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EXAMPLES MANUAL FOR PROGRAM MAVART

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P153521.PDF [Page: 5 of 124]

I+ UNLIMITED PISTRIBUTION

National Defence Defense nationale Research and Bureau de recherche Development Branch et developpement

.

EXAMPLES MANUAL FOR

PROGRAM MAVART

by

DREA CR/95/416

E.L. Skiba - G.W. McMahon - Z. Wozniak

ACRES INTERTEL Limited 5259 Dorchester Road

Niagara Falls, Ontario, Canada L2E 6W1

Scientific Authority ~~ 12.~ Christopher J. Purcell

W7707-4-2206/01-0SC Contract Number

January 1995

CONTRACTOR REPORT

Defence Research Establishment Atlantic

Prepared for

Centre de Recherches pour Ia Defense Atlantique

Canada

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P153521.PDF [Page: 7 of 124]

Examples Manual for ProgramMA V ART

by

E. L. Skiba, G. W. McMahon and Z. Wozniak

ABSTRACT

This document is the third in a set of documents that provides information on the Finite Element Model to Analyze the Vibrations and Acoustic Radiation of Transducers (MA V ART). The other two documents in the set are: 1. User's Manual for Program MA V ART, 2. Theoretical Manual for Pro~ram MAVART.

The program MAV ART has been developed for the Defence Research Establishment Atlantic (DREA) under research contracts to Canadian industry from 1976 to the present.

This manual provides a number of sample problems to aid users in model development and output interpretation. It demonstrates virtually all elements and features of MA V ART. It also provides a set of problems for validating modifications and new installations.

Ce manuel est le troisieme d'une serie de documents qui contiennent des renseignements sur le logiciel du Modele d'analyse des rayonnements acoustiques et des vibrations des transducteurs par elements finis (MA V ART). Les deux autres documents de la serie sont: 1. le Manuel de l'utilisateur du logiciel MAVART, et 2.le Manuel de theorie du logiciel MAVART.

MA V ART est un logiciel qui a ete developpe pour le Centre de recherches pour la defense/Atlantique dans le cadre de plusieurs contrats de recherche conclus avec l'industrie canadienne de 1976 a aujourd'hui.

Ce manuel donne des exemples de problemes concus pour aider les utilisateurs en vue du developpement de modeles, et de !'interpretation des resultats. Ce manuel demontre pratisuement tous les diverses composantes et fonctions de MAVART. ll contient aussi une serie de problemes pour la validation des modifications et des nouvelles installations.

11

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TABLE OF CONTENTS

Page

ABSTRACT........................................................................... 11

TABLE OF CONTENTS.......................................................... 111

~T OF FIGIJRES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

1 ~OI>llTCTIO~ .................................................................. 1-1

2 TEST PROBLEM I>EFINITIO~ .. . . . .. . .......... ....... ... . .. . . .. . .. . . .. .. ... 2-1 2.1 LIST OF PROBLEMS.............................................................. 2-1 2.2 PROBLEM I FEATURE MATRIX............................................ 2-2

3 GEMm.AL APPROACH TO PROBLEM ANALYSIS.................. 3-1 3.1 MODEL DEVELOPMENT....................................................... 3-1 3.2 PROBLEM SOLVING............................................................. 3-1 3.3 DATA PRESENTATION......................................................... 3-1

4 SERIES 1: PAR PIEZOELECTRIC SPHERICAL SHEIL............ 4-1 4.1 PROBLEM 1 - STATC, Pressure Load...................................... 4-5 4.2 PROBLEM 2 - EIGEN, Resonance........................................... 4-8 4.3 PROBLEM 3 - EIGEN, Antiresonance.. ..... .. ........ ....... ... .... .... ... 4-11 4.4 PROBLEM 4 - CAPAC, Frequency Sweep................................. 4-12 4.5 PROBLEM 5 - CAPAC, TRANsient.......................................... 4-15 4.6 PROBLEM 6 -DRIVE, Frequency Sweep.................................. 4-19

