Solving problems in the dynamics with ANSYS.pdf

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Solving problems in the dynamics with

ANSYS

Quick Start Guide

Yekaterinburg, 2002

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Table of contents

Предисловие……………………………………………………………………………… 4

1. HARMONIC ANALYSIS ........................................................ 5

1.1.Introduction .................................................................................... 5

1.2.Full method ................................................................................. 6

1.3.Truncated method ......................................................................... 6

1.4.Superposition method mod ................................................................. 7

2. Solution of the problem of harmonic analysis ............................................. 8

2.1 Application of loads and to obtain a decision ......................................... 9

2.2 Viewing results ..................................................................... 16

2.3 Постпроцессоры…………………………………………………………………. 16

2.4 Harmonic analysis of a simple system ................................................... 19

3. Truncated method ................................................................................. 24

3.1 Application of loads and getting truncated solutions .............................. 24

3.2 Viewing Results truncated solutions ............................................... 26

3.3 Expanding the solution .......................................................................... 26

3.3.1 Enhanced modes .............................................................................. 27

3.3.2. View Results expansion ..................................................... 29

4.Metod superposition modes ...................................................................... 30

4.1 Solution of the harmonic analysis method of the superposition modes ............ 31

4.2 Expanding the solution and view the results .......................................... 33

5. Harmonic analysis of prestressed structures ................ 33

6. Transient analysis ................................................34

6.1 Введение…………………………………………………………………………….. 34

6.2 Preparation for the dynamic transient analysis ..... 34

6.3. Three methods of solution .......................................................................... 35

6.3.1. Full method ................................................................................ 35

6.3.2. Reduced method ..................................................................... 36

6.3.3. Superposition method mod .................................................................. 37

6.4 Dynamic analysis of transients comprehensive method ....................... 38

6.5 Main options ................................................................................. 48

6.6 Options accounting nonlinearities .................................................................. 49

6.7 Output Control Options ................................................................... 49

6.8 Viewing results ........................................................................ 51

6.8.1 Using POST26 ....................................................................... 52

6.8.2 Use POST1 ...................................................................... 53

6.9 Reduced transient dynamic analysis .................. 57

6.9.1 Viewing the results of the reduced solutions .................................... 63

6.9.2 Expanding the solution ..................................................................... 63

6.10 Example transient analysis ................................................ 66

6.11 The method of superposition of modes in transient analysis ........................ 73

6.11.1 Preparation of modal solutions .......................................................... 73

6.11.2. Solution of the superposition modes ................................................... 74

6.12 Dynamic analysis of prestressed structures ............... 78

6.12.1 Dynamic analysis of prestressed structures

comprehensive method ................................................................................... 79


reduced by ....................................................................... 79


by superposition of modes .................................................................... 80

7. Other features of the transient analysis ................................ 80

7.1 Some guidance on the choice of the integration step ............................. 80

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7.2 Automatic time step ......................................................... 83

7.3 Демпфирование…………………………………………………………………… 83

8. Spectral analysis ............................................................. 87

8.1 Types of spectral analysis .............................................................. 88

8.1.1 Response Spectrum ............................................................................ 88

8.1.2 Method of dynamic design ............................................... 88

8.1.3 Method of spectral density ........................................................ 88

8.2 The procedure for solving the problem of spectral analysis ... 89

8.3 Expansion modes .............................................................................. 93

8.4 Combining fashion ........................................................................ 94

8.5 Viewing Results ....................................................................... 95

8.6 Example of calculating a simple system ........................................................ 96

8.7 How to perform an analysis of random fluctuations ......................................... 103

8.7.1 Disclosure modes .............................................................................. 103

8.7.2.Poluchenie spectral solutions ................................................... 103

8.7.3 Combining fashion ..................................................................... 107

8.7.4 Viewing results .................................................................... 107

8.7.5 Calculation of the output spectral density in POST26 ............................. 108

8.7.6 Calculation of covariance functions in POST26 ................................. 109

8.8 How to perform multiple response spectrum analysis .......................... 110

Part 2. Fundamentals of the theory............................................................. 111

9. Modal analysis ................................................................. 111 10. Superposition method MOU ............................................................... 113

10.1 Modal damping ............................................................ 118

11. Spectral analysis ............................................................ 119

11.1 Assumptions and Limitations .............................................................. 120

11.2 Description of analysis ......................................................................... 120

11.3 Odds ratios and modal contribution .............................. 123

12. ANALYSIS OF RANDOM VIBRATIONS ............................................... 127

12.1 Description of the method .......................................................................... 128

12.2 The spectral power density and standard response

response value .............................................................................. 130

12.3 Spectral cross members for partially

correlated input power spectral densities .................... 133

12.4 Correlation space ........................................... 134

12.5 Propagation of .................................................... 136

12.6 Method of multivariate spectral analysis (SPOPT, PSD) 136

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Roman classic, vintage,

Wonderfully long, long, long,

Preachy and sedate,

No romantic ploys.

AS Pushkin

Foreword.

The contents of this book can be divided into two parts. The first part contains

instructions for harmonic analysis, dynamic analysis of transients, spectral analysis. It

details the features of these programs, the possible options of their application, are

some examples. For example, in the chapters on harmonic analysis and the analysis

of transients considered three possible methods: full, truncated (reduced) using the

main degrees of freedom, and the method of superposition of modes. The advantages

and disadvantages of each of these methods.

The second part is devoted to some theoretical illustrations issues discussed in

the first part. For example, the theoretical basis of the method of superposition of

modes (in Russian literature it is called the method of expansion in their own forms).

Briefly reviewed random fluctuations. Briefly reviewed univariate and multivariate

analysis, as well as issues of cross-correlation spectral densities. Of course, this

review is in no way claim to be complete and only serves as the first part.

1. HARMONIC ANALYSIS

1.1. Introduction

Cyclic load will result in a harmonic response of mechanical systems.

Harmonic analysis is used to find the steady response of linear systems loaded with

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sinusoidal forces. The calculation is performed by finding the system response at

several frequencies and plot the frequency response. Maximum response found on the

schedule and will meet the maximum stresses in the structure. Harmonic analysis is

used for finding the maximum levels steady vibration. Transients are not evaluated in

this kind of analysis.

Harmonic analysis is a linear analysis. Some non-linearity, such as plasticity,

contact phenomena, or gaps will be ignored, even if they are defined in the system.

The harmonic analysis can be applied to prestressed designs, such as a violin string

(assuming that the voltage of the harmonic load substantially less than that of the

prestressing).

There are three methods of analysis: full, truncated and the method of

superposition of modes. (Fourth, relatively "expensive" - a method of transient

analysis for a harmonic load, represented as a function of time). Program ANSYS /

Linear Plus allows only the method of superposition of modes. Before learning how

to use these methods, consider the advantages and disadvantages of each method.

1.2 Complete Method

The complete method is the simplest of the three methods. It uses the full

system matrices to calculate the harmonic amplitudes of vibration. Matrices may be

symmetric or symmetric. Full advantage of the method are:

Easier to use because you do not have to worry about choosing the main

degrees of freedom, or their own forms;

It uses the full matrix, so that the approximation of the mass matrix is not

needed;

It allows you to use non symmetric matrices that are typical for some

applications, particularly for acoustic problems;

It calculates the displacements and stresses;

It supports all types of loads: forces at the nodes specified (non-zero)

displacement, load elements (temperature and pressure).

He makes good use of solid-state load.

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The disadvantage is that it does not allow for pre-loading. Another disadvantage of

this method is that it is relatively "expensive" compared with the other two methods.

1.3 Truncated method.

Truncated method allows you to reduce the size of the problem by using the

main degrees of freedom of the system matrices and acronyms. After calculating the

displacement degrees of freedom on the main solution can be extended to the

original, the full set of degrees of freedom. Advantages of this method:

It works faster and is less "expensive" than a complete method;

It takes into account the effect of the preload;

Disadvantages of this method:

In the first step we calculate only move along the main degrees of freedom.

The second step determines the complete displacement and stress;

Load elements (pressure, temperature) are not included;

All loads may be applied only to the principal degrees of freedom of the

user's choice. (This limits the use of solid-state load);

1.4 Superposition method mod.

The method of superposition of modes based on the summation of factors

eigenmodes of modal analysis. Its advantages are:

It is faster than the full and truncated methods for most tasks;

Load on the elements attached to the (previous) modal analysis can be

applied to harmonic analysis using the LVSCALE;

It allows you to "thicken" the decision to design ranges of natural

frequencies. This allows you to get a more accurate response curves;

Effects of preload can be addressed;

This method allows for the modal damping (as a function of frequency).

Disadvantages of the method:

Not specified zero displacement shall not apply;

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When the program is used to find the POWER DYNAMICS natural

frequencies (modal analysis) the initial conditions can not be pre-applied

loads.

All three methods have a common limitation:

All loads must be sinusoidal law changes over time;

All loads must have the same frequency;

Nonlinearity is not taken into account;

Transients are not calculated.

You can work around some of these restrictions using stationary dynamic

analysis with harmonic loads, represented as a function of time.

2. Solution of the problem of harmonic analysis.

First we give a detailed description of harmonic analysis using the full method,

and then give the difference for the truncated method and the method of superposition

of modes.

The procedure for solving the problem consists of three basic steps:

1. Construction of the model.

2. Application of loads and to obtain a decision.

3. Evaluation of the results.

Construction of the model. In this step you define the name of the task, then the task

header using the preprocessor define element type, real constants, material properties

and geometry of the model.

Note 1) When harmonic analysis takes into account only the linear behavior of the

model. Nonlinear elements are interpreted as linear. For example, if you switch

contact elements to the model, their rigidity are calculated at the initial stage, and

does not change thereafter.

Note 2) Must be carefully defined modulus EX (or stiffness in any other form), and

density (DENS) (or mass). Properties of the materials to be linear isotropic or

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orthotropic, temperature-dependent or constant. Nonlinear material properties are

ignored.

2.1. Application of loads and to obtain a decision.

In this step, you define the type of analysis and its options, a load is applied,

specify options load steps and initiate a solution.

The maximum response of the system will be at frequencies coinciding with its own

frequencies. Before the solution of the problem of harmonic analysis is useful to

determine the natural frequencies of modal analysis.

1. Enter into the solutions with the command / SOLU or in the interface: Main

Menu> Solution

2. Determine the type of analysis and its options (Table 2.1)

Type of analysis and its options.

Table 2.1

Options Team Actions in the interface

New Analysis ANTYPE Main Menu> Solution> Analysis Type-New Analys

Analysis Type:

Harmonic Response ANTYPE Main Menu> Solution>Analysis Type-New Analis>

Harmonic Response

Method of solution HR OPT Main Menu> Solution>Analysis Option

Output Format HR OUT Main Menu> Solution>Analysis Option

Matrix view of the

masses LUMPM Main Menu> Solution>Analysis Option

Solver EQSLV Main Menu> Solution>Analysis Option

.

Each of these options is discussed below.

Option New Analysis used if you need to attach the new harmonic force.

Option Harmonic Response determines the type of analysis-harmonic analysis.

Option method of solution allows you to select one of the following methods:

Complete Method (Fool method);

Truncated method (Reduced method);

The method of superposition of modes (Mode superposition method).

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Option output format (Solution Listing Format) allows you to define how the

harmonic movement will be printed (Jobname. OUT). You can choose to output the

real and imaginary parts, or amplitude and phase angle.

Option form of the mass matrix (LUMPM).

This option is used by default or consistent mass matrix approximation lumped

(in this case the mass matrix is diagonal). We recommend this option to accept the

default for most applications. For some problems, using "thin" system, such as thin

beams or very thin shell approximation of the mass matrix using lumped masses often

leads to better results. Mass matrix approximation using concentrated masses requires

less time and memory solutions.

After filling in all the fields of the dialog "box" options harmonic analysis,

click on the "OK" button and a dialog will appear the second "box", where you

choose the method of solutions.

Option "A method of solving" [EQSLV]. You can choose the frontal method

(default), the method of Jacobi joined gradients (JCG), The method of incomplete

Cholesky connected gradients (ICCG). Frontal method is recommended for most

problems in mechanics.

3. The application loads the model.

Harmonic analysis by definition implies that the applied loads vary

sinusoidally in time. For a complete description of harmonic loads using three

parameters: amplitude, phase angle, the frequency range (Figure 2.1).

Fig. 2.1.

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Amplitude - this is the maximum load value that can be selected by using the

commands shown in table 2.2.

Phase angle - is the measured time at which the load is delayed or ahead

relative to a given point in time. On the complex diagram (see Figure 2.1), this angle

is measured with respect to the real axis. The phase angle is only required if there are

multiple loads with different phases. For example, the unbalanced rotation of the

antenna shown in Fig. 2.2 would lead to vertical loads in four reference points with

different phase angles.

Fig.2.2

The phase angle can not be injected directly. For this it is necessary to

determine the real and imaginary component, and using the value VALUE1

VALUE2 in the appropriate command task load. Fig. 2.1 shows how to determine the

real and imaginary components.

Frequency range is a range of frequencies of the harmonic load. It is defined

later in the option of load steps with the command HARFRQ.

Note 1. Harmonic analysis of the reaction can not rely on the simultaneous

action of several forces having different frequency, for example from two machines

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with different angular velocities, operating simultaneously. Although you can use to

account for POST1 joint action of these forces.

Table 2.2 shows the load applicable for static analysis. The exception is the

inertial loads, which you can attach to a solid model (points, lines, surfaces), or finite-

element model (nodes and elements).

Loads applied to harmonic analysis.

Table 2.2.

Type of load Category Family

teams

Interface

Move UX, UY,

UZ, ROTX,

ROTY, ROTZ

Fixation D Main Menu> Solution> Load-Apply>

Structural-Displasement

Force, moment

FX, FY, FZ, MX,

MY, MZ

Energy F Main Menu> Solution> Load-Apply>

Structural-Force/Moment

Pressure (PRES) Surface loads SF Main Menu> Solution> Load-Apply>

Structural-Pressure

Temperature

(TEMP)

Wrap (FLUE)

Bulk load BF Main Menu> Solution> Load-Apply>

Structural-Temperature

Gravity, rotation,

etc.

Inertial loads - Main Menu> Solution> Load-Apply>

Structural-Other

In the process of solving the problem of load can be applied, moved, printed,

or they can be produced mathematical operations. Table 2.3 shows the operations

team with loads for harmonic analysis.

Operations team with loads

Table 2.3

Type of

load

Solid

model or

CEM

Objects Appli

cation

Destruction Print Operations Options

1 2 3 4 5 6 7 8

Movemen

t

Solid Point DK DKDELE DKLIST DTRAN -

Movemen

t

Solid Line DL DLDELE DLLIST DTRAN -

Movemen

t

Solid Surface DA DADELE DALIST DTRAN -

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Continued table. 2.3 1 2 3 4 5 6 7 8

Movemen

t

KEM Nodes D DDELE DLIST DSCALE DSYM

DCUM

Force Solid Point FK FKDELE FKLIST FTRAN -

Force KEM Nodes F FDELE FLIST FSCALE FCUM

Pressure Solid Line SFL SFLDELE LSFLLIST SFTRAN SFGRAD

Pressure Solid Surface SFA SFADELE SFALIST SFTRAN SFGRAD

Pressure KEM Nodes SF SFDELE SFLIST SFSCALE SFGRAD

SFCUM

Pressure KEM Elements SFE SFEDELE SFELIST SFSCALE SFGRAD

SFBEAM

SFFUN

SFCUM

Temperat

ure, flow

around

Solid Point BFK BFKDELE BFKLIST BFTRAN -

Temperat

ure, flow

around

Solid Line BFL BFLDELE BFLLIST BFTRAN -

Temperat

ure, flow

around

Solid Surface BFA BFADELE BFALIST BFTRAN -

To print the existing loads, use:

Utility Menu> List> Loads> Load type

4. Select options loading steps.

Options for the harmonic analysis are listed in Table 2.4.

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Options steps of loading

Table 2.4

Option Team Interface

Main options

The number of harmonics in

the solution NSUBST Main Menu> Solution> Load Step Opt> Time /

Friquens> Freq & Substep

Stepwise or gradual loading

(Step or ramped) KBC Main Menu> Solution> Load Step Opt> Time /

Friquens> Time & Time Step / Freq & Substep

Options dynamics

Frequency range forces HARFRQ Main Menu> Solution> Load Step Opt> Time /

Friquens> Freq & Substep

Damping ALPHAD

BETAD

DMPRAT

Main Menu> Solution> Load Step Opt> Time /

Friquens> Damping

Options presentation of results

Print OUTPR Main Menu> Solution> Load Step Opt> Output

Ctrls> Solu Printout

Database and the resulting file OUTRES Main Menu> Solution> Load Step Opt> Output

Ctrls> DB / Result File

Extrapolation of the results ERESX Main Menu> Solution> Load Step Opt> Output

Ctrls> Integration Pt

Key features include the following:

Number of harmonics solutions [NSUBST]

You can request a certain number of harmonics in the decision. Solution

(steps) will be placed within the selected frequency interval [HARFRQ]. For

example, if you select the 10 harmonics in the range of 30 to 40 Hz, the

program will calculate the responses to 31, 32, 33, ... 39 and 40 Hz. The

reaction was not calculated on the lower frequency band.

Stepwise or gradual loading [KBC].

Load can be stepwise or continuously. By default, it is smooth, this means that the

amplitude of the load is gradually increased with each step. Stepwise load [KBS, 1]

means that the amplitude of the load is kept constant for all steps in the frequency

range.

Frequency range forces [HARFRQ].

The frequency range must be specified in the number of cycles per unit time. Within

this range, you can then select the number of harmonics to be calculated.

Damping.

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Damping in the system must be carefully defined, otherwise can be infinite amplitude

at the resonance frequency. Frequency-dependent damping coefficients entered teams

ALPHAD (proportional to the mass matrix) and BETAD (Proportional to the

stiffness matrix),and frequency-independent damping coefficient by the command

DMPRAT.

RemarkIf damping is not defined in the direct method of harmonic analysis, the

program uses zero damping.

Options presentation of the results include the following:

Print Output [OUTPR] Use this option to include a variety of data results in the

output file (Jobname.OUT).

The database and the data in the result output file [OUTRES]. This option

controls the data in the output file (Jobname.RST).

Extrapolation of the results [ERESX]. Using this option allows you to view the

results on the elements of copying them to the nodes instead of extrapolating

(default).

5. Save a copy of the database in a file named. You can then re-ignite your model,

and introducing restarting ANSYS command RESUME.

Team SAVE. Interface: Utility Menu> File> Save as.

6. Running solutions. Team SOLVE. Interface Main Menu> Solution> Solve -

Current LS.

7. Repeat steps 3 to 6 for other additional loads and other frequency bands. If you

plan to pursue post-processing (POST26), the frequency ranges must not overlap

between load steps. Another method of increasing load steps is to maintain the

loading steps in the file, and then obtaining solutions using a macro.

