysis



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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 responsequickly, and some information is lost (phase).

Spectrum Analysis



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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 othererratic loads

Building frames and bridges

Spectrum Analysis



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** Covered in this seminar

Spectrum Analysis


ANSYS allows four types of spectrum analysis:

Single-point respo nse spectrum**

A single response spectrum excites all specified points in the model.

Mult i -point respon se spectrum **

Different response spectra excite different points in the model.

Dynamic d esign analys is m ethod (DDAM) A specific type of spectrum defined by the U.S. Naval Research

Laboratory to evaluate shock resistance of shipboard equipment.

Power Spectral Dens ity (PSD)**

A probabilistic approach used in random vibration analysis.


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Spectrum Analysis

B. Term inology & Concepts

Topics covered:

Definition of a spectrum

How a response spectrum is used to calculate a structures

response to the excitation

Participation factor

Mode coefficient Mode combination


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Spectrum Analysis - Terminology & Conc epts

Def in i tion of spect rum

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


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If the shaker table is excited at frequency f1

and the displacement response of the foursystems 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 aresponse

spectrum.

f

u

f

u

f

u


Def in i t ion of spect rum


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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 respo nse spectrum

curve and a direct ion o f excitat ion.


Def in i t ion of spect rum


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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.


Approach


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For each mode of the structure, a part ic ipat ion factorgi

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.


Approach - Part ic ipation factor


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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.

mode 3

mode 2

mode 1

Y

X


Approach - Part ic ipation factor


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The mode coefficient is the scale factor used to multiply the

mode shapes to get the maximum response.

The mode coef f ic ient Ai for each mode is Ai= Sigi *

Siis 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{ }i

*A different formula is used for acceleration, velocity and force spectra;

see the ANSYS Theory Manual.


Approach - Mode coef f ic ient


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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 theliterature. Usually each industrys regulating authority

recommends or enforces a technique most suitable for that

industry.


Approach - Mode combination


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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


Approach - Mode combination


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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 Structu ral Analysis Guide.

Spectrum Analysis

Terminology & Concepts


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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


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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 spectrums frequency content.

Expand all modes. Only expanded modes can be used for the

spectrum solution. Loads and BCs: For a base excitation, be sure to constrain the

appropriate DOFs.

Files: The .modefile contains the eigenvectors and is needed for the

spectrum solution.


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Sw itch to Spectrum Analys is Type

Build the model



Exit and re-enter Solution

New analysis: Spectrum

Analysis options: Discussed next

Damping: Discussed next


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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.


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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 (mo dal damping)

Some form of damping must be

specified for the CQC mode

combination method.

*Material property DAMP in this caseis damping ratio, not beta damping.


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Def ine the Response Spectrum

Build the model




Settings: type of spectrum and excitation direction

Table of spectral value versus frequency

Mode combination method


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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 aforce

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.


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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.


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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.




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Mode combinations (continued)

The signi f icance thresholdallows 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.




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Solve and Review Results

Build the model




Solve and review results Solve the current load step.

Mode combination calculations are written as

POST1 commands to the .mcomfile.

Review results: discussed next.


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Solve and Review Results

Review results:

Enter POST1 (general postprocessor).

Perform mode combinations

Commands to do this are written to .mcom file during solution.

Read the filejobnam e.mcom using Utility Menu > File > Read Input from...

Review deformed shape.

Plot and list stresses and strains.


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Response Spectrum Analysis Procedure

Build the model




Solve and review results


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D. Spectrum Analysis Gu idelines

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

If you have material dependent damping ratio, this should be specifiedin the modal analysis.


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Spectrum Analysis Guidelines

Spectrum analysis

Remember that no results file is written in a spectrum analysis.

Instead the instructions for mode combination are written to

jobname.m com.

Most combination methods involve squaring operations causing the

component stresses to lose their signs. Hence deriving equivalent o r

pr inc ipal stressesfrom 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 > Calc Options > Stress Options) before reading in

jobname.mcom. This causes direct operation on derived quantities

leading to more conservative results.


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Spectrum analysis

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 areincluded in the spectrum analysis.