5 SERIES 2: PT PIEZOELECTRIC RING... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1 5.1 PROBLEM 7- EIGEN, Fourier Mode 2 Resonance..................... 5-4 5.2 PROBLEM 8 - CAPAC, Frequency Sweep . . . .. . . . . .. . . . . . . . . . .. . . . .. .. . . . . 5-6 5.3 PROBLEM 9 - DRIVE, Resonance Search................................. 5-8

6 SERIES 3: PAR TRILAMINAR BENDER I>ISK........................ 6-1 6.1 PROBLEM 10- STATC, Electric Stress..................................... 6-4 6.2 PROBLEM 11 - STATC, Thermal Stress................................... 6-6 6.3 PROBLEM 12- CAPAC, SLIDER Element................................ 6-8

7 SERIES 4: SIIEIL 'I'IIIN'I>ISwruBE ... ... . ............ ........ .. ..... .... 7-1 7.1 PROBLEM 13- EIGEN, Resonances........................................ 7-5 7.2 PROBLEM 14- STATC, Temperature Differential..................... 7-8

8 SERIES 5: ANALYTICAL FLUID PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1 8.1 PROBLEM 15 - DRIVE, Piston .. . . .. . .. . . .. . . .. . .. .. . . .. .. .. .. . . .. .. .. . .. . .. .. 8-3

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TABLE OF CONTENTS (Cont'd)

Page

9 SERIES 6: COMPLIANT FLUID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-1 9.1 PROBLEM 16 - DRIVE, Spherical Air Bubble............................ 9-5

10 SERIES 7: R.IN"G. .. . . . . . . . . . .. . . . . . .. . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10-1 10.1 PROBLEM 17- EIGEN, RING-Stiffened SHELL........................ 10-2

11 SERIES 8: DAMPING APPLICATIONS.................................. 11-1 11.1 PROBLEM 18 - DRIVE, PT Ring Damping............................... 11-1

12 SERIES 9: RING ARRAY, RIGID NODE................................. 12-1 12.1 PROBLEM 19 - DRIVE, PT Ring Array.................................... 12-7 12.2 PROBLEM 20 - DRIVE, RIGID Node ... ... .. .... ... . .. . ... .. ... . ... . .. . .... 12-9

13 SERIES 10: TORSIONAL PROBLEMS . ... ... . . .. .. ... . ... ... . . ... . .. . .. . .. 13-1 13.1 PROBLEM 21 - STATC, Torsional Load.................................... 13-3 13.2 PROBLEM 22 - EIGEN, Torsional Modes.................................. 13-3

14 SERIES 11 PROBLEMS • NONLINEAR EFFECTS...................... 14-1 14.1 PROBLEM 23 - Static-nonlinear.............................................. 14-2 14.2 PROBLEM 24 - EIGEN, Nonlinear effects................................. 14-2

15 SUMMARY.......................................................................... 15-1

16 RECO]\f]\IEN]JATI ONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-1

17 ~CES....................................................................... 17-1

lV

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LIST OF FIGURES Page

TESI' PROBLElVI DEFINITION

2.2.1 Problem-Feature Matrix.................................................. 2-3

SERIES 1: PAR PIEZOELECTRIC SPHERICAL SHELL

4.0.1 4.0.2 4.0.3 (a) 4.0.3 (b) 4.0.4 (a) 4.0.4 (b)

4.1.1 4.1.2 4.1.3 4.1.4

4.2.1 4.2.2 4.2.3 4.2.4

4.4.1 4.4.2 4.4.3

Model Diagram............................................................... 4-2 Full F.E. Model............................................................... 4-2 Fluid Node Numbering.................................................... 4-3 Solid and Connecting Fluid Node Numbering..................... 4-3 FLUID and FTOF Element Numbering............................. 4-4 SOLID and SILV Element Numbering. ............................ 4-4

PROBLEM! Fixities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-6 Deformations.................................................................. 4-6 Axial Stresses................................................................. 4-7 Radial Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7

PROBLEM2 1st Mode Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9 2nd Mode Shape..................................... . . . . . . . . . . . . . . . . .. . . .. . .. 4-9 3rd Mode Shape .. .. . . . . . . .. ... . . .. ... .. ... . ... .. . . .. . .. .. . . .. . .. . . . . .. . ... . .. 4-10 4th Mode Shape............................................................... 4-10