8. Exit solutions.

2.2. View the results.

Harmonic analysis results are recorded in the results file Jobname.RST They

consist of the following data, each of which depends on the frequency for which the

solution was obtained:

Direct data.

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Move nodes (UX, UY, UZ, ROTX, ROTY, ROTZ).

Derived data:

Voltage at the nodes and elements;

Deformation at the nodes and elements;

Forces in the elements;

Reaction sites, etc.

If you define the system damping, her reaction will not be in phase with the

acting forces. The results of calculations are complex numbers are stored as real and

imaginary parts. Complete results will be obtained if the system is applied to several

forces that differ in phases.

2.3. Postprocessors.

You can view the results using the postprocessor or POST26 or POST1.

Usually first used to identify POST26 frequency at which the maximum displacement

will (or voltage), choose the most interesting points of the model and then used to

study objects POST1 pattern in these critical regimes.

POST1 is used to display the results in model objects at certain

frequencies;

POST26 allows you to view the results in some points of the model in

some frequency bands.

Some typical operations postprocessors for harmonic analysis are given

below. It must be remembered that:

To view the results in POST1 and POST26 database must contain the

same storage model for which the solution is obtained;

Must be received by the resulting file (Jobname.RST).

Using POST26

The result is a table of POST26. Each variable has its own reference

number, the first number is reserved for frequency.

1. Define variables using these options:

Teams:

NSOL to move nodes;

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ESOL for solutions in the elements (eg, stress)

RFORSE reactions (forces).

Interface:

Main Menu> TimeHist Postpro> Define Variables.

Team FORSE allows you to define the full force static components in the total

power damping forces or inertial components.

2. Drawing graphics variables in a function of frequency or other variable. Enter the

command PLCPLX to operate with an amplitude phase angle, the real or

imaginary part.

Teams: PLVAR, PLCPLX

Interface:

Main Menu> TimeHist Postpro> Graph Variables.

Main Menu> TimeHist Postpro> Setting> Graph.

3. Prints variables. To display the extreme values, use the command EXTREM, then

enter the command PRCPLX amplitude, phase angle, the real or imaginary part.

Teams: PRVAR, EXTREM, PRCPLX.

Interface:

Main Menu> TimeHist Postpro> List Variables / List Extremes

Main Menu> TimeHist Postpro> Setting> List.

In POST26 there are many other functions such as mathematical

transformations of variables with complex arithmetic, moving variables into arrays,

etc.

As a result of viewing the results, you can determine the critical frequencies for

future use postprocessor POST1.

Use POST1

1. Read the result of harmonic analysis. For these purposes, you can use the

command SET, But it will read either the real part of the solution, or imaginary.

The true magnitude of the result obtained by applying the command SRSS

(Square root of the sum of squares) of the real and imaginary parts.

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2. Draw a deformed state of the system, the contours of stresses, strains, etc. Or

vector [PLVEC]. For tabulated listing, use commands PRNSOL, PRESOLEtc.

Option of drawing strain state

Team PLDISP

Interface: Main Menu> General Postproc> Plot Results> Deformed Shape

Drawing Paths option.

Teams: PLNSOL or PLESOL.

Interface: Main Menu> General Postproc> Plot Results> Contour Plot-Nodal

Solu or Element Solu.

Use the options to display the results, such as stress (SX, SY, SZ ....), deformation

(EPELX, EPELY, EPELZ ...) and displacement (UX, UY, UZ ...).

Option drawing vector.

Team PLVEC

Interface:

Main Menu> General Postproc> Plot Results> Vector Plot-Predefined.

Option tabular listing.

Teams:

PRNSOL (Results of nodes)

PRESOL (Results of elements)

PRRSOL (Reaction)

NSORT, ESORT.

Interface Main Menu> General Postproc> List Results> Nodal Solution.

Main Menu> General Postproc> List Results> Element Solution.

Main Menu> General Postproc> List Results> Reaction Solution.

To sort the data before printing, use the command NSORT or ESORT.

POST1 postprocessor has many other functions, such as scaling results, the

conversion results in different coordinate systems, the combination of load cases, etc.

2.4. Harmonic analysis of a simple system.

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In this simple problem, we define the harmonic response of two mass system.

We define the amplitude and phase angles Xi ifor each mass mi. The system is under

the action of a harmonic force tF sin1 ,

operating on the weight m1.

Properties of the materials for this

inibmm /sec5.0 2

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inibKKK C /20021

The amplitude of force ibF 2001

The length of the springs is arbitrary and is used only to determine the

direction of deformation of the springs. Two main degrees of freedom defined by the

masses in the direction of the deformation of the springs. Selected frequency range

from zero to 7.5 Hz. Total accounted for 30 harmonics. Frequency step is 7.5/30 =

0.25 Hz

Entering a title problem.

1. Utility Menu> File> Change Title

2. Enter the text «Harmonic Response of Two Mass Spring System» and press

«OK»

Determining the types of elements.

1. Main Menu> Preprocessor> Element Type> Add / Edit / Delete.

2. Click Add. A dialog "box" type elements.

3. Read the list of items on the left side "box" and select «Combination».

4. Click once on the right side of the box on «Spring-damper 14."

5. Click APPLY.

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6. In the left pane, select «Structural Mass» and select them.

7. Click the mouse once on «3D mass 21" on the right side "of the box."

8. Click OK dialog "box" type element closes.

9. Click on the «Close».

Entering real constants.

1. Main Menu> Preprocessor> Real Constants

2. Click Add. Dialog box real constants.

3. Click Type 1.

4. Click OK. Dialog "box" for Combin14.

5. Enter 200 for the spring stiffness (K) and 0.1 for the damping coefficient

(CV1) and click OK.

6. Repeat steps 2-4 for the type of item 2, MASS21.

7. Enter 0.5 for for mass in the X direction, and press OK.

8. Click Close, then close the box real constants.

Construction of units.

1. Main Menu> Preprocessor> Modeling-Create> Nodes> In Active CS.

2. Enter 1 for the host.

3. Type 0, 0, 0 coordinates X, Y, Z, respectively.

4. Click Apply.


6. Type 1, 0, 0 coordinates X, Y, Z, respectively.

7. Click OK.

8. Utility Menu> PlotCtrl> Numbering. Open the control box numbered.

9. Click once «Node Numbers on»

10. Click OK.

11. Main Menu> Preprocessor> Modeling-Create> Nodes> Fill between

Nds

12. In the graphics window, click on the nodes 1 and 4 (the left and right side of

the screen). Small "box" opens around each node.

13. Click OK. Dialog box build assemblies between two nodes.

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14. By default, the node 2 is constructed. Click OK. Nodes 2 and 3 will appear

on the graphics window.

Constructing springs.

1. Main Menu> Preprocessor> Modeling-Create> Elements> Auto

Numbered-Thru Nodes. Open "peak" menu.

2. On the graphics window, click once on nodes 1 and 2.

3. Click Apply. A line appears between the selected nodes.

4. Touch once the nodes 2 and 3.

5. Click Apply. A line appears between the selected nodes.

6. Press once on nodes 3 and 4

7. Click OK. A line appears between the selected nodes.

Constructing the masses

1. Main Menu> Preprocessor> Modeling-Create> Elements> Elem

Attributes

2. Enter the number 2 as the element type.

3. Enter 2 as the number of real constants, and click OK.

4. Main Menu> Preprocessor> Modeling-Create> Elements> Auto

Numbered-Thru Nodes. Open "peak" menu.

5. On the graphics window, click once on node 2.

6. Click Apply.

7. On the graphics window, click once on the site click OK 3.

Selecting the type of analysis, the main degrees of freedom and load steps.

1. Main Menu> Solution> Analysis Type-New Analys

2. Click on «Harmonic» and click OK.

3. Main Menu> Solution>Analysis Option

4. Click on Full, choosing solution method.

5. Click once on the "Amplitude + phase» to select the output format.

6. Click OK on the dialog box Full Harmonic Analyze

7. Main Menu> Solution> Load Step Opt>-Output Ctrls> SoluPrintout.

8. Click on «Last substep» Print frequencies click OK.

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9. Main Menu> Solution> Load Step Opt> Time / Friquens> Freq &

Substep

10. Enter 0 and 7.5 for the frequency range.

11. Enter 30 for the number of steps.

12. Click on «Stepped»

13. Click OK.

Determination of loads and boundary conditions.

1. Main Menu> Solution> Load-Apply> Structural-Displasement> On

Nodes

2. Press «Pick All» «Select All"

3. Select UY.

4. Click OK.

5. Main Menu> Solution> Load-Apply> Structural-Displasement> On

Nodes

6. In the graphics window, click on the nodes 1 and 4.

7. Click OK.

8. In the dialog box, click on the UX and UY.

9. Click on OK.

10. Main Menu> Solution> Load-Apply> Structural-Force/Moment> On

Nodes

11. In the graphics window, click on the node 2.

12. Click OK.

13. In the dialog box, select the FX

14. Enter 200 for the real part and click OK.

Decision model.

1. Main Menu> Solution> Solve - Current LS.

2. See information in the status window and click Close.

3. Click OK.

4. When the solution is completed, a dialog box «Solution is done" Click

Close.

22

View the results.

For this example, we will see the results of solutions for nodes 2 and 3.

1. Defining a list of variables for output: Main Menu> TimeHist Postpro>

Define Variables. Dialog box appears.

2. Click Add. Dialog box appears.

3. The default is the nodal solutions «Nodal DOF». Dialog box nodal

displacements.

4. Enter 2 as the variable number. (No. 1 by default - frequency).

5. Enter 2 as the node number.

6. Enter 2UX, as the label of the user. (The graph will indicate the

corresponding curve with that label).

7. In the right part of the "box" to find «Translation UX»

8. Click OK.

9. Press the «Add» on selection dialog variables. Dialog box appears.

10. Click OK on the nodal solutions «Nodal DOF». Dialog box appears.

11. 3 Enter the number as a variable.

12. Enter 3 as the node number

13. Enter 3UX, as the label of the user.

14. In the right part of the "box" to find «Translation UX»

15. Click OK.

16. Press Close

17. Making graphics Utility Menu> PlotCtrl> Style> Graphs This opens the

control menu graphics

18. In the dialog box to select the type of grid X and Y.

19. Click OK.

20. Output graphics Main Menu> TimeHist Postpro> Graph Variables.

21. Enter 2 as 1st variable schedule.

22. Enter 3 as the 2nd variable schedule.

23. Click OK. Displayed graph (Fig. 2.2).

23

Fig.2.2

3. Truncated method

Truncated method uses truncated matrix for the harmonic analysis. Solution of

the problem consists of the following steps:

1. Construction of the model;

2. Application of loads and getting truncated solutions;

3. View Results truncated solutions.

4. Expanding the solution

5. View Results expanded solutions.

The first step of course, is exactly the same as for the full model.

3.1. Application of loads and getting truncated solutions.

In the method we mean truncated principal degree of freedom.

1. Enter processor solutions.

Team / SOLU Interface: Main menu> Solution

2. Determine the type of analysis and options. Options truncated analysis are

the same as for the full method.

Select the method for solving a truncated;

You can include the effects of prestressing [PSTRES]. This requires the

element files from the previous static or transient dynamic analysis.

24

3. Identify key degrees of freedom. Degrees of freedom are the main

determinants or dynamic degrees of freedom that define the dynamic

behavior of the system. For dynamic analysis of a truncated main degrees of

freedom are selected based on the location on the system where you want to

apply force or zero displacement.

4. Apply the load on the model. The harmonic load may be the same as that

described for the full method, except for some features:

Allowed only application of forces and displacements. Load elements,

such as pressure, temperature, acceleration is not supported.

Forces and non-zero displacement must be attached only to the main

degrees of freedom.

5. Select option loads steps. They are the same as that for the full method,

except for commands OUTRES and ERESX. Team OUTPR controls the

printing resolution of the major nodal degrees of freedom. [OUTPR,

NSOL, ALL (or NONE)].

6. Save a copy of the database.

Utility Menu> File> Save as.

7. Run the solution

Main Menu> Solution> Solve - Current LS.

8. Repeat steps 4 through 7, with additional strength and frequency bands. If

you plan to use a temporary frequency postprocessor POST26, frequency

ranges must not overlap.

9. Close the menu SOLUTION.

3.2 View Results truncated solutions.

Results truncated harmonic analysis are recorded in a truncated file

movements Jobname. RFRQ. It consists of movements on the main degrees of

freedom, which have a harmonic character strength for each frequency for

which the solution was obtained. As a method for complete these movements

will be complex if it was determined damping, or if forces were applied in

more than one phase. You can see the movement on the main degrees of

25

freedom as a function of frequency, using POST26. (POST1 can not be used

because the complete solution for all degrees of freedom is not received). The

procedure for using POST26 exactly the same as for the full method, except for

certain features.

Before defining variables for use POST26 team FILE to select the

data to be read from the file Jobname.RFRQ. For example, if

HARMONIC - name of the task team would be: FILE,

HARMONIC, RFRQ.

Taking into account only the data of the degrees of freedom of

nodes (corresponding to the main degrees of freedom), thus applies

only team NSOL to define variables.

3.3. Expanding the solution

Expansion of solutions designed to calculate the total displacements, stresses

and forces for all degrees of freedom. These calculations are carried out only for the

frequencies and phase angles that you chose. Before initiating extension, you should

review the truncated section using POST26 and identify critical frequencies and

phase angles.

Expanding the solution is not always required. For example, if you are

interested mainly movements in specific points of the structure, the truncated solution

can meet your needs. If you want to define the movement is not on the main degrees

of freedom, or you are interested in stress, you need to apply an extended solution.

It must be remembered that:

Files should be created RFRQ, TRI, EMAT and ESAV;

The database must contain the same model that produced a truncated

section.

3.3.1. Expansion modes

1. Restart processor solutions

26

/ SOLU

Main Menu> Solution.

2. Please enable expansion solution and its options.

Options Teams Interface Rasshienie On / Off EXPASS Main Menu> Solution>Analys Type-Expansion Pass

The number of solutions

to expand NUME XP Main Menu> Solution>Load Step Opt-Expansion

Pass>Range of Solu s Frequency range to

expand NUME XP Main Menu> Solution>Load Step Opt-Expansion

Pass>Range of Solu s Phase angle for expansion HREXP Main Menu> Solution>Load Step Opt-Expansion

Pass>Range of Solu s Stress analysis on / off NUME XP

EXPSOL

Main Menu> Solution>Load Step Opt-Expansion

Pass>Range of Solu s Nodal solutions

Output Format HROUT Main Menu> Solution> Analys Opt

Option Expansion Pass On / Off [EXPASS] (Enhanced On / Off)

Select ON

Option The number of solutions to expand [NUMEXP, NUM].

Select a number.

Option Frequency range expansion [NUMEXP, BEGRNG, ENDRNG].

3. Select the band. (See the previous example). If you do not want to expand

the number of solutions you can use the command EXPSOL for the identification of

a unique solution for the expansion (or by setting the step number of load or by its

frequency). If several solutions will be expanded in a certain frequency range, we

suggest that you request and the real and imaginary parts of the resulting expansion

[HREXP, ALL]. Thus, you can easily combine two parts POST26 for viewing peak

displacements, stresses and other results. On the other hand, for a single solution

obtained team [EXPSOL], You can select the phase angle. In this case, the peak

displacement can be obtained by using the [HREXP,angle].

Option stress evaluation On / Off. [NUMEXP or EXPSOL].

You can turn off the calculation of stresses and forces, if you want. By default,

they are calculated.

27

Optional output format nodal solutions [HROUT].

Determine how harmonic motion will be printed (Jobname. OUT). You can

choose as the real and imaginary parts (by default), and as the amplitude and phase

angle

Team OUTPR. Interface: Main Menu> Solutions> Load Step Opt-Output

Ctrls> Solu Printout.

The database and output files,

The following options are used to manage the data in the resulting output file

(Jobname. RST).

Team OUTRES. Main Menu> Load Step Opt-Output Ctrls> DB / Results

File.

Extrapolation of the results.

Using these options, you can view the results for the integral elements by

copying them to the nodes instead of extrapolating (default).

Note: FREQ field team OUTPR and OUTRES can have two values: ALL or

NONE.

Team ERESX. Interface: Main Menu> Load Step Opt-Output Ctrls>

Integration Pt.

4. Start expansion solution.


5. Repeat steps 2,3 and 4 for additional solutions. Each expansion solution is

stored as a separate step load in the results file.

6. Exit solutions, closing the menu Solution.

3.3.2. View Results expansion

The results of the extended solutions are written to a file Jobname.RST. It

consists of the following data calculated for each frequency for which the solution

was obtained.

Direct data:

Move nodes (UX, UY, UZ, ROTX, ROTY, ROTZ).

28

Derived data:




Reaction at the nodes;

Etc.

You can view them using POST1. If you've spent expansion solutions at

several frequencies, you can use POST26 to plot stress as a function of frequency.

The procedure for using POST1 or POST26 the same as that described for the

full method, except for one feature: if you want the phase angle [HREXP, angle],

The only one solution is possible for each frequency.

4. The method of superposition of modes

The method of superposition of modes based on the summation of the

coefficients of the eigenmodes obtained in modal analysis to calculate the harmonic

response. The procedure of the method consists of the following steps:


2. Solution of the problem on their own frequencies and mode shapes (modal

analysis);

3. Preparation method of superposition of harmonic response modes;

4. Expansion of the solution;

5. View the results.

Some remarks on modal analysis

Use the following methods: subspace (subspace), block (Block Lanczos),

truncated (reduced) or Power Dynamic. (Other methods, such as damped

unbalanced and do not apply for the superposition of modes). If the method

of Power Dynamic modal analysis, it is possible to use non-zero loads or

displacements.

Modal analysis must contain as many modes as necessary to obtain a

harmonic reaction.

29

When using the truncated method include those principal degrees of

freedom, to which were attached harmonic loads;

If you need a harmonically varying loads on the (pressure, temperature,

acceleration), you must select them in the modal analysis. Load ignored in

modal analysis, but a vector of loadings will be calculated and written to

file their own forms (Jobname. MODE). You can use a vector of loadings

in the harmonic solution.

Disclosure eigenmodes is not necessary for the method of superposition. (If

you want to see the mode shapes graphically, it needs their expansion);

These models, such as rotations of nodes should not change between modal

and harmonic analysis.

4.1. Solution of the problem of harmonic analysis method of the

superposition of modes.

In this step, the program uses the mode shapes obtained from modal analysis.

File eigenmodes (Jobname. MOD) should be created and the database must contain

the same model that was conducted modal analysis. If the modal analysis method was

used subspace method or block, and thus the mass matrix used by default, ie, the

diagonal mass matrix, can be obtained by a complete file (Jobname. FULL).

1. Enter into solutions.

/ SOLU or Main Menu> Solution

2. Determine the type of analysis and its options. This is the same as described

for the full method, except for some differences.

Select the method of superposition of modes [HROPT].

Select the mods that you want to use to solve [HROPT]. This determines

the accuracy of the harmonic solutions. Typically, the number of the modes

should be chosen so that their frequency response was 50% greater than the

frequency range of applied forces.