***** RESPONSE SPECTRUM CALCULATION SUMMARY

CUMULATIVE

MODE FREQUENCY SV PARTIC.FACTOR MODE COEF. M.C. RATIO EFFECTIVE MASS MASS FRACTION

1 2.37E-04 10 -1.18E-20 -5.34E-14 0 1.40E-40 3.07E-38

2 474 21.099 6.22E-02 1.48E-07 1 3.87E-03 0.85132

3 1182 10 1.14E-15 2.07E-22 0 1.30E-30 0.85132

4 1182 10 3.42E-16 6.20E-23 0 1.17E-31 0.85132

5 1881 10 -5.08E-16 -3.64E-23 0 2.58E-31 0.85132

6 2361 10 3.52E-11 1.60E-18 0 1.24E-21 0.85132

7 2361 10 -2.60E-02 -1.18E-09 0.007981 6.76E-04 1

8 3044 10 -4.39E-13 -1.20E-20 0 1.93E-25 1

9 3044 10 1.27E-12 3.48E-20 0 1.62E-24 1

10 4011 10 5.08E-12 8.00E-20 0 2.58E-23 1

SUM OF EFFECTIVE MASSES 4.55E-03

Spectrum Analysis Guidelines


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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 Workshopsupplement for details. (Response

Spectrum Works hop - Workbench Table, Page W-49.).
http://localhost/var/www/apps/conversion/tmp/ws_ppt/Workshop.ppthttp://localhost/var/www/apps/conversion/tmp/ws_ppt/Workshop.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_8/57%20Workshop.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_8/57%20Workshop.ppthttp://localhost/var/www/apps/conversion/tmp/ws_ppt/Workshop.ppthttp://localhost/var/www/apps/conversion/tmp/ws_ppt/Workshop.ppthttp://localhost/var/www/apps/conversion/tmp/ws_ppt/Workshop.ppthttp://localhost/var/www/apps/conversion/tmp/ws_ppt/Workshop.ppthttp://localhost/var/www/apps/conversion/tmp/ws_ppt/Workshop.ppthttp://localhost/var/www/apps/conversion/tmp/ws_ppt/Workshop.ppthttp://localhost/var/www/apps/conversion/tmp/ws_ppt/Workshop.ppt


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F. Random Vibrat ion Analysis

Topics covered:

Definition and purpose

Overview of ANSYS capabilities

ANSYS procedure


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Random Vibrat ion An alysis

Def ini t ion 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


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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 fun ct ion (PSD),a statistical representation

of the load time history.



Image from Random Vibrations Theory and Practice by Wirsching, Paez and Ortiz.


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What is a PSD?

A PSDrecords the mean squarevalue 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.




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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




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Input:

The structures natural frequencies and mode shapes

The PSD curve (explained next)

Output:

1sdisplacements 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.




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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


Overview of ANSYS Capab i l i t ies


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Solution:

Relative or absolute 1soutput

Option for calculating 1sforces/stresses etc.

Solution for complete structure i.e., results can be contoured.

Output in form of 1sdisplacements, velocities or accelerations



R d Vib i A l i


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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 RISKand FPASallow user to compute

equivalent stress / predict life.




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Random Vibrat ions Procedure

Six main steps:

Build the model



Define and apply the PSD excitation

Solve

Review results

R d Vib t i


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Random Vibrat ions

Bu i ld the Model

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 Model ing Con siderat ionsin Module 1.

R d Vib t i


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Random Vibrat ions

Obtain the Modal Solut ion

Build the model


Same procedure as a normal modal

analysis.

A few differences, discussed next.

R d Vib t i


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Random Vibrat ions


Mode extraction:

Only valid methods are Block Lanczos, subspace, or reduced.

Block Lanczos strongly recommended

Extract enough modes to cover the spectrums frequency content.

Expand all modes. Only expanded modes can be used for the

spectrum solution.

Random Vibrat ions


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Loads and BCs:

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.

Random Vibrat ions


R d Vib t i


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Random Vibrat ions

Sw itch to Spectrum Analys is Type

Build the model



Exit and re-enter Solution

New analysis: Spectrum

Analysis options: Discussed next

Damping: Discussed next

Random Vibrat ions


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Random Vibrat ions

Switch to Spectrum Analys is 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.