PROBLEM4 Deformations at 404Hz, Phase= 0. ............... ... ...... .. . ..... .... 4-13 Deformations at 404 Hz, Phase = n..................................... 4-13 Capacitance versus Frequency......................................... 4-14

PROBLEM5 4.5.1 Deformations at 0.0075 s.. ......... ............. ....................... .... 4-16 4.5.2 Deformations at 0.0080 s........... ......... ............................... 4-16 4.5.3 Deformations at 0.0085 s........... ...................... .............. .... 4-17 4.5.4 Deformations at 0.0090 s.. .................. ........................... .... 4-17 4.5.5 Deformations at 0.0095 s................................................... 4-18 4.5.6 Deformations at 0.0100 s.. ......... ......... .... ........................... 4-18

PROBLEM6 4.6.1 Admittance Components vs. Frequency............................. 4-20 4.6.2 Transmitting Voltage Response........................................ 4-20 4.6.3 Displacement of Solid Boundary........................................ 4-21 4.6.4 Directional Response at 330 Hz . . . . . . . . . . .. . . . . . . .. . . . . .. . . . . . .. . . . . . . . . 4-21 4.6.5 Near-Field Pressure Contours.......................................... 4-22 4.6.6 Detail of Near-Field Pressure ......... ... ... . . .. .. ... ... .. . . . .. .. .... .. . 4-22 4.6.7 Near-Field Pressure Gradient Contoure ............................ 4-23

v

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LIST OF FIGURES (Cont'd)

p~

SERIES 2: PT PIEWELECTBIC RING

5.0.1

5.0.2 5.0.3 (a) 5.0.3 (b)

5.1.1 5.1.2

5.2.1 5.2.2

5.3.1 5.3.2 5.3.3 5.3.4 5.3.5 5.3.6

Cross-Sectional Diagram of Free-Flooded Ring in Sea Water ....................................... . Element Numbering ....................................................... . Outer Node Numbering .................................................. . Inner Node Numbering Detail ......................................... .

PROBLEM7 1st Mode Shape ( m = 2 ) at 72.9 Hz .................................... . 1st Mode Shape ( m = 0 ) at 516 Hz ..................................... .

PROBLEMS Deformations at 516 Hz .................................................... . Capacitance vs. Frequency .............................................. .

PROBLEM9 Deformation at 375 Hz ..................................................... . Tangential Stress Contours ............................................. . Admittance Components vs. Frequency ............................ . Transmitting Voltage Response ....................................... . Directivity at 37 5 Hz ........................................................ . Near-Field Pressure Contours at 375 Hz ........................... .

5-2 5-2 5-3 5-3

5-5 5-5

5-7 5-7

5-9 5-9

5-10 5-10 5-11 5-11

SERIES 3: TRILAMINAR BENDER DISK

6.0.1 6.0.2 . 6.0.3 6.0.4 (a) 6.0.4 (b)

6.1.1 6.1.2

6.2.1 6.2.2

6.3.1 6.3.2 6.3.3 6.3.4

Cross-Sectional Diagram................................................. 6-1 F.E. Model...................................................................... 6-2 Element Numbering........................................................ 6-2 Node Numbering, Aluminum Layer................................. 6-3 Node Numbering, Piezoelectric Layers.............................. 6-3

PROBLEM 10 (Electrical Stress) Electrical Deformation with a Hard SLIDER...................... 6-5 Radial Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5

PROBLEM 11 (Thermal Stress) Thermal Deformation with a Hard SLIDER....................... 6-7 Radial Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-7

PROBLEM 12 (Sliders) Deformations without Slider Softening . ... .. .. .. . . ... .. .. .. ...... 6-9 Deformations with Slider Softening................................... 6-9 Deformations with Partial Slider Softening........................ 6-10 First Mode Shape ............................... ,............................ 6-10

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Page

SERIES 4: SHEIL THINDISKITIJBE

7.0.1 Disk Radial Cross-Section................................................ 7-2 7.0 .2 Tube Radial Cross-Section................................................ 7-2 7.0 .3 Disk Node Numbering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 7-3 7.0.3 Disk Element Numbering................................................ 7-3 7.0.5 Tube Node Numbering..................................................... 7-4 7.0.6 Tube Element Numbering................................................ 7-4