Usually the decision "thickens" design around the natural frequencies

[HROUT] To obtain a more accurate response curve.

30

Usually at each frequency printed summary table of coefficients

contribution of each mode to the overall response of the system. [HROUT].

3. Attach the load to the model. Harmonic loading is the same, except for the

following features:

Only force, acceleration and load vector created in modal analysis are

correct. Use the command LVSCALE application load vector of modal

analysis.

If you are using modes obtained by truncated modal analysis, force must be

applied to the main degrees of freedom.

4. Select the option of load steps. This is the same as that described for the

truncated method, except that you can choose a modal damping [MDAMP].

Using the [NSUBST] You can choose a number of solutions on each side of

the resonance peak, if you select [HROUT]. The default is calculated 4

solutions, but you can choose another number from 2 to 10.

5. If you used the method of subspace or block method, you can use the nodal

components using the OUTRES, NSOL to limit the data movement in the

reduced displacement file Jobname. RFRQ. Solution of the extension will

give the correct results for those nodes and the elements in which all

element nodes are written to a file with the extension RFRQ.

6. Save a copy of the database.

Utility Menu> File> Save as.

7. Run a task on a solution.


8. Repeat steps 3 through 6 for additional loads and frequency ranges. If you

plan to use a temporary frequency postprocessor POST26, frequency ranges

must not overlap.

9. Exit solutions.

4.2. Expanding the solution and view the results

Expansion procedure solutions are the same as that described for the truncated

method. File Jobname. TRI required only if you used the method of disclosure of the

31

reduced modes. Results consist of a harmonically varying displacements, stresses and

strains response for each frequency. You can view these results using POST26 or

POST1

5. Harmonic analysis of prestressed structures.

Harmonic analysis of prestressed structures calculates harmonic response

systems such as a violin string. This is only possible method for solving a truncated

or method of superposition of modes. It is assumed that the amplitude of the voltage

harmonic component is much smaller than the magnitude of the prestressing.

Procedure reduced solutions prestressed structures is the same as that for any

reduced test, except for the fact that you must first carry out static analysis.

1. Build the model and get the static calculation, taking into account the effect

of the preload. [PSTRES, ON].

2. Restart «SOLUTION» now reduced harmonic solution with the inclusion of

the effects of pre-load. [PSTRES, ON]. This should be prepared files

Jobname. EMAT and Jobname. ESAV of static analysis.

3. To account for the effect of preloading method of superposition of modes,

you must first perform a modal analysis based on prestressing. The results

of this analysis will be considered in the method of superposition of modes.

6. Transient analysis

6.1. Introduction.

Transient analysis (sometimes called an analysis time) is used to find the

dynamic response of the system under the action of some of the load-dependent time.

You can use this type of analysis to determine the time dependence of the

displacements, strains, stresses and forces in the system as a response to some

combination of static, transient and harmonic forces. Calculation of the time scale

means that the effects of inertia and damping are important. If these effects are not

taken into account, it is sufficient static analysis.

The basic equation of motion is

)(tFuKuCuM ,

32

where M -Mass matrix, С -Damping matrix, K -Stiffness matrix (u ) Vector

accelerations of nodes u Vector velocity nodes u Vector displacement nodes.

)(tF Load vector.

At some given time t, this equation can be interpreted as a set of "static"

equilibrium equations considering inertia forces and damping forces. ANSYS uses

the Nyumarka procedure for time integration. Increment of time given time steps.

6.2. Preparing for dynamic transient analysis.

You will use the same set of commands for building models and generating

dynamic transient analysis that you use for other types of finite element analysis.

Dynamic transient analysis is more time consuming than static analysis, and requires

large computer resources and very large your "engineering" of resources (eg, time

spent). You can save some of these resources performing some preliminary work for

the understanding of the physical nature of the problem. For example, you can:

1. Initially, the simplest problem to solve. Model consisting of beams, weights

and springs can give a good approximation to the true solution with

minimal time. This simple model can fully satisfy you in solving the

problem of finding the dynamic response of the system.

2. If you turn the system nonlinearity, first try to understand how these effects

influence the system under static load. In some cases, they do not make

sense to include in a dynamic analysis.

3. Try to understand the dynamics of the system. It is useful to carry out a

modal analysis to determine the natural frequencies and vibration modes.

This will help you to understand the response of the system under the action

of loads. In addition, knowledge of the natural frequencies will help you

determine the right size of time steps.

4. For nonlinear problems using technology subsystems (Substructuring) linear

part of the model to reduce solution time.

6.3. Three methods of solution.

There are three methods of solving the problem of dynamic transient

analysis: full, reduced and the method of superposition of modes. Program

33

ANSYS / LINEAR PLUS allows only the method of superposition of modes.

Before a detailed study of each of these methods consider the advantages and

disadvantages of each.

6.3.1. Full method.

Full method uses full system matrices for finding the dynamic response of the

system. He is the most powerful of the three methods because it takes into account all

types of nonlinearities (plasticity, large displacement, large deformations, etc.). If you

do not want to take into account the non-linearity, you should consider the

appropriateness of the other two methods as a complete method - the most

"expensive" of these three methods.

Advantages of the full method:

It is most simple, as it requires no major degrees of freedom of choice or

mode shapes.

It takes into account all types of nonlinearities.

It uses the full matrix so that the matrix mass of approximation is required.

All displacements and stresses are determined in one step.

It takes into account all types of loads: nodal forces are not zero

displacements (although this is not recommended), load elements (pressure

and temperature).

He makes good use of solid-state load.

The main drawback of the method is that it requires more memory and more time

solving.

6.3.2. Reduced method.

Reduced method condenses the size of the problem by using the main

degrees of freedom and the reduced matrices. After calculating the displacement

along the main degrees of freedom, ANSYS solution extends to the full set of degrees

of freedom. Benefits reduced method:

It works faster and is less "expensive" method.

Disadvantages reduced method:

34

In the initial decision determined only move along the main degrees of

freedom. The second step, known as the expansion step is required for all

displacements, stresses, forces. Sometimes the expansion step is not

required.

Load elements (pressure, temperature) is not supported but allowed

acceleration.

All loads must be attached to the main degrees of freedom selected by the

user. This limits the use of solid loadings.

Time steps should be permanent, as the automatic determination of the steps

is not supported.

Allowed nonlinearity type contact conditions between nodes or gaps.

6.3.3. Superposition method mod.

In the method of superposition of modes summed coefficients of mode shapes

of modal analysis. This method is the only permissible method for the program

ANSYS / LINEAR PLUS. Its advantages are:

It works faster and is less "expensive" method and compared with the

reduced and full methods.

Load elements enclosed in modal analysis, can be applied by using the

LVSCALE.

This method supports the modal damping (damping ratio as a function of

the mode number).

Disadvantages of the method:

Time step size should be constant, as automatic selection step is not

supported.

Allowed nonlinearity type contact conditions between nodes or gaps.

It can not be used for a "floating" or not fixed system.

When you use the Power Dynamics, the initial conditions can not be pre-

applied loads or displacements.

Method does not support the use of non-zero displacement.

35

6.4 Dynamic analysis of transient complete method.

The procedure for solving the problem of dynamic transient analysis full

method consists of three main steps:


2. Application loads and obtain a decision;


Construction of the model. In this step you define the name of the task, then the

task header using the preprocessor define element type, real constants, material

properties and geometry of the model.

You can use both linear and nonlinear elements;

Modulus of elasticity or stiffness EX in any form and density (DENS) or

mass in some form must be specified. Material properties can be linear or

non-linear, isotropic, orthotropic, or temperature-dependent.

Some observations on the density of the partition.

Density partition should be fine enough to represent the highest form of the

desired fashion;

Regions model where you are interested in stress and strain must be broken

thinner than the regions where you are only interested in moving;

If you want to include non-linearity, the partition must be such as to

manifest the effect of these nonlinearities. For example, plasticity requires

reasonable density integration points (hence the thin partition) to surfaces

with high gradient of plastic deformation;

If you are interested in the effects of wave propagation, the partition should

be sufficient to represent the wave. Usually it takes about 20 elements per

wavelength along the direction of propagation.

Application of loads and to obtain a decision. In this step, you define the type of

analysis and its options, a load is applied, choose the option of load steps and initiate

the finite element solution.

1. Enter into the "solution" command / SOLU or via interface:

36

Main Menu> Solution

2. Determine the type of analysis and its options (Table 6.1).

Table 6.1


New Analysis ANTYPE Main Menu> Solution> Analysis Type-New Analys

Analysis Type:

Transient Dynamic ANTYPE Main Menu> Solution>Analysis Type-New Analys>

Transient Dynamic

Method of solution TR NOPT Main Menu> Solution>Analysis Option

Matrix view of the

masses LUMPM Main Menu> Solution>Analysis Option

The effect of large

deformations NLGEOM Main Menu> Solution>Analysis Option

Hardening effect SSTIF Main Menu> Solution>Analysis Option

Options Newton-

Raphson

NROPT Main Menu> Solution>Analysis Option

Solver EQSLV Main Menu> Solution>Analysis Option

Each of these options will be discussed in detail below.

Option New Analysis [ANTYPE]. This option allows you to re-analyze the problem.

Using this option is necessary if a) you want to give the system a pre-pre-static

voltage or full transient analysis you want to extend the time limits. . . .

[

37

Program - chosen (by default);

Full (Full);

Modified (Modified);

Initial stiffness (Initial Stiffnes).

Option method for solving (Equation Solver) [EQSLV].

Select one of the following methods:

Frontal method (default for linear systems);

Jacobi method coupled gradients (JCG);

Method JCG out - of - memory;

Incomplete Cholesky method coupled gradients (ICCG);

Method gradients connected with the preconditions (PSG);

Method PSG out-of-memory;

Iteration (with automatic selection; For linear static / full transient analysis or

transient thermal analysis step) - recommended;

Sparse (SPAR) (default for nonlinear analysis when enabled SOLCONTROL)

For large models, we recommend the method of PSG.

3. Application of the load model.

Transient analysis to determine the uses of load, time-dependent. Choosing

such loads, you need to divide curves "load-time" at the appropriate time steps. Each

"corner" in the time curve could be one load step, as shown in Fig. 6.1.

Figure 6.1

3

1

2

4

5

Time

Force

1

2

3 4

5

Time

Force

38

The first time step is typically used to set the initial conditions. Then you must

determine the load and load steps option for subsequent time steps. For each time

step, you need to determine the amount of load and the value of time, to determine

the load options, such as the jump or smooth increase of load, the use of automatic

step selection, etc. Then you have to write each time step in the file and solve all the

time steps together.

Accounting for initial conditions.

First time step in the applied load is used to set the initial conditions, ie, at t =

0. Dynamic transient analysis requires two sets of initial conditions (since the

equations of motion are second-order equations): initial move 0u and the initial

velocity 0u . If they are not specified, they are considered to be zero. Initial

acceleration 0u always taken to be zero, but you can choose nonzero initial

acceleration of the relevant application acceleration on a small time interval.

Zero initial displacement and zero initial velocity - This is the default initial

conditions, ie if 000 uu you should not determine anything. You can attach the

load corresponding to the first "corner" curve "load-time" in the first time step.

Nonzero initial displacement and nonzero initial velocity you can enter these

initial conditions by using the IC or via:

Main Menu> Solution>-Loads Apply> Initial Condit `n> Define

Caution: You can not define the initial conditions are incompatible. For

example, you have identified an initial velocity in a single degree of freedom, and the

initial velocity in all other degrees of freedom equal to zero, which leads to conflict

of initial conditions. In most cases, the initial conditions are entered in all loose

degrees of freedom model. If these initial conditions are not the same for all degrees

of freedom, then you need to enter them as shown below. (See description of the

commands TIMIND andIC).

39

Zero initial displacement and nonzero initial velocity Nonzero velocity

introduced application of small displacements within a small time interval for part of

the structure where the rate is to be determined. For example, if 25.00 u , You can

enter the movement for equal time interval 0.001 0.004, as shown below.

TININT, OFF | Effect integration time off

D, ALL, UY, 0.001 | Introduced small displacement (velocity is along the Y)

TIME, 0.004 | initial velocity = 0.001 / 0.004 = 0.25

LSWRITE | Write to the file load steps (Jobname.S01)

DDELE, ALL, UY | Destruction applied displacements

TININT,ON | The effect of the integration time is included

Nonzero initial displacement and nonzero initial velocity

This is the same case as shown above, except that instead of small

displacements introduced their real values. For example, if 0,10 u , 5,20 u You must

enter the displacement of 1.0 to 0.4 during the time interval.


D, ALL, UY, 1.0 | Introduced movement (velocity is along the Y)

TIME, 0.4 | initial velocity = 1.0 / 0.4 = 2.5


DDELE, ALL, UY | Destruction applied displacements

TININT,ON | The effect of the integration time is included

Nonzero initial displacement and zero initial velocity

This requires two steps [NSUBST, 2] In increments of selected movements [KBS, 1].

Without predetermined displacement increment will vary directly with time, leading

to a non-zero initial velocity. The example below shows how to enter 0,10 u and

00 u .


D, ALL, UY, 1.0 | Introduced movement (velocity is along the Y)

TIME, 0.001 | Small timeslot

NSUBST, 2 | Two Steps

KBC, 1 | Races load

40


Transient analysis

TIMINT,ON | The effect of the integration time is included

TIME, .... | Real time step

DDELE, ALL, UY | Delete specified displacement

KBC, 0 | Smoothly increasing load (if needed)

Continued normal transient analysis procedures

Nonzero initial acceleration

This may be approximated by selecting the desired acceleration [ACEL] In a small

time interval. Commands are used to input the initial acceleration of 9.81 are shown

below.

ACEL,, 9.81 | Initial acceleration in Y direction

TIME, 0.001 | Small timeslot

NSUBST, 2 | Two Steps

KBS, 1 | Races load


Transient analysis

TIME, .... | Real time step

DDELE, .. | Delete specified displacement (if required)

KBC, 0 | Smoothly increasing load (if needed)

Continued normal transient analysis procedures

Application of loads for dynamic transient analysis

Table 6.2 shows the possible load for the dynamic analysis of transients. Choosing

inertial loads, you can identify them and solid model and finite element model. You

can also define complex boundary conditions array of table type.

Table 6.2

Load transient analysis

Type of load Category Teams Interface

Displacement (UX,

UY, UZ, ROTX,

ROTY, ROTZ

Fixing D Main Menu> Solutions>-Loads-Apply-

Stuctural-Displasiment

Forces, torques (FX,

FY, FZ, MX, MY,

MZ)

Energy F Main Menu> Solutions> -Loads-Apply-

Stuctural-Forse/Moment

Pressure (PRES) Surface loads SF Main Menu> Solutions>-Loads-Apply-

Stuctural-Pressure Temperature (TEMP)

Wrap (FLUE)

Bulk load BF Main Menu> Solutions>-Loads-Apply-

Stuctural-Temperature

41

Gravity, rotation Inertial loads - Main Menu> Solutions>-Loads-Apply-

Stuctural-Other

The application loads using commands

Table 6.3 lists the commands for dynamic loads are transient analysis.

Table 6.3

Type of

load

Tvedotelnay

a or finite

element

model

Objects Application Removal Print Operations Options

1 2 3 4 5 6 7 8

Move

Solid Point DK DKDELE DKLIST DTRAN -

Solid Line DL DLDELE DLLIST DTRAN -

Solid Surface DA DADELE DALIST DTRAN -

Finite

element

Nodes D DDELE DLIST DSCALE DSYM

DSUM

Energy

Solid Point FK FKDELE FKLIST FTRAN -

Finite

element

Nodes F FDELE FLIST FSCALE FSUM

Pressure

Solid Line SFL SFLDELE SFLLIST SFTRAN SFGRAD

Solid Surface SFA SFADELE SFALIST SFTRAN SFGRAD

Finite

element

Nodes SF SFDELE SFLIST SFSCALE SFGRAD

SFCUM

Finite

element

Elements SFE SFEDELE SFELIST SFSCALE SFGRAD

SFBEAM

SFFUN

SFCUM

Continued tabl.6.3

1 2 3 4 5 6 7 8

Temperatur

e, Wrap

Solid Point BFK BFKDELE BFKLIST BFTRAN -

Solid Line BFL BFLDELE BFLLIST BFTRAN -

Solid Surface BFA BFADELE BFALIST BFTRAN -

Solid Volumes BFV BFVDELE BFVLIST BFTRAN -

Finite

element

Nodes BF BFDELE BFLIST BFSCALE BFSUM

Finite

element

Elements BFE BFEDELE BFELIST BFSCALE BFSUM

Inertia

- - ACEL

OMEGA

DOMEGA

CGLOC

CGOMGA

DCGOM

IRLF

- - - -

Application of the load using the interface.

The application of loads on the line.

Main Menu> Solutions>-Loads-Apply-Stuctural-Displasiment> On Lines

Printing loads

Utility Menu> List> Loads>Load type

42

Possible options for dynamic load steps transient analysis are shown in Table 6.4.

Table 6.4

Option Team Actions in the interface

Dynamic options

Time integration

effect

TIMINT Main Menu> Solutions>-Loads Step Opt-

Time/Frequense> Time Integration

Transient

integration

parameters

TINTP Main Menu> Solutions>-Loads Step Opt-

Time/Frequense> Time Integration

Damping ALPHAD

BETAD

MP,

DAMP

Main Menu> Solutions>-Loads Step Opt-

Time/Frequens> Damping

Main options

Time TIME Main Menu> Solutions>-Loads Step Opt-

Time/Frequens> Time & Time Step or Time & Substep Hopping or smoothly

varying load KBC Main Menu> Solutions>-Loads Step Opt-

Time/Frequens> Time & Time Step or Time & Substep

Continued table. 6.4

Step time

integration

NSUBST

DELTIME Main Menu> Solutions>-Loads Step Opt-

Time/Frequens> Time & Substep or Time & Time Step

Automatic selection

step

AUTOTS Main Menu> Solutions>-Loads Step Opt-

Time/Frequens> Time & Substep or Time & Time Step

Accounting options nonlinearities Maximum number of

iterations NE QIT Main Menu> Solutions>-Loads Step Opt-Nonlinears>

Equlibrium Iter Convergence tolerance CNVTOL Main Menu> Solutions>-Loads Step Opt-Nonlinears>

Convergence Crit

Option

compensation

prediction

PRED Main Menu> Solutions>-Loads Step Opt-Nonlinears>

Predictor

Options research

lines

LNSRCH Main Menu> Solutions>-Loads Step Opt-Nonlinears>

Line Search

Criterion of "creep" CRPLIM Main Menu> Solutions>-Loads Step Opt-Nonlinears>

Creep Criterion

Options closure

solutions

NCNV Main Menu> Solutions>-Loads Step Opt-Nonlinears>

Criteria To Stop

Output Control Options

Output to a printer OUTPR Main Menu> Solutions>-Loads Step Opt-Output


Database and the

resulting file

OUTRES Main Menu> Solutions>-Loads Step Opt-Output

Ctrls> DB / Results File

Extrapolation of the

results

ERSX Main Menu> Solutions>-Loads Step Opt-Output

Ctrls> Integration Pt

43

3.1 Dynamic options include:

The effect of time integration [TIMINT]

The effect of time integration can be included to account for the effect of inertia and

damping (if we get off static solution). By default, the time integration effect is

enabled. This option is useful to start the transition from the initial static solution.