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Random Vibrat ions

Define and App ly the PSD Exc itat ion

Build the model




Specify PSD settings Define PSD versus frequency table

Apply excitation at desired nodes

Random Vibrat ions


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Random Vibrat ions

Define and App ly the PSD Excitat ion

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.

Random Vibrat ions


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PSD versus frequency table

Specify table number (usually 1).

Then enter frequency and PSD value pairs.

Random Vibrat ions


Random Vibrat ions


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PSD versus frequency table (continued)

Graph the PSD table to verify the input.

Random Vibrat ions


Random Vibrat ions


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Apply the PSD

Procedure depends on the type of

PSD.

Acceleration, velocity, or

displacement PSD:

These are base excitations and

can be applied only at previouslyconstrained nodes.

Apply as a constraint in UX, UY, or

UZ (excitation direction) with a

value of 1.0.

Pick nodes...

Random Vibrat ions


Random Vibrat ions


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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 applythe pressure PSD excitation.

Set value to 1.0 or desired scale

factor.

Random Vibrat ions


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Random Vibrat ions

Solve

Build the model




Solve

Activate PSD mode combination

method

Specify items to be calculated*

Calculate participation factors*

Initiate PSD solution*

*Discussed next


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a do b a o s

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.

Random Vibrat ions


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Review Results

Build the model




Solve

Review results

Plot and list 1squantities (POST1)

Generate a response PSD (POST26)

Calculate covariance between two quantities (POST26)

Life prediction

Random Vibrat ions- Review Results


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Review 1-Sigma Stresses

Random vibration results are 1s

quantities: 1sdisplacements, 1sstresses, 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 Umaxwill

be 0.15 or less. It also indicates:

a 95% probability (2s) that Umaxwill

be 0.15x2 = 0.3 or less.

a 98% probability (3s) that Umaxwill

be 0.15x3 = 0.45 or less.

1s

2s

3s

Gaussian

(normal)

Distribution



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To review 1sdisplacements & stresses:

Enter POST1 (General Postproc).

Read results from load step 3, which is where 1sresults are stored on the

results file.

Note: 1svelocities and 1saccelerations, if requested, are stored in

load steps 4 and 5, respectively.

Then plot and list the desired quantities.




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1s

results are typically used for:

Fatigue calculations

In PSD analyses, the average frequency of excitation (number of

cycles/second) is given by 1svelocity / 1sdisplacement.

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 usingusual S-N diagram procedures.




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Response PSD

Gives engineers an idea of how a response quantity (stress, for

example) varies with frequency.

Results file contains 1svalues, which is the square root of the

area under the PSD curve.

POST26, the time-history postprocessor, is used to calculate

response PSD.

Response PSD



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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.

Response PSD



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2. Identify results quantities for which response PSD is to be

calculated. TimeHist Postpro > Variable Viewer

Can be any nodal or element result item.

Choose category,

then pick node...

Response PSD



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3. Calculate and plot the response PSD.

TimeHist Postpro > Calc Resp PSD...

TimeHist Postpro > Graph Variables

Response PSD



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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.

Covariance



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To calculate covariance:

1. Reset or exit and re-enter POST26.

2. Identify the two response quantities for which covariance is to be

calculated.

Covariance



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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...

Covariance

Random Vibrat ions


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Build the model




Solve

Review results

Procedure

G Workshop Random Vibrat ion (PSD)


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Training ManualG. Workshop Random Vibrat ion (PSD)

In this workshop, you will determine the displacements and

stresses in a model airplane wing due to an acceleration PSD.

See your Dynamics Workshopsupplement for details.

Random Vibrat ion Works hop - Model A irp lane Wing , Page W-55
http://localhost/var/www/apps/conversion/tmp/scratch_8/Dynamics_70_workshops.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_8/Dynamics_70_workshops.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_8/Dynamics_70_workshops.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_8/Dynamics_70_workshops.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_8/Dynamics_70_workshops.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_8/Dynamics_70_workshops.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_8/Dynamics_70_workshops.ppt

Dynamics 70 M6 Spectrum

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