PROBLEM13 7 .1.1 Disk, 1st Flexural Mode Shape.......................................... 7-6 7.1.2 Tube, 1st Mode Shape, Fixed Ends..................................... 7-6 7 .1.3 Tube, 1st Mode Shape, Free Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7 7.1.4 Tube, 1st Mode Shape, Fixed Rotations ................ .. .... ........ 7-7

PROBLEM14 7 .2.1 Disk Thermal Deformations............................................. 7-9 7 .2.2 Disk Forced Deformations................................................ 7-9

SERIES 5: ANALYTICAL FLUID PROBLEM

. 8.0.1 8.0.2 8.0.3

8.1.1 8.1.2

Diagram of Piston in Infinite Baffle................................... 8-1 Node Numbering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-2 Element Numbering........................................................ 8-2

PROBLEM15 Piston Directivity at 12,000 Hz........................................... 8-4 Near-Field Pressure Contours at 12,000 Hz........................ 8-4

SERIES 6: COMPLIANT FLUID

9.0.1 9.0.2 (a) 9.0.2(b) 9.0.2 (c) 9.0.3 (a) 9.0.3 (b)

9.1.1

Air Bubble in Water ........................................................ . Outer Fluid Node Numbering .......................................... . Middle Fluid Node Numbering ........................................ . Compliant Fluid Air Node Numbering ............................. . Outer Element Numbering .............................................. . Inner Element Numbering ............................................. .

PROBLEM16 Far-Field Pressure and Radial Displacement of a Resonant Air Bubble in Water .................................... .

9-2 9-2 9-3 9-3 9-4 9-4

9-6

SERIES 7: RING

10.0.1 SHELL Tube with RING Stiffening.................................... 10-1

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Page

SERIES 8: DAMPING APPLICATIONS

PROBLEM18 11.1.1 11.1.2 11.1.3 11.1.4 11.1.5 11.1.6 11.1.7

Transmitting Voltage Responses with Material Damping . . . 11-2 Efficiencies with Material Damping.................................. 11-2 TVR with Viscous Pressure Damping (Stiffness)................ 11-3 TVR with Viscous Pressure Damping (Mass) . . .. . . .. . . .. . . . .. . . . 11-3 TVR with Viscous Pressure Gradient Damping (Stiffness).. 11-4 TVR with Viscous Pressure Gradient Damping (Mass)....... 11-4 Efficiencies with Viscous Damping................................... 11-5

SERIES 9: RING ARRAY, RIGID NODE

12.0.1 Cross-Sectional Diagram ................................................ . 12.0.2 F.E. Mesh ..................................................................... . 12.0.3 (a) Outer Node Numbering .................................................. . 12.0.3 (b) Middle Node Numbering ................................................. . 12.0.3 (c) · Inner Fluid Node Numbering .......................................... . 12.0.3 (d) Solid Node Numbering .................................................... . 12.0.4 (a) Outer Element Numbering .............................................. . 12.0.4 (b) Inner Element Numbering ............................................. .

12.1.1 12.1.2 12.1.3

12.2.1

PROBLEM19 Displacement at 6000 Hz with SOLID Spacer ..................... . Far-Field Directivity at 6000Hz with SOLID Spacer ............ . Near-Field Pressure Contours at 6000 Hz with SO LID Spacer ......................................................... .

PROBLEM20 Pressure Contours at 6000 Hz with RIGID Node as Spacer ............................................ .

SERIES 10: 'OORSIONAL PROBLEMS

12-2 12-3 12-3 12-4 12-4 12-5 12-6 12-6

12-7 12-8

12-8

12-9

13.0.1 13.0.2 13.0.3

Solid Cylinder Model for Torsional Problems...................... 13-1 Element Numbering........................................................ 13-2 Node Numbering . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . . . . . 13-2

SERIES 11 PROBLEMS- NONLINEAR EFFECTS

14.0.1 Solid Cylinder Model for Nonlinear Effects......................... 14-1

viii

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1 INTRODUCTION

MA V ART is a two dimensional, axisymmetric, dynamic finite element code developed for Defence Research Establishment Atlantic (DREA) under research contracts with Canadian industry. Its purpose is to provide a computer aided design facility for axisymmetric electroacoustic transducers. MA V ART's name is an acronym for the code's function, that is, a Model to Analyze the Vibrations and Acoustic Radiation of Transducers.