The first step is done with option turned off effect of time integration.

Transition parameters of integration [TINTP]

Transient integration parameters control the operation of the program integration by

Newmark. The default method of average acceleration. See section on the basics of

the theory.

Damping

Damping is present in some form in most systems and must be considered in solving

the problem. You can choose four forms of damping in the full dynamic analysis:

Alpha (damping is proportional to the mass) [ALPHAD];

Beta (damping proportional to stiffness) [BETAD];

Damping-dependent material properties [MP, DAMP];

Damping elements (for example, an element COMBIN7).

6.5 Main options.

Key features include the following:

Time [TIME]

This option specifies the time at the end of the load step.

Hopping or smoothly varying load [KBS].

This option indicates whether the load varies smoothly for step [KBC] Or abruptly

[KBC, 1]. The default is a smooth change in load for static analysis and stepwise for

the full dynamic analysis when SOLCONTROL option enabled.

Step time integration [DELTIM or NSUBST].

Integration step is the time increment of time by integrating the equations of motion.

You can directly [DELTIM], Or determined by the number of substeps [NSUBST].

Time step size determines the accuracy of the solution: the smaller the step size, the

44

higher the accuracy. Recommendations on the choice of the time step size are given

later.

Automatic selection step [AUTOTS].

This option, known as the optimization of the time step in transient analysis,

increases or decreases the step size depending on the response of the system. For

most purposes, we recommend that you enable automatic selection of the upper and

lower limits of the step sizes. These limits are selected using commands DELTIM or

NSUBST. By default, this option is enabled, when the option SOLCONTROL.

6.6 Accounting options nonlinearities.

Accounting options nonlinearities are used only if the system includes nonlinearity

(plasticity, contact elements, creep, etc.). These include:

The maximum number of iterations [NEQIT];

Accuracy convergence [CNVTOL];

Option correction prediction [PRED];

Option research line [LNSRCH];

Criterion creep [CRPLIM];

Option closure solutions [NCNV].

More information on the use of these options, see "Guidelines for nonlinear

problems."

6.7. Options control the output.

Control options include the following output:

Output to a printer [OUTPR]

Using this option allows you to enable some of the results in the output file

(Jobname. OUT).

The database and output files, [OUTRES]

This option allows you to control the data in the resulting file (Jobname. RST)

Extrapolation of the results [ERESX].

Using this option allows you to view the results of the integration points of the

element by copying them to the nodes instead of extrapolation. (Default).

45

Attention: By default, only the last time step is written to the output file with the full

method of dynamic analysis. To record all the steps set FREQ field team OUTRES

position «ALL». Maximum 1000 steps written to the output file by default. To

increase memory resources, use the command /CONFIG, NRES.

Sometimes the proper use of commands OUTRES or OUTPR may lead to some

difficulties.

Example load file is shown below.

TIME, ..... | Time end of the first load step

LOADS. .. | Meaning of load over time

KBC, | Continuous or intermittent load

LSWRITE | Write to the file load step

TIME, .... | Time end of the second load step

LOADS ... | load value at a time over

KBC, ... | Continuous or intermittent load


TIME, ..... | Time end of the third load step

LOADS ... | load value at a time over

KBC | Continuous or intermittent load


And so on

4. Save the configuration load each load step through the interface or the

command LSWRITE:

Main Menu> Solution> Write LS File.

Repeat steps 3 and 4 for each corner of the load-time curve. You may want to

have an extra load step, extending the last point on the curve over time to account for

the reaction system after the load.

5. Save a copy of the database in the task file. You can then restart ANSYS

and use the command RESUME.

Utility Menu> File> Save As

6. Run the task solution. Additional features of building and problem solving

with numerous steps loads (array method parameters) listed in section 3.10

procedures manual ANSYS.

Team LSSOLVE.

Interface: Main Menu> Solution> Solve-From Ls Files

7. Close the menu SOLUTION.

46

6.8 View Results

The results of the dynamic analysis of transients recorded in the output file Jobname.

RST. They consist of the following data, which are functions of time:

Direct data;

Nodal displacements (UX, UY, UZ, ROTX, ROTY, ROTZ).

Derived data:




Reaction forces at the nodes;

Etc.

You can view these results using a postprocessor POST26 or POST1:

POST26 to view the results of specific points of the model as a function of time;

POST1 is used to view the results of the full model at some point in time.

Some typical operations postprocessor for transient analysis are given below.

It must be remembered:

To view the results in POST1 POST26 or database must contain the same model

for which the solution is obtained;

Must be received by the file Jobname. RST.

6.8.1 Using POST26.

POST26 working table of results as a function of time, known as variables. Each

variable has its own reference number. Variable number 1 reserved for time.

Teams variable definitions:

NSOL (Direct data, move nodes);

ESOL(Derived, Solutions elements such as voltage);

RFORCE (Reaction forces);

FORCE(Full strength, static, damping, or inertial component);

SOLU (The size of the time steps, the maximum number of iterations, the frequency

response, etc.).

Interface:

47

Main Menu> TimeHist Postpro> Define Variables

Note: In the reduced method and the method of superposition of modes are taken into

account only static forces in the team FORCE.

Drawing or printing variables. Viewing the results as a function of time, you can

choose the critical moments for future use postprocessor POST1.

Teams:

PLVAR (Graphical variables);

PRVAR, EXTREM (print variables).

Interface:

Main Menu> TimeHist Postpro> Graph Variables

Main Menu> TimeHist Postpro> List Variables

Main Menu> TimeHist Postpro> List Extreme

In POST26 has many functions such as mathematical operations, moving variables in

the array of parameters, moving variables in the parameter array.

6.8.2 Using POST1.

1. Reading data from the database:

Team: RESUME.

Interface: Utility Menu> Resume from.

Read the desired result set. Use the command SET identification data or the number

of time steps in time.

Team SET.

Interface:

Main Menu> General Postpro> Read Result-By Time / Freq

Note: If you define the time for which there is no result, the result will be obtained by

linear interpolation between the two closest points in time.

Option strain state

Team PLDISP .

Interface: Main Menu> General Postproc> Plot Results> Deformed Shape

Print Option efforts reactions and moments

Team PRRSOL.

48

Interface: Main Menu> General Postproc> List Results> Reaction Solu.

Note: The PRRSOL prints the forces and moments in the fixed nodes. To show the

forces of reaction type: / PBS,RFOR,, 1 , then to display the desired node or element

[NPLOT or EPLOT]. Use the command RMOM instead RFOR reaction moments.

Print Option nodal forces and moments

Team PRESOL, F (Or M).

Interface: Main Menu> General Postproc> List Results> Element Solution.

You can print the sum of the nodal forces and moments for the selected set of nodes.

Team FSUM.

Interface: Main Menu> General Postproc> Nodal Calcs> Total Force Sum.

You can check the total force and the total points for each selected node. From the

equilibrium condition, the total force must be zero, except for the point of application

of external forces and units, where there are reactions.

Team NFORCE .

Interface: Main Menu> General Postproc> Nodal Calcs> Sum @ Each Node.

Team FORCE (Main Menu> General Postproc> Options for Outp) requires you

which component you want to display:

Full (default);

Static;

Damping;

Inertial.

From the equilibrium condition, the total force must be zero, except for the point of

application of external forces and units, where there are reactions.

Option results in the line elements (construction diagrams).

Team ETABLE

Interface:

Main Menu> General Postproc> Element Table> Define Table.

For elements arranged on the lines, such as beams, rods, tubes using this option

increases the visibility of the derived data (stress, strain, etc.). The resulting data are

49

identified by combinations of marks and reference numerals or names of the

components in the command ETABLE. (See tabulation elements).

Print option error approximation.

Team PRERR.

Interface: Main Menu> General Postproc> List Results> Persent Error.

For linear static analysis using volume or shell elements application team PRERR

allows us to estimate an error decision with respect to the granularity. This command

calculates the percentage of energy of the system error norm (SEPC), which

represents the relative sampling error in the division.

Option of drawing the contours of approximation errors.

Team PRESOL,SERR

Interface: Main Menu> General Postproc> Plot Results> Contour Plot-Element

Solu

Using the PRESOL,SERR error allows you to draw in the energy system from

element to element. Model regions with a high rate of energy mistakes - good

candidates for revision level partitions.

Option drawing contours.

Team: PRNSOL or PRESOL .

Interface: Main Menu> General Postproc> Plot Results> Contour Plot-Nodal

Solu or Element Solu.

Using this option allows you to draw such results as voltage (SX, SY, SZ ...), strains

(EPELX, EPELY, EPELZ ...) and displacement (UX, UY, YZ ...).

You can picture data table outline elements (diagrams) for the elements arranged on

the lines.

Team: PLETAB, PLLS

Interface: Main Menu> General Postproc> Element Table> Plot Element Table

Main Menu> General Postproc> Plot Results> Contour Plot-Line Elem Res

Note: Derived data, such as stress and strain are averaged at nodes by using the

command PLNSOL. This result of "contaminated" in ulah with different materials in

the elements in the shells of different thicknesses or other features. In order to avoid

50

these errors, before drawing choose elements having the same material, the same

thickness of the shell, etc. Or use the Power Graphic team AVRES (Main Menu>

General Postproc> Options for Outp) to turn off the averaging.

Optional vector drawing.

Team PLVEC (vector drawing), PRVEC (print vectors).

Interface:

Main Menu> General Postproc> Plot Results> Vector Plot-Predefined

Main Menu> General Postproc> List Results> Vector Data.

Vectors show (not to be confused with the vector modes) effective trajectory vector

quantities, such as translation, rotation and strain.

Print option tabulated.

Teams:

PRNSOL (Result of the decision in the nodes);

PRESOL (Result of an item)

PRRSOL (Reactions)

NSORT, ESORT

Interface:

Main Menu> General Postproc> List Results>solution option

Main Menu> General Postproc> List Results> Sorted Listing-Sort Nodes or Sort

Elems.

In POST1 provides many other functions: scaling results, the combination of load

cases, etc.

Below is an example of input and command tasks.

Construction of the model

/ FILNAM, .... | Name of the task

/ TITLE, .... | Title

/ PREP7 | Log in preprocessor

--------

-------- | Construction of the model

FINISH

Application of loads and to obtain a decision

/ SOLU | mode solutions

ANTYPE, TRANS | Transient analysis

TRNOPT, FULL | Complete Method

D, | Binding

51

F, | Force

SF,

ALPHAD, | Damping proportional to the masses

BETAD, | Damping proportional stiffness

KBC, | Continuous or intermittent load

TIME, | end time load step

AUTOTS, ON | Automatic selection of the time step

DELTIM, | the time step size

OUTRES, | Options resulting file

LSWRITE | Record of the first load step

-----------

----------

---------- | Loads, time, etc. for the second load step

LSWRITE | Record of the second load step

SAVE

LSSOLVE, 1,2 | Initialization solutions with multiple load steps

FINISH

View Results

/ POST26

SOLU,

NSOL, | Accumulation of results across nodes as variables

ESOL,,,, | Collect results on elements like variables

RFORCE | Accumulation reactions as variables

PLVAR, | Drawing graphs

PRVAR | Print variables

FINISH

/ POST1

SET, | read results in a database

PLDISP, | deformed state

PRRSOL, | Efforts reactions

PLNSOL, | Drawing nodal results

PRERR, | Estimated percentage of errors

.........

FINISH

6.9 Reduced dynamic transient analysis

Reduced method, as its name implies, uses a reduced matrix to calculate the

dynamic response. It is possible to program ANSYS / Multiphysics, ANSYS /

Mechanical, ANSYS / Structural. You should consider the application of this method,

if you do not include the non-linearity (no more than a contact node to node) in the

problem.

Reduced dynamic analysis procedure consists of five basic steps:


2. Getting reduced solutions;

3. View the results of the reduced solution;

52

4. Expanding the solution;

5. View Results expanded solutions.

The first step is the same as for the full method, except that it does not take into

account the nonlinearity (can be considered the only contact between the two nodes,

defined in terms of the form of the gap, instead of introducing the corresponding

element). Follow the steps detailed below.

Reduced solution is obtained in relation to major degrees of freedom. The

procedure for obtaining the following solutions:

1. Enter into the solutions with the command / SOLU or via interface:

Main Menu> Solution.

2. Determine the type of analysis and its options. This is the same as for the

full method, except for the following differences:

Select the method for solving the reduced [TRNOPT];

Nonlinearities accounting options are not possible [NLGEOM, SSTIF,

NROPT];

You can include the effects of prestressing [PSTRES]. This file requires elements

from the previous static or transient analysis;

Restart not possible [ANTYPE].

3. Identify key degrees of freedom. It should mainly characterize the dynamic

behavior of the system. Reduced for transient analysis, the main degrees of

freedom necessary to choose where you want to define the conditions of the

gap, to contribute or not zero displacements. You can print selected major

degrees of freedom, or to destroy them.

Teams:

M

MGEN

TOTAL

MLIST

MDELE

Actions in the interface:

53

Main Menu> Solution> Master DOFs> Define / Copy / Program Selected

Main Menu> Solution> Master DOFs> List All

Main Menu> Solution> Master DOFs> Delete.

4. Define the conditions of the gap if necessary.

Team GP

Main Menu> Solution> Dynamic Gap Cond> Define

You can print or delete conditions

Teams:

GPLIST

GPDELE

Main Menu> Solution> Dynamic Gap Cond> List All

Main Menu> Solution> Dynamic Gap Cond> Delete

Terms gap

Terms gap can only be defined between the two main nodes or between the

main nodes and support as shown in the figure.

Terms gap similar to the element type is selected and the gap between the

surfaces between which is supposed to contact (impact) during the transition

process. ANSYS considers efforts in the gap, which develop when the gap is

closed using the equivalent nodal load vector. Some recommendations for

determining the conditions of the gap are given below.

Use sufficient conditions for obtaining a uniform distribution of contact

stress between the contacting surfaces.

54

Determine reasonable contact stiffness. If the stiffness is too small contact

surfaces overlap too much. If the hardness is too high, a very small time

step is required for the time of impact. The main recommendations selection

stringency gap is one or two orders of magnitude higher than the rigidity of

the adjacent elements. Can determine the rigidity of adjacent elements using

the ratio AE / L, where A-effective contact area, E-modulus softest material

in the gap, L - depth of the first layer of elements in the gap.

5. Enter the initial conditions for the model. Following values are possible in

the reduced transient analysis:

Only possible movement, strength, and translational acceleration (such as

gravity). Load acceleration is not supported if the model contains some key

coordinates of the nodes with the rotation of the nodal coordinate systems.

Forces and non-zero displacement must be attached only to the main degrees of

freedom.

As mentioned for the full method, numerous steps load usually require the definition

of "load history" in transition analysis. The first step is used to establish initial

conditions. The second and subsequent load steps are used to enter values.

Set the initial conditions. Basic conditions may be set explicitly only for

displacements ( 0u ), The initial velocity and acceleration must be zero.

)0,0( 00 uu . Movement can not be eliminated in the subsequent steps load,

nevertheless, they can not be used for determining the initial velocity. In the

reduced transient analysis of static solution is always considered as a

preliminary ruling used to determine 0u .

Identify options load steps for the first step. The following options for the

first step (see Table. 6.5).

55

Table 6.5

Options for the first load step


Dynamic options

Integration

options TINTP Main Menu> Solutions>-Loads Step Opt-Time/Frequens>

Time Integraton Damping ALPHAD

BETAD

MP, DAMP

Main Menu> Solutions>-Loads Step Opt-Time/Frequens>

Damping

Main options

Step time

integration DELTIM Main Menu> Solutions>-Loads Step Opt-Time/Frequens>

Time & Time step


Prints OUTPR Main Menu> Solutions>-Loads Step Opt-Output Ctrls>

Solu Printout

Dynamic option.

The dynamic features include the following:

Integration parameters [TINTP].

This parameter controls the integration algorithm Nyumarka. The default method of

average acceleration.

Damping.


the problem. You can choose four forms of damping in the reduced dynamic analysis:




Damping element (e.g., element COMBIN7).

Main options

Integration step [DELTIM]

Integration step is assumed constant.

If you enter the TIME command for the first load step, it will be ignored. The first

solution is always static at T = 0.

Options control the output.

56

Output control options include the following:

Printer output [OUTPR].

This option displays the movement on the main degrees of freedom.

6. Write down the first load step file (Jobname. S01).

Team: LSWRITE.

Interface: Main Menu> Solutions>-Solve-Write Ls File

7. Ask loads and load steps option for transient analysis, recording every step

load file [LSWRITE].

The following steps are supported for load transient analysis:

Main options

Time (Time) (determined during the end of each load step).

Hopping [KBC1] or smooth load [KBC].

Output Control

Output to the printer [OUTPR].

Reduced file movements [OUTRES].

For all of these commands is the correct label NSOL (decision nodes). By default, the

[OUTRES] Writes every fourth point in time for the reduced displacement file (if the

conditions for clearance, writes every point solutions).

8. Save a copy of the database to a file named.

Team SAVE.

Interface: Utility Menu> File> Save as

9. Run the solution.

Team LSSOLVE

Interface Main Menu> Solution> Solve - From LS File.

10. End solutions.

Team FINISH.

6.9.1 Viewing the results of the reduced solution.

Results reduced transient analysis written in the reduced displacement file

Jobname. RDSP. It consists mainly of movements according to coordinates as a

57

function of time. You can see the movement on the main degrees of freedom as a

function of time using the POST 26.

The procedure for using POST 26 is the same as described for the full method,

except for the following differences:

Before defining the variables in POST 26, use the command FILE

(Main Menu> TimeHist Postpro> Setting> File) To select the data to be

read from a file Jobname. RDSP. For example, if the name of the task -

TRANS, it will be a full team FILE FILE, TRANS, RDSP.

Degrees of freedom only data nodes (main degrees of freedom) are

possible to view.

6.9.2 Expanding the solution.

Advanced solution starts after the reduced solution and expects to complete

displacement, voltage and power in all degrees of freedom. This solution will be

obtained for only one point in time that you choose. Before starting the extended

solutions you need to see the results of the reduced solutions using POST26 and

select a critical point in time.

Note. Preparation of extended solutions is not always required. For example, if

you mainly interested in moving to some points of your design, the reduced solution

can satisfy you. Of course, if you want to determine not move in the main degrees of

freedom, or you are interested in voltage and power, you should get an extended

solution.

Files with the extensions. RDSP,. EMAT,. ESAV,. DB and. TRI should be

obtained in the reduced solution.

The database must contain the same model that received the reduced solution.

1. Restart the decision.

Team / SOLU

Main Menu> Solution

2. Please enable expansion solution and its options.


Expansion

solution on / off EXPASS Main Menu> Solution>-Analis Type-Expansion

58

Pass Non-extensible

solutions NUMEXP Main Menu> Solution> Load Step Opts-Expansion

Pass> Range Of Solus Expansion of a

unique solution EXPSOL Main Menu> Solution> Load Step Opts-Expansion

Pass> Single Expand-By Time / Freq

Option Expansion solution on / off. Select the «ON».

Option Non-extensible solutions. Choose rooms. These solutions will be

expanded in the selected value of time. Solutions near lying about this time will

expand. We must determine whether to consider the voltage and power (default is

considered both).

Option Expansion of a unique solution. Use this option to identify a unique solution

for the expansion, if there is no need to expand several solutions. You can select it in

step load and the number of time step or time.

3. Select the option of load steps. Suitable output control option to extend the

dynamic analysis.

Optional output to the printer [OUTPR]

Use this option to enable some of the results in the output file (Jobname. OUT).

Database and the resulting file [OUTRES]

This option controls the data in the output file (Jobname. RST).

Extrapolation of the results [ERESX]

Use this option to view the results of integration points of copying them to the

nodes instead of extrapolating them (default).

Note: FREQ field in teams OUTPR and OUTRES can be ALL or NONE.

4. Start expansion solution.

Team SOLVE.

Interface

Main Menu> Solution>-Solve-Current LS.

5. Repeat steps 2, 3 and 4 for additional expansion decisions. Each expansion step is

saved as a separate load step in the resulting files.

6. Exit solutions.

59

The results of the extended solutions are written to the output file Jobname.

RST. They consist of the following data calculated for each time point, for which the

reduced solution.

Direct data;

Nodal displacements (UX, UY, UZ, ROTX, ROTY, ROTZ).

Derived data:




Reaction forces at the nodes;

Etc.

You can view these results using POST1. If you got an extended

solution for multiple time points, you can use POST26 to plot stresses, deformations

or forces as a function of time. The procedure for using POST1 or POST26 the same

as for the full method.

6.10 Example transient analysis.

This example shows a transient analysis using the reduced method at a constant

force to the finite time of its rise. In this problem, a steel beam with a concentrated

mass is exposed to dynamic loading.

Steel beam length l supports a concentrated mass m. Acts on the beam dynamic

load F (t) c time ascending rt and a maximum value 1F . If the weight of the beam is

not taken into account to determine the time at which you want the maximum

displacement maxt and the maximum displacement value maxy . Still need to determine

the maximum stress in the beam.

The beam is used in this decision, and its area is taken as a unit. Full-time 0.1

allows weight to achieve maximum displacement. Home degree of freedom takes on

mass in the transverse direction. Static solution is obtained on the first load step. The

time of maximum response (0,092 s) was chosen to expand solutions.

Conditions of the problem

60

E = 30 * 103 ksi, 0259067.0m kips sec2/in, J = 800.6 in4 h= 18 in, l = 240 in

(20ft), F1 = 20 kips

rt = 0.075.

Fig. 6.3 Calculation scheme beams

Definition of title problems.


2. Enter the text «Transient response to a constant force with a finite rise

time»

3. Click OK.

Determining the type of elements

1. Main Menu> Prepocessor> Element Type> Add / Edit / Delete

Dialog "box"

2. Click on Add. Dialog "box" library elements.

3. In the left pane, click on the «Structural Beam»

4. In the right pane, click on the «2D elastic 3" and click on the Apply

5. In the left pane, click on the «Structural mass»

6. In the right pane, click on «3D mass 21" and click "OK"

7. In the "box", click once on the «Type2» and click on «Options».

8. In the dialog box options for inertia in rotation to find «2Dw / o rot iner»

and select it.

9. Press "OK" and «Close» in the dialog box type elements.

h

F

(t)

l

l / 2

Y

0,075 0,092 0.1

F (t)

t,

with

20

Step 3 Load

Step 2 Load

Step 1 Load

61

Determination real constants.

1. Main Menu> Preprocessor> Real Constants

2. Click on Add. Dialog box types of items for real constants.

3. Click OK. The dialog for BEAM3

4. Enter 1 for the area, 800.6 for IZZ and 18 for the height of the beam.

5. Click on OK.

6. On the dialog box, click on the real constants «Add»

7. Click on type 2 MASS21 and click on OK. Dialog box for real constants for

MASS21 opens.

8. Enter 0.0259087 for 2D mass and click on OK.

9. Click on the Close dialog real constants.

Defining the properties of materials.

1. Main Menu> Preprocessor> Material Props>-Constant-isotropic properties

dialog box materials.

2. Click OK. Opens a second dialog box.

3. Enter 30e3 for the elastic modulus EX and click OK.

Definition of nodes

1. Main Menu> Preprocessor> Modeling-Create> Nodes> in Active CS Dialog

box build assemblies in the active coordinate system.

2. Enter 1 for the host and click «Apply» to define the site 0,0,0.


4. Enter 240,0,0 coordinates and click OK.

5. Building between two specified nodes. Main Menu> Preprocessor> Modeling-

Create> Nodes> Fill between Nds. The corresponding menu.

6. Press once on nodes 1 and 3, and press OK.

7. Click OK, accept the default settings.

Identify elements

1. Main Menu> Preprocessor> Modeling-Create> Elements> Auto Numbered>

Thru Nodes.

2. Click on nodes 1 and 2 and click Apply

62

3. Click on nodes 2 and 3 and click OK.

4. Main Menu> Preprocessor> Modeling-Create> Elements> Elem Attributes.

Attributes dialog box elements.

5. Select 2 for elements of MASS21

6. Select 2 for a set of constants, and click OK.

7. Main Menu> Preprocessor> Modeling-Create> Elements> Auto Numbered>

Thru Nodes. Peak open menu items.

8. Press once to node 2 and click OK.

Determining the type of analysis and analysis options.

1. Main Menu> Solution> Analysis Type-New Analys

2. Click on «Transient» and click on OK.

3. Main Menu> Solution> Analisis Option. Dialog box transient analysis.

4. Click on «Reduced» and click OK. Dialog box of the reduced analysis.

5. In the pop-up menu to select the damping effects «Ignore».

6. Click OK.

Defining the main degrees of freedom

1. Main Menu> Solution> Master DOFs> User Selected-Define. Open "peak"

menu.

2. Click on node 2 and click OK. Dialog menu.

3. In the pop-up menu for the first degree of freedom to choose «UY»

4. Click OK.

Setting options load steps.

1. Main Menu> Solution> Load Step Opts-Time/Frequenc> Time - Time Step.

Dialog box appears.

2. Enter 0.004 as the size of the time step and click OK.

Application of the load on the first load step

1. Securing the beams at the nodes 1 and 3. Main Menu> Solution> Loads

Apply>-Structural-Displacement> On Nodes. Open "peak" menu.

2. Click on one node and click «Apply». Dialog box appears.

3. Select «UY» and click on «Apply».

63

4. Click on the node 3 and click OK. Dialog box appears.

5. Click on «UX», «UY» is already selected. Click OK.

6. Application of the load. Main Menu> Solution> Loads Apply> -Structural-

Force/Moment> On Nodes. Peak open menu.

7. Click on node 2 and click OK. Dialog box appears.

8. In the pop-up menu to choose the direction of the forces «FY», enter 0 for the

initial static solutions.

9. Click on OK and press SAVE_DB

Setting output options.

1. Main Menu> Solution> Load Step Opts>-Output Ctrls> DB / Results File.

Dialog box appears.

2. Click on the «Every substep» (Every step) and click OK.

Decision of the first load step.

1. Main Menu> Solution>-Solve-Current LS.

2. Review the information in the status window and click «Close».

3. Click OK on the dialog box, the current load step to start the solution.

4. Click on the «Close», when the decision is received.

Appendix next load step.

1. Main Menu> Solution> Load Step Opts-Time/Frequenc> Time - Time Step.

Dialog box options time and time steps.

2. Enter 0.075 for (Time) the end of load step, click OK.

3. Main Menu> Solution> Loads Apply> -Structural-Force/Moment> On

Nodes. Peak open menu.

4. Click on node 2 and click OK. Dialog "box".

5. Enter 20 as the value of the force and click OK.

Solution next load step.


2. Review the information in the status window and click «Close»



64

Install and load the next step solution.

1. Main Menu> Solution> Load Step Opts-Time/Frequenc> Time - Time Step

Dialog box options time and time steps.

2. Enter 0.1 for (Time) the end of load step, click OK.


4. Review the information in the status window and click «Close»



7. Main Menu> Finish

Expanding the solution

1. Main Menu> Solution> Analysis Type-Expansion Pass. Turn the «Expansion

pass» in position «ON» and click OK.

2. Main Menu> Solution> Load Step Opts-Expansion Pass> Single Expand -

By Time / Freq. Dialog menu.

3. Enter 0.092 for time point and click OK.


5. Review the information in the status window and click «Close».



View results in POST26.

1. Main Menu> Time Hist Postpro> Setting> File. Dialog menu.

2. In the menu, select "file.rdsp» and click OK.

3. The choice of variables. Main Menu> Time Hist Postpro> Define Variables.

Dialog box appears.

4. Click on the "Add» (Add). Dialog box variables.

5. Select (default) decision nodes (Nodal DOF result) and click OK. Dialog menu

data nodes.

6. Select 2 (default) as the reference number of the variable.

7. Enter 2 as the node number.

8. Enter NSOL as the label of the user.

65

9. In the right pane, select «Translation UY».

10. Click OK, then «Close» to dialog variables.

11. Main Menu> Time Hist Postpro> Graph Variables.

12. Enter 2 for the first variable schedule and click OK. There dates in the graphics

window.

13. Main Menu> Time Hist Postpro> List Variables.

14. Enter 2 for the first variable, and click OK.

15. Review the information and click «Close».

View results in POST1.

1. Main Menu> General Postproc> Read Results-First Set

2. Main Menu> General Postproc> Plot Results-Deformed Shape

3. Press «Def + undeformed» and click OK.

Exit from ANSYS.

1. Press QUIT

2. Click on SAVE, if necessary

6.11 The method of superposition of modes in transient analysis.

The method of superposition of modes based on the summation of the

coefficients of the modes obtained from modal analysis to calculate the dynamic

response. The procedure of the method consists of five basic steps:


2. Getting modal solutions (finding natural frequencies and forms);

3. Solution of transient analysis method of superposition of modes;

4. Expanding the solution;


The first of these steps, exactly the same as for the full method.

6.11.1 Preparation of modal solutions.

You should keep in mind the following features:

66

Methods of solving the problem may be the following: reduced, block, subspace,

or Power Dynamic (the other two methods and damped unbalanced do not apply

to the method of superposition of modes). You can use the Power Dynamic, only

if there is no initial static load (all loads are zero in the first step);

Disclose all the modes required for transient analysis;

Method for reduced disclosure events, including the main degrees of freedom of

the nodes in which the forces and conditions specified gap.

Enter fixing. These consolidation will be ignored if they are specified in the

transition analysis by superposition of modes instead of modal analysis.

If you must enter the load elements (pressure, temperature, acceleration, etc.) in a

transient dynamic analysis, you must determine their modal analysis. Load

ignored in modal analysis, but the load vector will be calculated and recorded in

the form file mode (Jobname. MODE). You can then use this load vector for the

transient analysis.

These models should not change between modal and transient analysis.

There is no need to disclose the method of superposition of fashion events. If you

want to view from the reduced form of fashion solutions, you have to open them.

6.11.2. The solution by superposition of modes.

In this step, the program uses the mode shapes obtained in the modal analysis. It must

be remembered that:

File mode shapes to be produced (Jobname. MODE).

The database must contain the same model for which the solution obtained modal

analysis.

1. Enter the command / SOLU or via interface: Main Menu> Solutions.

2. Determine the type of analysis and its options. This is the same as that described

for the full method, excluding the following features:

Select the method of superposition of modes [TRNOPT].

Determine the number of modes to be used in decision [TRNOPT]. This

determines the accuracy of the transient solutions. At a minimum, you must use all

the modes which, in your opinion, affect the dynamic response of the system. For

67

example, if you turn to the consideration of the high frequencies, the number of

modes must be taken into account include high fashion. By default, all modes

obtained in the modal analysis.

Nonlinear options [NLGEOM, SSTIF, NROPT] Are not considered.

Restart not possible [ANTYPE].

3. Define the conditions of the gap if necessary. They can only be defined between

the two major nodes or between the master node and the "ground."

Team GP . Interface: Main Menu> Solutions> Dynamic Gap Cond> Define

4. Attach the load to the model. The following load method of superposition of

modes:

Only possible forces and translational acceleration vector loads generated in the

modal analysis. Not zero displacements are ignored. For an application, use the

command load vector LVSCALE (Main Menu> Solutions> Loads - Apply>

Load Vector> For Mode Super).

If the mode shapes obtained from the reduced modal analysis, force may be

applied only to the main degrees of freedom.

Multiple load steps usually require the definition of "loading history" in

transition analysis. The first step load is used to determine the initial conditions, and

the second and subsequent steps are used for the analysis of transients.

Set the initial conditions. The initial conditions, which can be accurately set this

initial move. Static solution using the method of superposition of modes is

always considered as the first solution at a given load. If modal analysis program

was used POWER DYNAMICS, any load or displacement is not taken into

account, ie, in this case only accepted zero initial conditions.

Options load steps for the first step


Dynamic options

Integration options TINTP Main Menu> Solution> Load Step Opts-

Time/Frequenc> Time Integration

Load Vector LVSCALE Main Menu> Solutions> Loads - Apply>

Load Vector> For Mode Super

68

Damping ALPHAD

BETAD

DMPRAT

MP, DAMP

Main Menu> Solutions>-Loads Step Opt-

Time/Frequens> Damping

Main options

Step time integration DELTIM Main Menu> Solutions>-Loads Step Opt-

Time/Frequens> Time & Time step


Prints OUTPR Main Menu> Solutions>-Loads Step Opt-Output


Dynamic options

The dynamic features include the following:

Integration parameters [TINTP].

This parameter controls the integration algorithm Nyumarka. The default method of

average acceleration.

Load vector [LVSCALE]

This option allows you to apply the load vector, built in modal analysis. You can use

this application for the load vector on the load elements (pressure, temperature, etc.)

Damping.


the problem. You can choose four forms of damping in the reduced dynamic analysis:



Constant damping ratio [DMPRAT];


Modal damping [MDAMP];

Main options

Integration step [DELTIM]

Integration step is assumed constant. By default, the integration step is taken

equal to 1 / (20f), where f is the lowest frequency considered in the decision. Team

[DELTIM] Is seen only on the first load step, and is ignored in subsequent steps.

Options control the output.

69

Output control options include the following:

Printer output [OUTPR].

This option displays the movement on the main degrees of freedom.

5. Write down the first load step file (Jobname. S01).

Team: LSWRITE.

Interface: Main Menu> Solutions>-Solve-Write Ls File

6. Ask loads and load steps option for transient analysis, recording every step load

file [LSWRITE].

The following steps are supported for load transient analysis:

Main options

Time (Time) (determined during the end of each load step).

Load vector [LVSCALE]

Hopping [KBC1] or smooth load [KBC]. The default is smooth load.

Output Control

Output to the printer [OUTPR].

Database and the resulting file [OUTRES].

For all of these commands is the correct label NSOL (decision nodes). By

default, the [OUTRES] Writes every fourth point in time for the reduced

displacement file (if the conditions for clearance, writes every point solutions).

7. If the method is a subspace or block method for modal analysis (MODOPT,

SUBST, or MODOPT, LANS) you may use the nodes using the command

OUTRES,NSOL to limit the movement of data to be written to the reduced image

Jobname. RDSP. Solution with the extension will give the correct results only for

those nodes and for those elements in which all nodes are written to the file Jobname.

RDSP. To use this option, first disable logging command OUTRES,NSOL, NONE,

then select the object OUTRES,NSOL,freq, cjmponent. Repeat the command for

some additional detail components and record RDSP file. Can only be one output

frequency. ANSYS uses the last frequency selected team OUTRES.

8. Save a copy of the database to a file named.

Team SAVE.

70

Interface: Utility Menu> File> Save as


Team LSSOLVE

Interface Main Menu> Solution> Solve - From LS File.

10. End solutions.

Team FINISH.

In the method of superposition of modes in the analysis of transients recorded

reduced file movements, no matter what method was used for the modal analysis. If

you are interested in the design voltage, then you need to carry out expansion

solution. Expansion procedure is the same decision that the reduced method. File

Jobname.TRI necessary in the event that the method used for the reduced modal

solutions. Result of the decision is contained in the file extension Jobname. RST.

The results include displacements, stresses, reaction forces for each time point,

which had received extensive solution using postprocessors POST26 and POST1.

6.12 Dynamic analysis of prestressed structures

Dynamic analysis of prestressed structures allows to calculate the dynamic

response of systems, for example, take into account the thermal expansion of the

residual thermal stresses. Dynamic analysis of prestressed structures depends on the

type of transition.

6.12.1 Dynamic analysis of prestressed structures complete method.

You can include the effects of prestressing full dynamic analysis using the

application preloads static load step without removing them in the next steps. The

procedure consists of two main steps:

1. Build your model, enter the SOLUTION, and define the type of analysis

(Transient analysis) [ANTYPE, TRANS].

Attach all preloads;

Turn off the effects of time integration [TIMINT, OFF];

Turn on the strengthening effects [SSTIF, ON];

Set a time equal to the value of some small dummy [TIME];

Write down the first load step file Jobname. S01.

71

If preloading effects develop as a result of the nonlinear behavior of structures

(as in the case of residual thermal stresses in the casting), it may take several steps to

determine the load prior static. In the case of geometric nonlinearity (large

displacement effect) you can take into account the effect of prestressing team

[NLGEOM, ON].

2. For all subsequent load steps include effects of integration [TIMINT, ON]

and spend a full dynamic analysis, as described above. Writing down all the

steps to load the file [LSWRITE] you can initiate a solution with multiple

load steps.

6.12.2 Dynamic analysis of prestressed structures reduced method.

Dynamic analysis of prestressed structures reduced method requires that you

first loaded up your model in a separate static analysis. It is assumed that the voltage-

dependent time (transient) will be substantially less than the voltage on the preload. If

it is not, then you need to use the full dynamic analysis.

1. Build the model and get the static solution to the effects of

prestressing [PSTRES, ON].

2. Reenter SOLUTION now the solution of the reduced transient

analysis when the accounting effect of prestressing [PSTRES, ON].

The files Jobname. DB, Jobname. EMAT and Jobname. ESAV must

be obtained.

6.12.3 Dynamic analysis of prestressed structures using superposition modes.

When taking into account the effect of pre-stress in the method of

superposition of modes, you must first carry out a modal analysis from the

prestressing. If the modal analysis results given prestressing retrieved, transient

analysis is performed as described above.