MAVART was first implemented in 1976, under a contract with Acres International Limited of Niagara Falls, Ontario. It has been revised and improved a number of times over the years, both at Acres and elsewhere. The current version (12.0) resulted from a contract with Acres International.

This document is a revised edition of the 1987 Examples Manual, issued as part ofDREA Contractor Report CR/87/442 [Ref.1.1]. Companion Theoretical [1.1b] and User's Manuals [1.1a] complete the manual set.

The purposes of this revised Examples Manual are:

• to aid MA VART users in model development, and data input and output interpretation;

• to provide comprehensive new user training; • to exercise and demonstrate all elements and features of MA V ART; • to provide a set of problems to validate proper operation of the program after

modifications are made; • to help in establishing timing benchmarks for reference after modifications

are made.

In addition to the tests reported in this Examples Manual, some special testing of MA V ART's capabilities and limitations have been conducted. These tests are reported separately and are referenced in this document when they are relevant to an example problem.

All of the programs in the MA V ART package are written in Fortran 77. and will run on DEC VAX, HP 9000, Apple Macintosh, Silicon Graphics, and IBM 386 PC compatible platforms without further modifications.

1-1

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2 , TEST PROBLEM DEFINITION

Most test problems come from DREA experience with simple, typical transducer designs that have special features or are of special interest. They have been selected so as to exercise all MA V ART features.

2.1 LIST OF PROBLEMS

The problems that have been included in the manual are grouped below by principal element type and are individually numbered.

Series 1 Problems PAR Piezoelectric Spherical Shell

1 STATC 2 EIGEN 3 EIGEN 4 CAPAC 5 CAPAC 6 DRIVE

pressure load (hydrophone sensitivity, stresses) resonance antiresonance frequency sweep transient frequency sweep (stresses at resonance)

Series 2 Problems Pr Piezoelectric Ring

7 EIGEN 8 CAPAC 9 DRIVE

resonance ( m = 2,0) frequency sweep ( m = 0 ) resonance search, with and without extra DOF's carried

Series 3 Problems PAR, SOLID and SLIDER Trilaminar Bender Disk

10 STATC 11 STATC 12 CAPAC

electric stress thermal stress deformations, with and without slider softening

Series 4 Problems SHELL Thin Disk!fube

13 EIGEN 14 STATC

Series 5 Problem

15 DRIVE

Series 6 Problem

16 DRIVE

Series 7 Problem

17 EIGEN

flexural and extensional resonances transverse temperature differential and nodal load

Analytical FLUID and Fl'OS

piston in infinite hard baffle

SOLID (Compliant Fluid), FLUID, and Fl'OS

Compliant fluid modelling of a resonating spherical air bubble

RING and SHELL

RING-stiffened SHELL with added nodal mass

2-1

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Series 8 Problem DAMPING Applications

18 DRIVE complex modulus and viscous damping in PT ring

Series 9 Problems Rigid Node Application

19 DRIVE ID DRIVE

Two Free-flooding PT Rings and SOLID Spacer RIGID node replacing spacer

Series 10 Problem Torsional Applications

21 STATC 22 EIGEN

SOLID cylinder with torsional loading (m=O) SOLID cylinder in torsional resonance

2.2 PROBLEM I FEATURE MATRIX

Figure 2.2.1 provides a cross reference chart of significant program features and test problems. It allows the MA V ART user to quickly find which example problem demonstrates a given problem type, element type, node type, or other program feature. The listed problems attempt to exercise all of the features to an adequate degree, biased towards the normal use of the program.

Most features of GRAFl are also exercised by these example problems, although no cross reference chart is presented.