7. Other features transient analysis.

7.1. Some guidance on the choice of the integration step.

72

As described above, the accuracy of solution transients depends integration

step: a step smaller, the more accurate solutions. Too big step integration will make a

mistake in the reaction of the higher modes. Too small step integration bude lead to

too much computation time, and need a great memory. To calculate the optimal time

step, we present five recommendations:

1. Determine the frequency response. The time step should be substantially

less than the resulting motion of the system. Since the dynamic response of

the system can be represented by a combination of modes, the time step

should allow to analyze the highest fashion of this combination. For the

integration procedure for Nyumarku found that an acceptable time step

should be a fraction of the twentieth period of higher frequency.

The step size = 1/20fMAX.

Requires a smaller time step, if the result of the calculations necessary to

determine the acceleration.

The figure shows the dependence of the oscillation period increase depending

on the time step for the one-mass system. The graph shows that when the step size

1/20 of an increase in the period of the oscillation period is less than 1%.

2. Determine the curve of load versus time. The time step must be small

enough to pass a function of load. The trend seen in the reaction with

respect to some lag applied forces, especially after a sudden change in load,

as shown in the figure. Sudden change of load requires small time steps. In

this case it is recommended to take the time step is smaller than 1/180f.

73

Ris7.1

3. Determine the frequency of contact. In problems with contact elements

(impact) the time step should be considerably shorter than the interaction

between the two contacting surfaces. In other words, will attend apparent

losses of energy shot is perfectly elastic. Step integration may be

determined from the frequency of contact.

Integration step = 1/Nfc,

Where mkf c 21

Here k gap stiffness, m the effective mass in the gap, N-number of points per

cycle. To minimize energy losses usually take N = 30. Make sure to take a

greater value if you want to find the acceleration. For the reduced method and

the method of superposition of modes N must be at least not less than 7.

You can use more than 30 steps per cycle during impact when the contact

between the contact and the mass much less than the total time and the transient

weight so that the effect of energy loss in the entire system can be small.

4. Determine the wave propagation. If you are interested in the effects of wave

propagation, the time step should be less time meme propagation through

the element.

5. Determine nonlinearity. For most nonlinear problems, the required time

step depends strongly on the type of nonlinearity. For example, the system

tends to increase the rigidity under load (e.g., a large problem when the

displacement of the flexural model to the membrane). This should be an

assessment of the highest frequency of the system.

After evaluating the time step using these recommendations, use the minimum

value for the calculation. Using the automatic time step will allow you to

entrust the problem of choosing the step program.

7.2 Automatic time step.

Automatic selection of the time step allows you to change the integration step,

based on the frequency response and nonlinear effects. The main benefit of this

program is that the total number of integration steps can be reduced. If present in the

74

problem of non-linearity, the use of automatic step has additional advantages. You

can activate the automatic selection of the step team AUTOTS.

Although the obvious advantage of the automatic step selection for all types of

analysis, there may be situations in which you may not be benefits from the use of

automated steps.

Tasks having only a localized dynamic behavior (such as turbine blades and hub

assembly), where part of the system containing the low-frequency energy, the

high frequency area may prevail.

Problems with permanent-magnet (for example, seismic load), wherein the time

step has a tendency to change continuously, as excited by different frequencies.

Problems with kinematic excitation (as solid), where the contribution of the

oscillations as a rigid body can dominate.

7.3 Damping.

Damping present in most systems must be within the dynamic analysis. In

ANSYS are used five types of damping:

Alpha and beta damping (Rayleigh damping);

Damping, depending on the material properties;

Constant damping ratio;

Modal damping;

Damping elements.

When using ANSYS / Linear Plus is possible to use only a constant

damping ratio and modal damping. You can select more than one form of damping in

the model. The program will generate damping matrix [C], as the sum of the selected

forms of damping.

Table 7.1 lists the types of damping applicable to various kinds of

analysis.

Table 7.1

Type of analysis Alpha and

Beta damping

[ALPHAD,

BETAD]

Damping-

dependent

material

properties

[MP, DAMP]

Constant

damping ratio

[DMPRAT]

Modal

dlempfirovani

e

[MDAMP]

Damping

elements3

[COMBIN7

etc.]

75

Statics No No No No No

Continuation of Table 7.1 Modal

Without damping

Modal damping

Net5

Yes

Net5

Yes

Net5

Yes

No

Yes

No

Yes

Harmonic

Full

Truncated

Super. Pos Mod

Yes Yes Yes No Yes

Yes Yes Yes No Yes

Yes Yes4 Yes Yes No

Transition

Full

Reduced

Super. Pos Mod

Yes Yes No No Yes

Yes Yes No No Yes

Yes Yes4 Yes Yes No

Spectral

SPRS

MPRS2

Yes1 Yes Yes Yes No

DDAM Yes1 Yes Yes Yes No

PSD Yes No Yes Yes No

Stability No No No No No

Substructure Yes Yes No No Yes

Notes to Table 7.1

1. Only damping

2. Damping is only used for the combination of modes, but not for the

calculation of the coefficients of the modes.

3. Including damping matrix Superelement.

4. If converted to modal damping opening modes

5. Effective damping ratio is determined for spectral analysis.

Alpha and beta damping matrix is used to determine the damping Rayleigh.

Cushioning Matrix matrix is determined using mass and stiffness matrices:

KMC

Teams and ALPHAD BETAD used to determine and as decimal numbers.

Values and Usually not known in advance, but they can be determined from the

modal damping ratios i . i -The ratio of actual damping to the critical damping of

76

the i-th individual fashion. If i - Own the angular frequency of the i-mode, the

and must satisfy the relation:

22 iii

In many practical cases, alpha damping (or mass damping) can be ignored (=

0). In these cases, you can identify of known values i and i as

ii 2 .

To determine the coefficients and at a given damping it is usually

assumed that the amount and members of approximately constant in the

frequency range. (Fig. 7.2). Thus, for a given and the frequency range of 1 to 2 -

Coefficients and can be determined from the system of two equations.

Figure 7.2

Alpha damping may lead to erroneous results if a large mass was artificially

introduced into the model. One common example is when a large mass of artificially

introduced into the base model to be able to input range acceleration. (You can use a

large range of mass to convert accelerations range forces). Damping coefficient

alpha, which is multiplied by the mass matrix will produce artificially inflated

damping force in such systems, which leads to errors in the input spectrum and

determining the reaction system.

Beta damping and damping material may lead to distorted results in the

nonlinear analysis. These damping coefficients are multiplied by the stiffness matrix,

which is constantly changing in nonlinear problems. The resulting change in damping

can sometimes be the opposite to the actual change in damping, which may distort the

77

physical nature of the problem. For example, it is known from experience that in

physical systems, which are softened during the plastic deformation, respectively,

typically in the development of improved damping plastic flow. The model ANSYS,

having beta damping will decrease damping the development of plastic deformation.

Damping-dependent material properties allows you to define as beta damping

properties of the material [MP, DAMP]. It should be noted that the spectral analysis

is introduced into a damping ratio depending on the material properties Instead .

Another note that for items in which there are several materials, such as SOLID46,

SOLID65, SHELL91, SHELL99, can be determined for the entire element rather

than for each element in the material. In this case, determined from the material

properties of elements using the MAT.

Constant damping ratio - The easiest way to determine the damping system. It

represents the ratio of actual damping to the critical value and is entered as a decimal

fraction team DMPRAT. This is only possible in the spectral analysis, harmonic

analysis and the method of superposition of modes in transient analysis.

Modal damping gives you an opportunity to enter a different attitude damping

for different modes of vibration. This is only possible for the spectral analysis method

and the method of superposition of modes in transient analysis.

Damping elements introduced elements with viscous damping, such as

COMBIN7, COMBIN14, COMBIN37, COMBIN40 etc.

8. Spectral analysis.

Spectral analysis - a solution to the problem using the results of modal analysis

to calculate the displacements and stresses in the model. It is mainly used instead of

timing analysis to determine the response at random loads or loads, time-dependent,

such as an earthquake or wind loads, the load from the sea waves, the impact of a jet

engine, etc.

Spectrum is a graph of the spectral magnitude as a function of frequency,

which determines the intensity and spectral composition of the dynamic loads. There

are three types of spectral analysis:

78

Response spectrum;

-The only response spectrum (SPRS)

Multiple-response spectrum (MPRS)

The method of dynamic design (DDAM)

Method of power spectral density (PSD).

8.1 Types of spectral analysis.

8.1.1 Response Spectrum.

Range of reaction is the reaction systems with one degree of freedom to the

load, depending on the time. This reaction graph as a function of frequency, where

the reaction may be adopted displacement, velocity, acceleration or force. Two types

of spectral analysis of the reaction: the only response spectrum and multiple-type

reactions.

When analyzing single response spectrum you choose one spectral curve (or

family of curves) in several points of the model, for example in the support.

When analyzing Multiple response spectrum you choose different spectral

curves at various points.

8.1.2 Method of dynamic design.

The method of dynamic design - a technique used to evaluate the impact

resistance of marine equipment. Technology is essentially a spectral analysis of the

reaction, where the spectrum is obtained from a series of empirical equations and

shock design tables, obtained from the American National Research Laboratory.

Report NRL-1396.

8.1.3 Spectral density method.

Power spectral density - this statistic, for which the limit is defined as the rms

value of the random variable. It is used in random vibration analysis, in which the

instantaneous value of the reaction can only be determined by probability distribution

functions, showing the probability of finding a random variable in a certain range.

Graph of the spectral density function can be built in to the frequency of

displacement, velocity, acceleration and force. Mathematically, the area under the

curve equal to the variance (or mean square).

79

8.2 The procedure for solving the problem of spectral analysis.

1. Construction of the model

2. Getting modal solutions

3. Preparation spectral solutions

4. Expansion modes

5. Combination of modes


Modal solution must be obtained before the calculation of the spectrum,

mode shapes and frequencies are used in the spectral analysis. Producing

spectral solution before deployment modes, you can open only those modes

that are required for a final decision.

Construction of the model.

In this step, you define the name of the task, the task header, then using a

preprocessor define the element type, real constants, material properties and

geometry of the model.

Cover only the linear behavior in the spectral analysis. Nonlinear

elements (if they exist) are represented as linear. For example, if you turn the

contact elements and their rigidity are calculated initially, and is not changed.

Modulus of elasticity (or stiffness in some form) and the density (or

weight) must be carefully set. Material properties can be linear, isotropic or

orthotropic, or permanent depending on the temperature, but the nonlinear

properties are ignored.

Getting modal solutions.

Natural frequencies and mode shapes are required for obtaining a

spectral analysis.

Use the subspace block or reduced. Other methods are not suitable

for spectral analysis.

The number of received events should be sufficient to characterize

the response of the system in the required frequency range.

80

If you are using an interface method, choose NO to the dialog menu

disclosure modes [MODOPT], So that the modes do not unfold in

time but may be disclosed separately in the respective step solutions.

In this case you should choose YES for the disclosure of all modes.

If you are using damping depending on the properties of materials,

you must select it in the modal analysis.

Need to consolidate the power of freedom, where you want to attach

the base of the input spectrum.

Preparation spectral solutions.

File mode shapes (Jobname.MODE) of modal analysis should be obtained, and the

database should contain data of this model.

1. Enter into the decision:

Team / SOLU or interface Main Menu> Solutions

2. Determine the type of analysis and its options.

Table 8.1

Type of analysis and options

Options Teams Actions in java.lang.Object

New Analysis ANTYPE Main Menu> Solutions> Analys Type> New

Analys

Analys Type:

Spectrum ANTYPE Main Menu> Solutions> Analys Type> New

Analys> Spectrum

Type

spectrum:

SPRS

SPOPT Main Menu> Solutions> Analys Options

The number of

modes for

solutions

SPOPT Main Menu> Solutions> Analys Options

Accuracy of the solution depends strongly on the number of modes used: the

larger the number of modes, the higher the accuracy.

3. Identify options load steps. The following options for univariate spectral analysis.

Options load steps

Table 8.2


81

Options range

Type of response

spectrum

SVTVP Main Menu> Solutions> Load Step Opt-

Spectrum> Single-point-Settings

Exciting direction SED Main Menu> Solutions> Load Step Opt-

Spectrum> Single-point-Settings

Frequency

dependence of the

spectrum (the

spectral curve)

FREQ, SV Main Menu> Solutions> Load Step Opt-

Spectrum> Single-point-Freq Table / Spectr

Values

Damping (dynamic option)

Beta (damping

proportional to

stiffness

BETAD Main Menu> Solutions> Load Step Opt>

Time / Frequens> Damping

Constant damping

ratio

DMPRAT Main Menu> Solutions> Load Step Opt>


Modal damping MDAMP Main Menu> Solutions> Load Step Opt>


Options range.

Options include the following range:

Type spectrum [SVTYP].

Type of spectrum can be displacement, velocity, acceleration, force or spectral

density. All spectra except forces represent a seismic spectra and act on the supports.

Range of forces applied to the support units not using the commands F or FK and

direction labels FX, FY, FZ. Spectral Density [SVTYP, 4] is converted into the

internal displacement and the spectrum bandwidth is limited.

Direction of the impact.

Spectral curve [FREQ, SV].

Teams FREQ, SV used for determining the spectral curve. You can define a family

of spectral curves, each curve for various damping ratio. Use the STAT command to

print the current values of the curves. Another team ROCK allows you to determine

the spectrum span.

Damping (dynamic option).

If you have selected more than one form of damping, ANSYS will calculate

the effective damping ratio at each frequency. Spectral value of the effective

damping ratio is then calculated by interpolation on the spectral curve. If damping is

not specified, will be used spectral curve with the highest damping.

82


Main Menu> Solution> Solve-Current LS

Output solutions includes a table of operating factors. Table contains the

coefficients of the effective factors modes (with respect to the lowest accepted by

damping) and the mass distribution for each mode. To get the maximum response of

each mode (modal response) must be multiplied by the modal form fashion factor.

You can do this by reduction coefficients mod team GET and using them as a scale

factor in the team SET.

5. Repeat steps 3 and 4 for additional spectra of the reaction, if necessary. Note that

the solution is not written to a file with RST at this time.

6. Get out of the processor.

8.3 Expansion modes.

1. Click on the «YES» to expand the dialog box mod (Expansion Pass).

Team MXPAND.

Interface: Main Menu> Solution> New Analys-Modal Expansion Pass-On> LS

Opt-Expansion Pass> Single Expand-Modal-Expand Modes

You must disclose fashion, no matter what method is used: subspace, block or

reduced.

Selectively disclosed, may only be significant fashion. (See Use of field SIGNIF

in the team MXPAND). If you want to use the interface and selectively disclose

fashion, Select «NO» in the dialog box options for disclosure modes modal

analysis [MODOPT].

Only disclosed fashion combinations used for the operation modes.

If you are interested in stress, you are here to ask for stress analysis. By default,

when a voltage is not determined by the disclosure of modes, which means that

the voltage is not determined by the spectral analysis.

If you want to open all the fashion, you can include a step of disclosure modes in

modal analysis team MXPAND. Jobname

8.4

83

Main Menu> Solutions.

2.

3.

DSUM

Interface:

ACEL)

Note:


Jobname JobnameFile Jobname

If you choose to move the reaction mode, the displacements and stresses are

combined for each mode;

If you choose to speed the reaction mode, the speed and voltage are combined

for each mode;

84

If you have chosen as the type of acceleration response, the acceleration voltage

and combined for each mode;

5. Get out of the processor.

8.5 View the results.

The results of spectral analysis of a file written in Jobname. MCOM in the

form of instructions postprocessor POST1. These commands are calculated limit of

the reaction system, a combination of maximum modal responses in some form (in

accordance with the selected method). Limit reactions consist of limiting movements

(or velocity or acceleration), and if placed in the output file at step disclosure events,

limiting stress, deformation and reaction forces.

You can use the general postprocessor POST1 to view the results.

Note: If you want to directly combine operating voltage (S1, S2, S3, SEQV,

SI) of the resulting file, enter the following command SUMTYPE, PRIN before

reading file Jobname. MCOM.

1. Read commands from the file Jobname. MCOM.

Team /INPUT, for example, / INPUT, FILE, MCOM.

Interface: Utility Menu> File> Read Input From

2. See the results.

Option: show strain state.

Team PLDISP.

Interface: Main Menu> General Postproc> Plot Results> Deformed Shape.

Option viewing results in outline form.

Teams PLNSOL or PLESOL.

Interface: Main Menu> General Postproc> Plot Results> Contour Plot>

Nodal Solu or Element Solu.

Option viewing results in vector form.

Team PLVEC

Interface: Main Menu> General Postproc> Plot Results> Vector Plot>

Predefined

Option tabular listing.

85

Teams: PRNSOL (decision nodes);

PRESOL (solution elements);

PRRSOL (reaction forces).

Interface:

Main Menu> General Postproc> List Results> Nodal Solution

Main Menu> General Postproc> List Results> Element Solution

Main Menu> General Postproc> List Results> Reaction Solution

In POST1 postprocessor provides many other functions such as scaling results

in different coordinate systems, the combination of load cases, etc

8.6 Example of calculating a simple system.

Description of the problem.

Bar with hinged legs and a length l, a mass m, distributed over the length of

exposed vertical vibration at both poles. Vibration is defined as the range of motion.

Determine the movement of nodes and response efforts for the solution elements.

The original data.

The modulus of elasticity E = 30 * 106psi

Mass m = 0.2 ibsec2/in2

Sectional moment of inertia J = 1000/3 in4

Cross-sectional area A = 273.9726 in2.

Beam length l = 240 in

Section height h = 14 in.

Range of effects

Frequency, Hz Moving, in

0.1

800

0.44

0.44

Set the type of analysis.


2. Enter the text «Seismic response of beam structure» and click OK.

86

Determining the types of elements

1. Main Menu> Preprocessor> Element Type> Add / Edit / Delete Dialog

menu.

2. Click on Add. Dialog box opens library elements.

3. In the left pane select «Structural Beam»

4. On the right side click on «2D elastic3».

5. Click OK. The dialog box closes elements.

6. Click on the Close window element types.

Determination real constants.

1. Main Menu> Preprocessor> Real Constant. Dialog box real constants.

2. Press Add. Dialog box types of items for real constants.

3. Click OK. Dialog box for the item BEAM3.

4. Enter 273.9726 as cross-sectional area.

5. Enter the (1000/3) for the moment of inertia.

6. Enter 14 for the height of the beam.

7. Click on Close.

Determination of material properties.

1. Main Menu> Preprocessor> Material Props> Constant-Isotropic. The

properties dialog box material.

2. Click on OK, choosing the default material number 1.Otkroetsya dialog isotropic

material properties.

3. Enter 30e6 for the elastic modulus.

4. Enter 73E-05 for density click OK.

Determination of points and lines.

1. Main Menu> Preprocessor> Modeling-Create> Keypoints> In Active Cs.

Dialog box constructing points.

2. Enter 1 for the number of points.

3. Press Apply selecting default coordinates X, Y, Z equal to zero.

4. Enter 2 as the point number.

5. Enter 240,0,0 coordinates X, Y, Z.

87

6. Click OK.

7. Utility Menu> PlotCtrl> Numbering. Dialog box numbering.

8. Click on the "keypoint numbering on»

9. Click OK.

10. Main Menu> Preprocessor> Modeling-Create> Line> Straight Line. Peak

open menu

11. Click on point 1, then point 2. There will be a straight line between two points.

12. Click OK.

Installation of partitions.

1. Main Menu> Preprocessor> Meshing-Size Cntrls> Global - Size. Dialog box

appears.

2. Enter 40 as the number of divisions and then click OK.

3. Main Menu> Preprocessor> Meshing-Mesh> Lines

4. Click ALL.

Setting the boundary conditions.

1. Main Menu> Solutions> Loads-Apply> Structural-Displacement> On Nodes

2. In the graphics window, click on the node at the left end of the beam.

3. Click OK.

4. Click on UY.

5. Click OK.

6. Repeat steps 1-3 and select a node on the right end of the beam.

7. In the box click on UX and now UX and UY are highlighted.

8. Click OK.

Determining the type of analysis and options.

1. Main Menu> Solutions> Analys Type> New Analys

2. Select "Modal» and click OK. Dialog box appears.

3. Main Menu> Solutions> Analys Options Dialog box modal analysis.

4. Select the default "Subspace» as a method of disclosure events.

5. Enter 10 as the number of modes and press OK.

6. Click on the disclosure of events [MXPAND] and select «NO».

88

7. Click OK. Dialog box modal analysis.

8. Click OK.

Modal analysis solution.

1. Main Menu> Solution> Solve-Current LS

2. Review the information in the status window, click Close.

3. Click OK on the Solve Current Step.

4. When the decision is received, there will be a message «Solution is done"

Setting the spectral analysis.


2. A dialog box with the inscription preduprezhayuschey "Changing the type of

analysis is correct only in the first time step" Click OK. Click CLOSE.

3. Click on «Spectrum» and click OK.

4. Select only the default response spectrum (SPRS) and enter 10 for the number of

modes in the decision.

5. Click on the interactive button, selecting YES for Stress. Click OK.

6. Main Menu> Solutions> Load Step Opt-Spectrum> Single-point-Settings.

Dialog box appears.

7. Select «Seismic displac".

8. Enter 0,1,0 to select the direction of influence.

Determination of the frequency spectrum of the table.

1. Main Menu> Solutions> Load Step Opt-Spectrum> Single-Point-Freq Table.

2. Enter 0.1 for FREQ1 and 800 for FREQ2 and click OK.

3. Main Menu> Solutions> Load Step Opt-Spectrum> Single-Point-Spectr

Values. Dialog box value spectrum - Attitude damping.

4. Click OK, choosing the default without damping. Dialog box spectrum values.

5. Enter 0.44 and 0.44 for FREQ1 and FREQ2.

6. Click OK.

Solution of spectral analysis.


2. Review the information in the status window, click Close.

89

3. Click OK on the Solve Current Step.


Setting expansion step modes.


A dialog box with the inscription preduprezhayuschey "Changing the type of analysis

is correct only in the first time step" Click OK. Click CLOSE.

2. Click on «Modal» and click OK.

3. Main Menu> Solutions> Expansion Pass

4. Click on the dialog button to the ON position and press OK.

Disclosure modes.

1. Main Menu> Solution> Load Step Opts-Expansion Pass> Single Expand-

Expan Modes.

2. Enter 10 for the number of modes and enter 0.005 for the threshold of importance.

3. Click on the results of the elements, selecting YES.

4. Click OK.

Performing disclosure modes.


2. Review the information in the status window, and click Close.

3. Click OK.


Installing a combination of modes for spectral analysis.

1. A dialog box with the inscription preduprezhayuschey "Changing the type of

analysis is correct only in the first time step" Click OK. Click CLOSE.

2. Click on the Spectrum and click OK.

3. Main Menu> Solutions> Analys Options. Dialog box spectral analysis.

4. Select only the default response spectrum (SPRS). Click OK.

The choice of method of disclosure events.

1. Main Menu> Solutions> Load Step Opt-Spectrum> Single-Mode Combine.

Dialog box appears.

2. Select SRSS method of combining both modes.

90

3. Enter 0.15 as the threshold of significance.

4. Select travel as the output type. Click OK.

Combining modes


2. Review the information in the status window, and click Close.

3. Click OK.


Postprocessor: printing nodal and element data reactions.

1. Main Menu> General Postproc> List Results> Results Summary. A window

opens listing.

2. Review the information and click Close.

3. Utility Menu> File> Read Input From. Dialog menu reading files.

4. On the left side of the dialog box to select the directory containing your results.

5. In the right pane, select the file Jobname.mcom.

6. Click OK.

7. Main Menu> General Postproc> List Results> Nodal Solution.

8. On the left side of the dialog box to select the default DOF solution, and the right

side «All DOFs DOF»

9. A window opens listing. After viewing click Close.

10. Main Menu> General Postproc> List Results> Element Solution. Dialog box

solutions elements.

11. On the left side of the dialog box select «Line Elem Result» and select «Structural

ELEM» on the right side. Click OK.

12. Opens listing. After viewing click Close.

13. Main Menu> General Postproc> List Results> Reaction Solution.

14. In the left menu select «All struc fors F» and click OK.

15. Opens listing. After viewing click Close.

8.7 How to perform an analysis of random vibrations.

The analysis procedure consists of random fluctuations of the six basic steps:


91

2. Getting modal solutions;

3. Disclosure of events;

4. Getting spectral solutions;

5. Combining modes;


The first two steps - the same as that for the spectral analysis. Random

vibration analysis is not provided for the program ANSYS / Linear Plus.

When using the interface in the dialog box modal analysis [MODOPT]

Contains options for opening modes [MXPAND]. Press «YES» to disclose events.

Here the procedure for obtaining modal analysis and disclosure of events combined

in one step.

8.7.1 Disclosure modes.

You must reveal the modes obtained by subspace block method and reduced

method. It is necessary to take into account some special features:

Only disclosed fashion recorded in step combining modes;

If you are interested in the calculation of stresses arising from the action of

the spectrum, they need to "order" here.

Disclosure events may be a separate step solutions, or can be included in the

modal analysis.

At the end of step disclosure modes exit the "solution" command FINISH.

8.7.2.Poluchenie spectral solutions

For spectral solutions database should contain data model and the following

files from the modal analysis: Jobname. MODE,. ESAV,. EMAT,. FULL,. RST and

DB.

1. Log in solution.


2. Determine the type of analysis and options.

For the type of spectrum [SPOPT] To select the power spectral density

(PSD);

Choose stress analysis (ON), if it interests you.

92

3. select the option of load steps. The following options are available:

Spectrum Data

Type spectral density.

The spectral density can be defined as the displacement, velocity, force,

pressure, acceleration. It will be either base excitation, or excitation nodes in steps 4

and 5. If a spectral density in the form of pressure, the pressure must be defined in the

modal analysis.

Team: PSDUNIT.

Main Menu> Solution> Spectrum>-PSD-Setting.

Table spectral density versus frequency.

Teams: PSDFRQ, PSDVAL.

Interface: Main Menu> Solution> Spectrum>-PSD-PSD vs Freq.

In step 6 describes how to apply an additional excitation of the spectral density.

Damping (dynamic option).

The following forms of damping ALPHAD,BETAD,MDAMP.

Alpha damping proportional to the mass matrix.

Team ALPHAD.

Main Menu> Solution> Load Step Opts-Time/Frequenc> Damping.

Beta damping proportional to the stiffness matrix.

Team BETAD.


Constant damping ratio.

Team DMPRAT.


Frequency-dependent damping ratio.

Team MDAMP.


4. Attach the excitement of the power spectral density of the selected nodes.

Use a value of 1.0 to indicate the points where the excitation is applied. A

value of 0.0 may be used to remove the specification. Exciting direction is

93

selected using tags UX, UY, UZ team D (at the base excitation) and labels

FX, FY, FZ F command at excitation nodes. When excited nodes value

greater than 1.0 may be used for the scale factor of participation. For the

spectral density in the form of pressure necessary to move the load vector of

modal analysis [LVSCALE]. You can apply base excitation only those

nodes that were enshrined in the modal analysis.

Teams D or DK, DL, DA to excite the base.

F or FK excitation nodes. LVSCALE dTo pressure.

Interface: Main Menu> Solution>-Loads-Apply> Spectrum>-BasePSD

Excit-On Nodes.

5. Calculate the participation factor for the excitation spectral density. Use

field TBLNO (Table number) which is the spectral density of the table will

be applied and field Ecxit for to indicate whether excited or base unit.

Team PFACT.

Interface: Main Menu> Solution>-Load Step Opts - Spectrum>-PSD-

Calculate PF

6. If you need to make some impact in the form of spectral densities for the

same model, repeat steps 3, 4, 5 for each additional spectral density. Then

decide if it is necessary degree of correlation between the excitations, using

the following commands:

COVAL for mutual spectral density;

QDVAL for quadratic spectral values;

PSDSPL for spatial relationships;

PSDWAV relations for the propagation of the wave.

Interface: Main Menu> Solution>-Load Step Opts - Spectrum>-PSD-

Correlation

When you use the command PSDSPL or PSDWAV accordingly you should

use SPATIAL or WAVE in the team PFACT.

7. Selecting control output parameters.

94

Once the team PSDRES provides right control output and record the resulting

file. May be performed for calculating three types: displacement, velocity,

acceleration. Each of them may be defined relative to the chassis or completely.

Team PSDRES.

Interface: Main Menu> Solution>-Load Step Opts - Spectrum>-PSD-Calc

Controls.

Table 8.3 shows the possible configuration solutions.

Table 8.3

Decision Options Form

Displacement (DISP tag team

PSDRES)

Displacement, stress, strain,

force

Relative, absolute, or both

Speed (label VELO team

PSDRES)

Speed speed voltage, etc. Relative, absolute, or both

Acceleration (ACEL tag team

PSDRES)

The acceleration voltage and

the acceleration etc.

Relative, absolute, or both


Main Menu> Solution> Solve-Current LS.

9. Exit solutions.

8.7.3 Combining modes.

1. Log in solution.


2. Determine the type of analysis.

Option NEW Analysis [ANTYPE]

Select the type of analysis - spectrum.

3. Only PSD mode combination method is valid for the analysis of random

fluctuations. This method allows to calculate displacement in the form of a

sigma-voltage, etc. in the model. If you enter the command PSDCOM,

program will not assume that the parameters in the form of one sigma.

Team PSDCOM

Interface: Main Menu> Solution>-Load Step Opts - Spectrum>-PSD-Mode

Combin.

95

Field SIGNIF and COMODE often used to reduce the number of modes. If you

want to try this option to print the modal covariance matrix for the initial

receipt of the relative combination of modes.


Main Menu> Solution> Solve-Current LS.

5.Vyydite mode solutions.

8.7.4 View the results.

Results of the analysis of random fluctuations are recorded in the output file

Jobname.RST. They consist of the following data:

1. Disclosed mode shapes of modal analysis.

2. Static solution when excited base. [PFACT,, BASE].

3. The following data, if at step combination modes introduced team

[PSDCOM] and made the installation command [PSDRES].

1 movement (stress, strain, force);

1 speed, etc.

1 acceleration, etc.

You can view these results in the postprocessor POST1, then calculate the

output spectral density in the postprocessor POST 26.

To view the results in POST1, you must first understand how the data is

organized in the resulting file. File organization is shown in Table 8.4.

Table 8.4

Step load Substep Comments

1 1

2

3

Opened fashion 1

Opened Fashion 2

Opened fashion 3

2

Base Excitation

1

2

3

Single static solution for PSD table 1

Single static solution for PSD table 2

3 1 1 sigma movement

4 1 1 sigma speed

5 1 1 sigma acceleration

Note: In step 2, you can select the excitation nodes.

96

1. Read a set of results in a database. For example, reading one displacement

type SET, 3,1.

Interface: Main Menu> General Postproc>-Read Results-First Set

2. You can use the same options for spectral analysis.

Note: Averaging voltages at the nodes using the PLNSOL may not reflect the

analysis of random vibrations, because the voltage is understood in a statistical

sense.

8.7.5 Calculation of the output spectral density in POST26.

You can calculate and draw the output spectral density of displacement,

velocity or acceleration, if the received files Jobname. RST and Jobname. PSD.

1. Log in postprocessor POST26.

Team / POST26

Interface: Main Menu> TimeHist PostPro

2. Prepare a frequency vector. NPTS The number of points on the frequency

axis are added to both sides of the natural frequency to obtain a smooth

curve. (By default, -5). Frequency vector is taken as a variable number 1.

Team STORE, PSD,NPTS

Interface: Main Menu> TimeHist PostPro> Store Data.

3. Specify the variables (displacements, stresses, etc.).

Teams NSOL,ESOL.

Interface: Main Menu> TimeHist PostPro> Define Variables.

4. Calculation of the spectral density of the output and save it in the selected

variables.

Team RPSD.

Interface: Main Menu> TimeHist PostPro> Calc Resp PSD.

8.7.6 Calculation of covariance functions in POST26

97

You can calculate the covariance between the two variables recorded in the

output file (displacements, velocities, accelerations), if there are files Jobname. RST

and Jobname. PSD.

1. Log in postprocessor POST26.

Team / POST26

Interface: Main Menu> TimeHist PostPro

2. Define the variables that contain information of interest to you.

Main Menu> TimeHist PostPro> Define Variables.

3. Calculate the contribution of each component (relative to the absolute

reaction) and save them in a selected variable. Team PLVAR can be used

for drawing modal contributions (relative to the reaction), and mixed

pseudostatical part in the covariance function.

Team CVAR.

Main Menu> TimeHist PostPro> Calc Covarince.

4. Preparation of covariance.

Team * GET, NameVAR1, n, EXTREM, CVAR.

Interface: Utility Menu> Parameters> Get Scalar Data

8.8 How to parse multiple response spectrum.

The procedure to determine multiple response spectrum is the same as that for

random vibration with the following differences:

Select MPRS instead of as a type of PSD spectrum in the team [SPOPT];

Table spectral density versus frequency spectrum is represented as a table

of values as a function of frequency;

You can choose some degree of correlation between the spectra, or consider

them to be uncorrelated;

Determined only by the relative performance (relative to the base

excitation), the absolute values are not defined.

Supports all methods of combining modes except PSDCOM.

98

The analysis results are recorded plural spectral analysis file Jobname.

MCOM in the form of instructions postprocessor POST1. Provides

commands calculate the limiting reactions of the combination of maximum

modal responses. Limit reactions consist of limiting displacement limit

stresses, deformations and limit the forces of reaction.

Part 2. Fundamentals of the theory.

9. Modal Analysis

Modal analysis is used to determine the natural frequencies and mode shapes

design. It is assumed that the free undamped oscillations occur:

}{0[K]{u}}'[M]{u' (9.1)

Note that the stiffness matrix [K] May include the effect of pre-loading.

For a linear system free vibrations are harmonic:

tcosω}{{u} ii (9.2)

where i}{ eigenvector iTh waveform;

iω iTh en circular frequency (radians per unit time);

t time.

Thus, the matrix equation (9.1) takes the following form:

{0}}[K]){[M]ω( 2 ii (9.3)

99

This equation has a solution, apart from the trivial }0{}{ i only if the determinant

of the system [K])[M]ω( 2 i is zero, i.e.:

0[K][M]ω2 i (9.4)

The last equation is the task of the eigenvalues. Solution of equation (4), if n - order

of the matrix is the characteristic polynomial nSecond order, which has n roots:

21ω , 2

2ω ... 2nω Wherein n number of degrees of freedom. These roots are the

eigenvalues of the equation. Eigenvectors i}{ Is obtained by substituting zeros

received 2ωi in equation (3). Eigenvalue 2ωi determine the natural frequency of the

system 2ωi And eigenvector i}{ corresponding waveform (moving system).

Values of natural circular frequency )ω( and natural frequencies (f) are related

as follows:

2π

ωf i

i (9.5)

where if iTh natural frequency (cycles per unit time).

Usually eigenvector i}{ called normalized if the following equation

(reflecting the orthogonality property forms of natural vibrations)

1}]{[}{ ii M (9.6)

that is

}I{}]{[}{ M

100

In another case, the eigenvector i}{ normalized such that the largest of its

components is equal to one. Orthogonality condition waveforms can be explained as

the vanishing of the inertial forces iSecond waveform on the displacements kThe

second waveform.

When using the condensation frequency (reduction degree of freedom) n

eigenvectors can then be deployed on the stage of "extension" to the full set of

degrees of freedom modal design:

ii }ˆ]{[][}{ sm1

sss KK (9.7)

where i}{ s - Vector excluded (subsidiary) degrees of freedom i-Th mode (auxiliary

degrees of freedom are those degrees of freedom which will condense

to reduce the dimensionality of the system);

][],[ smss KK - Submatrix stiffness auxiliary degrees of freedom and

communications subsidiary of degrees of freedom of the

retained respectively;

i}ˆ{ - Held vector (basic) degrees of freedom i-Th mode.

10. Superposition method MOU

Superposition method uses data about the modes of natural frequencies and

mode shapes obtained through modal analysis, to assess the dynamic response of

structures from non-stationary or stationary harmonic disturbances.

The equations of motion can be expressed in the following form:

}F{}u]{K[}'u]{C[}''u]{M[ (10.1)

where [M] - mass matrix of the system;

[C] - damping matrix of the system;

[K] - stiffness matrix of the system;

101

{U''} - vector of nodal accelerations;

{U '} - vector of nodal velocities;

{U} - vector of nodal displacements;

{F} - vector of the applied forces, variables over time, which can be painted

as:

{F} = {Fnd} + s {Fs} (10.2)

where {Fnd} - vector of time-varying nodal forces;

s - the scale factor load vector;

{Fs} - vector of loadings obtained by modal analysis (see below).

Load vector {Fs} is calculated in the analysis waveforms. It is formed in the

same way as the load vector substructure.

Next development like this in the book of Bath [2].

We define a set of modal coordinates yi such that

n

1i

ii y}{}u{ (10.3)

where }{ i - Waveform i-Th mode;

n - number of used modes (entered via commands TRNOPT or HROPT).

Note that equation (10.3) preclude the use of a non-zero input displacement after yi

not determine directly, but through {u}. The inverse relationship exists (Eq. (3.10))

for the case where all the offsets are known, but not when only a few are known.

Substituting equation (10.3) to (10.1), we obtain:

102

}F{y}{]K[y}{]C[y}{]M[n

1i

ii

n

1i

ii

n

1i

ii

(10.4)

Multiplying a typical waveform }{ j

}F{}{y}{]K[}{y}{]C[}{y}{]M[}{ j

n

1i

iij

n

1i

iij

n

1i

iij

(10.5)

From the orthogonality condition of natural modes that

ji0}]{K[}{

ji0}]{M[}{

ij

ij

(10.6) and (10.7)

In the method of superposition of modes, or only allowed Rayleigh damping constant

due to the fact that

ji0}]{C[}{ ij (10.8)

Applying these conditions to equation (10.5) will remain only in terms ji

}F{}{y}]{K[}{y}]{C[}{y}]{M[}{ jjjjjjjjjj (10.9)

Odds jy , jy and jy are defined as follows:

A. Factor jy :

Provided valuation (17.3-6) - 1}]{[}{ ii M

1}]{[}{ jj M (10.10)

V. Factor jy :

103

Member of the damping is based on processing of modal coordinates, as

separate systems with one degree of freedom (ie independent equations 15.11-1) for

which

2jjjj C}]{C[}{ (10.11)

and

1M}]{M[}{ 2jjjj (10.12)

Figure 10.1 - oscillator with one degree of freedom

Equation (10.12) may determine j :

j

jM

1 (10.13)

Of Tsi (68)

jjjj MK2C (10.14)

where j - The share of critical damping for the j-th mode

and

)MK( jjj

where j - Own circular frequency for the j-th mode

Combining equation (10.13) with (10.15) and (10.11)

104

jj

2

j

jjjjj 2M

1MK2}]{C[}{

(10.16)

C. Factor yj

From equation (10.3), that is, from {0}}[K]){[M]ω( 2 ii

}]{M[}]{K[ j2jj (10.17)

Multiply on the left j}{

}]{M[}{}]{K[}{ jj2jjj (10.18)

Substitute equation (10.10) for the masses - 1}]{[}{ jj M

2jjj }]{K[}{ (10.19)

We introduce the symbol, let

}F{}{f jj (10.20)

represents the right-hand side of equation (10.9). Substitute equation (10.10), (10.16),

(10.19) and (10.20) into (10.9), and we obtain the equation of motion for the modal

coordinates:

jj2jjjjj fyy2y (10.21)

Since j is any fashion, equation (10.21) is n independent equations in n

unknowns yj. The advantage of the system is decoupled in that all computationally

105

expensive matrix algebra was made in the proper solver and long transients to be

analyzed can in principal inexpensive coordinates using equation (10.21). In

harmonic analysis, the frequency can be scanned faster than the method of reduced

harmonic response. Yi values converted back to geometric displacement {u}

(response of the system to the load), using equation (10.3). That is, the individual

modal responses yi are superimposed on each other to get the actual response, hence

the name "the superposition of modes." If the modal analysis was performed using

the method of condensation (MODOPT, REDUC), The matrices and load vectors in

the above equations will contain the main (held) degrees of freedom (ie, {u}).

10.1 Modal damping

Modal damping, i - Combining multiple inputs damping ANSYS.

mjjjj 22 (10.22)

where - Inertial damping factor (enter the command ALPHAD)

- Structural damping factor (enter the command BETAD)

- Coefficient of damping constant (by the command DMPRAT)

mj - Modal damping ratio (by the command MDAMP)

Because of the assumption in equation (10.8), explicit damping elements such as

COMBIN14 not allowed superposition procedure waveforms.

11. SPECTRAL ANALYSIS

Two types of spectral analysis: deterministic method and the spectral response

of a non-deterministic method of random fluctuations. Allowed excitation in the

supports (e.g., seismic action) and agitation at points remote from the supports. There

are four types of spectral analysis:

106

Univariate spectral analysis;

multivariate spectral analysis;

computational and experimental method of dynamic analysis;

method of power spectral density.

Univariate spectral analysis involves the use of one and the same spectral

curve (for example, changes in the force depending on time) for all the excited pixels.

Multivariate spectral analysis assumes the task in different points of excitation

spectral curves.

Computational and experimental method dynamic analysis is used to assess the

strength of the ship's equipment. The method consists in obtaining the system

response spectrum which is obtained based on empirical equations and tables based

on impact, presented in the report NRL-1369 Laboratory research Navy USA.

Method of power spectral density is a probabilistic approach to finding the

spectral response and is known as random vibration analysis.

11.1 Assumptions and Limitations

1. The design is linear.

2. For univariate spectral analysis and calculation and the experimental method of

dynamic analysis of the perturbation is given to the design spectrum with known

direction and frequency components influencing uniformly on all pivot points or

held by the specified (main) does not support the degree of freedom (when using

the frequency of condensation).

3. For multivariate spectral and power spectral density analyzes, the design may

resent different input spectra with their application to different reference points

107

and the reference (free from bonds) nodes. You can define up to ten different input

spectra.

11.2 Description of the analysis

Spectral analysis is possible when choosing the type of analysis (ANTYPE,

SPECTR) and if the pre-modal analysis was performed. If necessary association

modes, the desired mode can be obtained by "extending" (recover), as described in

section 17.3 (when using frequency-condensation method).

Supports four options - univariate spectral analysis (SPOPT, SPRS),

computational and experimental method for dynamic analysis (SPOPT, DDAM), the

method of power spectral density (SPOPT, PSD) and multivariate spectral analysis

(SPOPT, MPRS). Each option is described in detail hereinafter.

11.3 Univariate spectral analysis (SPOPT, SPRS)

Two types of perturbations defined on the support (kinematic disturbance, eg,

seismic) and in areas remote from the supports (power outrage). All the required data,

see Table 11.1

Table 11.1

Spectral types of load

Options perturbation

Reference perturbation Indignation remote from the

supports

The spectrum of the input

Table characterizing the

spectrum (team FREQ andSV)

Table amplitude factors

(Team FREQ andSV)

Orientation load Direction vector, introduced

team SED

Directions X, Y and Z,

selected as options FX, FY,

FZ team F

Load distribution Uniform at all pivot points

The amplitude in the

direction X, Y, Z is selected

as VALUE Team F

108

Input Type Speed, Acceleration,

Displacement Force

KSV from the team

SVTYP

0 2 3.4

1

Damping

Damping is taken into account for each mode and defined as:

iNMAT

m

s

m

NMAT

m

s

mm

ci

i

E

E

1

1

2 (11.1)

where i - The effective damping coefficient for i-Th mode;

- Structural damping factor (input value VALUE Team BETAD);

iω - Adjusted not own circular frequency for i-Th mode;

c - DC damping coefficient (input value RATIO Team DMPRAT);

m - Damping factor depending on the properties of the material number m

(Input value DAMP Team MP);

imism KE

21 - Strain energy for material number m;

i - Displacement vector for i-Th mode;

][ mK - The stiffness matrix of the structure of a material number m;

i - Modal damping factor for i-Th mode (command MDAMP).

Note that depending on the material of the damping contribution is calculated in step

"expansion" (Recovery) modes, so that this contribution of the damping has been

turned back.

11.3 Odds ratios and modal contribution

109

Odds contribution for a given direction of the perturbation is defined as:

D]M[

ii - For options based on kinematic perturbation (eg,

seismic);

F

ii - Option is based on the perturbation force;

where i - Contribution rate for i-Th mode;

i - Eigenvector normalized by equation (10.6) (Nrmkey on team MODOPT

has no effect);

}D{ - Vector defining the direction of the disturbance;

}F{ - A force vector at the inlet;

]M[ - Mass matrix.

Vector defining the direction of the perturbation has the form:

...DDD}D{ 321aaa

(11.2)

where ajD - Perturbation degrees of freedom j towards and (and may be equal to

or X or Y, or Z); value aD can be determined in one of two ways.

1. For values of D based on the team SED:

B

SxxD

B

S yyD

B

SzzD

110

where Sx, Sy, Sz - input values SEDX, SEDY andSEDZ, Respectively, for the team

SED

2z

2y

2x )(S)(S)(SB (11.3)

2. For values of D based on the commands SED and ROCK:

xxx RSD

yyy RSD (11.4)

zzz RSD

R is defined as:

z

y

x

z

y

x

z

y

x

r

r

r

C

C

C

R

R

R

(11.5)

where Cx, Cy, Cz - input values OMX,OMY and OMZRespectively in the team

ROCK;

- The symbol for the vector product;

rx = Xn - Lx

ry = Xn - Ly

rz = Xn - Lz

Xn, Yn, Zn - coordinates of the node n

Lx, Ly, Lz - position of the center of rotation (the input values CGX,CGY

and CGZ in the team ROCK).

Vector displacement, velocity and acceleration for each mode is calculated from the

eigenvector using the "modal factor":

iimii }{A}r{ (11.6)

111

where

ACEL ìåòêà åñëè2

VELO ìåòêà åñëè1

DISP ìåòêà åñëè0

m

label - three field teams in association modes (SRSS, CQC, GRP, DSUM,

NRLSUM);

i - Own the angular frequency;

Ai - modal factor (see below).

Modal factor is calculated in five different directions, depending on the type of

disturbance (team SVTYP).

1. For SVTYP0 (perturbation is applied to the support in the form of speed)

i

ivii

SA

(11.7)

where Svi - spectral velocity for i-Th mode (found in the input velocity spectrum at

the frequency fi and effective damping ratio i );

fi - iTh natural frequency (number of cycles per unit time = i/ 2);

i - iTh en circular frequency (radians per unit time).

2. For SVTYP1 (a perturbation applied force):

2i

ifii

SA

(11.8)

where Sfi - spectral power for i-Th mode (input table is of amplitude frequency

multipliers fi and effective damping ratio i );

3. For SVTYP2 (a perturbation applied to the support in the form of

acceleration):

112

2i

iaii

SA

(11.9)

where Sai - spectral acceleration for i-Th mode (found in the input acceleration

spectrum at the frequency fi and effective damping ratio i );

4. For SVTYP3 (a perturbation applied to the support in the form of

displacement):

iuii SA (11.10)

where Sui - spectral relocation i-Th mode (found in the input spectrum at the

frequency fi movement and effective damping ratio i );

5. For SVTYP, 4 (power spectral density (PSD), read Vanmarke (34) in the

list of literature theoretical guidance)

2

1

0

21

4

i

dSSA PiPi

i

i

i

(11.11)

where Spi - power spectral density for i-Th mode (derived from the spectral density

of the input frequency fi and effective damping ratio i );

- Damping factor (the input value RATIOCommands DMPRAT, 0.01

default);

The integral in the above equation can be approximated as:

i

iL

1j

sj0

p fSdS (11.12)

where Li = fi (in integer form);

Spj - estimated power spectral density at the frequency (f) equal to j (in real

form);

113

f - the effective bandwidth for fi = 1.

When Svi, Sfi, Sai, Sui, or Spi required between the tabulated values of input

frequencies, logarithmic interpolation is performed.

Spectral values and modal coefficients are displayed in the table "RESPONSE

SPECTRUM CALCULATION SUMMARY" given input curve with the smallest

damping coefficient, without effective damping coefficient.

12. ANALYSIS OF RANDOM VIBRATIONS

The method allows applying multiple perturbations in the form of power

spectral density (PSD) to the inputs of the system (up to ten), which can be:

1. Fully correlated,

2. Uncorrelated, or

3. Partially correlated.

The procedure is based on statistical calculations of each modal response, and

then combining them (summation or combination). It is assumed that the excitation -

stationary random processes.

12.1 Description of the method

For partially correlated force and kinematic perturbations full equation of

dynamic behavior of structures are divided into two parts for the unknown

displacements and given respectively:

0

F

u

u

KK

KK

u

u

CC

CC

u

u

MM

MM

r

f

rrrf

frff

r

f

rrrf

frff

r

f

rrrf

frff

(12.1)

114

where {uf} - vector of unknown displacement (from power disturbances);

{Ur} - vector set of displacements (excited by random load is given by a single

movement);

{F} - set the force perturbation (force value can be different from unity, taking

into account the scale of the coefficients of the deposit);

{0} - reaction force vector corresponding to the vector specified displacements

{ur}.

Unknown displacement can be subdivided into pseudo-static and dynamic

parts:

(12.2)

Pseudo-static displacement can be obtained from equation (12.1) except for the

first two terms in the left side of the equation and replacing the vector of unknown

displacements {uf} a vector of pseudo-static components of {us}

(12.3)

where ][][][ fr1

ff KKA .

Elements in i-Th column [A] Are pseudo-static displacement from single

predetermined movements excited iTh a power spectral density (PSD). Substitute

equation (12.3) and (12.2) into (12.1) and add a slight damping. The first row of the

matrix equation (12.1) can then be rewritten as follows:

(12.4)

The second term on the right side of the above equation is equivalent to the

forces due to kinematic perturbations.

115

When using the method of superposition of modes of equation (10.3) can be written

again as:

(12.5)

The above equation unrelated to each other and can be written as n separate

equations:

(12.6)

where n - The number of waveforms selected for evaluation;

yj - generalized displacements;

j and j

j

j

116

Transfer functions for the type of system with one degree of freedom takes

different forms depending on the type of input PSD and the desired type of

response. Forms of transfer functions to move as output listed below for different

occasions.

1. At the entrance - force or acceleration:

(12.13)

2. At the entrance - move

(12.14)

117

3. At the entrance - speed

(12.15)

where - The frequency of the excitation

j - Own circular frequency j-th mode

i = -1 - The imaginary unit.

Thus, the analysis of random fluctuations can be used to show the absolute rms

response unknown displacements:

(12.16)

where | | Re - denotes the real part of the argument;

2

iS - Dispersion iTh pseudo-static displacement;

),(ii dSv uuC - Covariance between static and dynamic

movements

General formulation described above provides simplified equations for some

situations that usually encountered in practice. To fully correlated power disturbances

and given identical displacements Subscripts l and m would exclude from the

equation (12.10) to (12.12). When there is only power disturbances last two terms in

equation (12.16) does not apply, and only the first term within the large parentheses

in equation (12.10) must be taken into account. For uncorrelated power and kinematic

perturbations, mutual PSDs (i.e. when l m) Are zero, and only the terms for which l

= m in equations (12.10) through (12.12) must be taken into account.

Equations (12.10) to (12.12) can be rewritten as:

118

n

j

n

k

jkjkijd RSi

1 1

)()( (12.17)

r

l

r

m

lmimils RAASi

1 1

)()( (12.18)

n

j

r

l

ijilijsd RASi

1 1

)()( (12.19)

where )(),(),( jllmjk RRR - Modal PSD members who are inside the large

parentheses equations (12.10) to (12.12)

Solution in closed form is obtained for the piecewise linear PSD (PSD) in a

logarithmic scale on both axes, is used to calculate each of the integral in equation

(12.16) (Chen and Ali (193), and Harichandran (194)). Subsequently, the dispersion

becomes:

(12.20)

(12.21)

(12.22)

Modal covariance matrix jkQ , lmQ and jlQ~

identified. PSD file. Note that

equation (12.20) to (12.22) are the union of events for analysis of random vibrations.

Dispersion for voltages nodal forces or reactions may be calculated from

equations similar to (12.20) to (12.22). If it is desirable to obtain a dispersion of

voltage waveforms replace (ij) And static displacement ( iA iA ) Modal voltages

( ij ) And static voltages ( iA ). Likewise, if a node is desired dispersion force

waveform and replacing static displacement modal nodal forces ( ij ) And static

119

nodal forces ( iA ). Finally, if desirable the dispersion reaction waveforms replace

static modal displacement reactions (ij) And static reactions ( iA ). Furthermore, the

dispersion of the first and second derivatives with respect to time all the variables

mentioned above may be calculated using the following relations:

(12.23)

)()( 4 uu SS

12.3 Not diagonal spectral terms for partially correlated input power

spectral density

For perturbations defined by more than one function of the power spectral

density (PSD), the cross terms that define the degree of correlation between the

different functions of the MTA is defined as:

)()()()()(

)()()()()(

)()()()()(

)(

3323231313

2323221212

1313121211

SQiCQiC

QiCSQiC

iQCiQCS

S (12.25)

where )(nnS -PSD spectra of the input that have relationships with each other;

)(nmC - Mutual spectral density, which constitute the real part of the cross-

members; )(nmQ - Components of the spectra, which represent the imaginary part of

the cross-members.

Mutual power spectral density function (PSD) is the coherence function:

)()(

)()()(

2

2

mmnn

nmnm

nmSS

QiC

(12.26)

where 1)(0 2 nm .

Although the above example shows the cross-correlation of the input spectra

for 3, this matrix can range in size from 2 x 2 to 10 x 10 (I.e., the maximum number

of tables - 10).

120

For the special case in which all the cross terms are zero, the input spectra are

believed to uncorrelated. Note that the correlation between power and kinematic

perturbations are not allowed.

12.4 Correlation space

The degree of correlation between the excited nodes can be specified.

Depending on the distance between the excited nodes and values RMIN and RMAX,

the impact of the power spectral density can be set so that the excitation of the nodes

can be uncorrelated, partially correlated or fully correlated. If the distance between

the nodes of the excited - less than RMIN, these two nodes are correlated completely;

if the distance is greater than RMAX, then these two uncorrelated Node; if the

distance is between RMIN and RMAX, partially correlated excitation and based on

the actual distance between nodes. The figure below specifies how RMIN, RMAX

and correlation.

For two excited points 1 and 2, the power spectral density would be:

1

1)()(

12

12

0a

aSS (12.27)

where:

MAX

MIN

MAXMIN

MINMAX

MAX

RеслиD

RеслиD

RDеслиRRR

DR

a

12

12

1212

12

0

1

D12 - the distance between two excited points 1 and 2;

So () - Set the input PSD.

121

Figure 12.1 Spheres of influence, reflecting the space correlation of disturbances

specified power spectral density

Excitation node i fully correlated with the excitation of the node j.

Excitation node i partially correlated with the excitation of the node k

Excitation node i uncorrelated with the excitation node l.

12.5 Wave Propagation

To include the effects of wave propagation in design from the random

loading, outrage PSD defined as:

lmdi

lm eSS

)()( 0 (12.28)

where:

delayV

VDd imlm

2 (Delay time).

imim XXD - Difference vectors excitation points i and m.

V- The speed of wave propagation along the axes X, Y and Z.

lX -coordinates perturbed node l.

Allowed more than one simultaneous wave or spatially correlated input power

spectral densities when the operation input [S ()] Reflects the influence of two or

more uncorrelated input spectra. In this case, a partial correlation of the inputs of

predetermined power spectral density currently not permitted.

122

12.6 Method of multivariate spectral analysis (SPOPT, PSD)

Analysis of the spectral response supports multifactor kinematic and power

disturbance, allowing up to 10 different spectral tables that may be unrelated

(uncorrelated) among themselves.

Many components to perform multivariate analysis of the spectral response of

the system is discussed in the previous sections devoted to the method of random

vibration. Assuming that the contribution of the coefficients j can be calculated

(e.g., from equation (12.8)), the coefficients for the modal lThe second table are as:

jjj SB (12.29)

where jS - Interpolated input spectral response for lSecond table j-th natural

frequency;

Modal coefficients together with the square root of the sum of squares (SRSS):

21

2

3

2

2

2

1 ... jjji BBBA (12.30)

Waveforms, modal voltages and other results after multiplication by modal

coefficients can then be combined with any of the available methods of combining

modes: (square root of the sum of squares (SRSS), complete quadratic combination

(CQC), the double sum (DSUM) , group (GRP) or summation method (DDAM)

Laboratory (NRL) Navy USA) as described in the previous sections on univariate

spectral analysis.

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