2-2

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Figure 2.2.1: Problem I Feature Matrix

Problem Number FEATURE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ID 21 222324

p R EIGE • • • • • • • 0 STAT • • • • • • B CAPA • • • • I DRIV • • • • • • • D FOUD •

PhaseTRAN • E SOLID(Q) • • • • • • • L PAR(Q) • • • • • • • • • E PT(Q) • • • • • • M FLUID(Q) • • • • • • • E MEMB N FTOS • • • • • • • • T FTOF • • • • • • • s SILV • • • • • •

SLIDER • • • SHELL • • • • RING • VISC •

N s • • • • • • • • • • 0 R • D A • • • • • • • • • E H • • • • s E • • • • • •

F • • • • • • • • Load • • Mass •

Fourier m>O • Torsion m=O • • Damping • Thermal

Load • • Frequency

Sweep • • • Resonance

Search • NonLinear • •

2-3

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3 GENERAL APPROACH TO PROBLEM ANALYSIS

The problem analysis followed the sequence:

• preprocessing, i.e. finite element model and input data preparation; • execution in MA V ART; and • postprocessing, to interrogate results of the analysis.

For this edition of the Examples Manual, the problems have been rerun on DREA's J.LVAX 3900. Some of the problems' input data were changed slightly for various reasons, and these changes are documented where they occur.

3.1 MODEL DEVELOPMENT

The path taken to prepare the finite element (F.E.) model for a problem is:

• generation of the finite element mesh using program GRID; • preparation of material property data using program MATERL; • addition of data file title, control and other data; and • minimization of the model's matrix profile, using program PROFIL,

if the backing store solver is being used (not required for SP ARSP AK).

In addition, the mesh may be modified using program MOD, for subsequent parametric studies.

3.2 PROBLEM SOLVING

Each MA VART execution produces one or more text output files §RESxx.DAT, where "§" represents a user identification character input on startup, and "xx" is the version number. A binary data file §PLTxx.DAT, required for GRAFl input, is also generated. A portion of the data in §RESxx.DAT is output to the terminal, or to a Batch LOG file so that the operation of the job can be monitored.

3.3 DATA PRESENTATION

Each MA V ART input file created at the preprocessing stage bears a name compatible with the problem annotation, e.g. for Problem 9 the corresponding input file is D09.DAT. Some problems have multiple versions of the data file, generally with letters A,B,etc appended to the file name.

Each section of the following chapters contain details pertinent to a particular problem solved. Significant results of the analysis are shown in the form of plots generated using GRAFl, such as:

• model grids; • stress and strain contours;

1

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• pressure and pressure gradient contours; • displacements of solid boundaries; • directional response of acoustic output; and • plots of various parameters versus frequency.

Most of the graphic output presented in this edition was generated by DREA's GRAFl program, but some minor editing was done in the MacDraw II application on the Macintosh computer.

2

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4 SERIES 1: PAR PIEZOELECTRIC SPHERICAL SHELL

A cross-sectional diagram of the radially poled piezoelectric spherical shell, made of Clevite PZT-4 material [4.1] and immersed in sea water, is shown in Figure 4.0.1. The inner and outer radii of the shell are 0.911 m and 1.0 m, respectively. The outer radius of the near-field sea water sphere used in the analysis is 2.0 m.

Due to symmetry of the problem, only the upper hemisphere has been analyzed. The mesh of the F.E. model is shown in Figure 4.0.2. The Z and R axes of the model system of cylindrical coordinates coincide respectively with the Z and X axes of the coordinate system shown. Element numbers are printed just above the centroid of the element.

Figures 4.0.3 (a,b) and 4.0.4 (a,b) give details of node and element numbering of the F.E. model. Note that the fluid node numbers are printed by GRAF1 just above the node position and solid node numbers are printed just below.

The elements grouped along the outer boundary of the fluid are FTOF (Type 7) elements. The outer surface of the shell is "covered" with SILV (Type 8) elements. These are necessary for antiresonance analysis of a PAR piezoelectric structure, but can be useful whenever Type A (PAR) nodes must be connected together electrically. Also along the outer surface of the shell are FTOS (Type 6) elements that connect the solid structure to the fluid.

The horizontal symmetry plane requires proper boundary conditions for solid elements; the cross plane displacements for Nodes 1 ,3 and 4 are restrained in the Z direction, i.e., JFIX(1)=1.

4-1

P153521.PDF [Page: 24 of 124]

Figure 4.0.1: Series 1 Problems- Model Diagram

z

(m)

Figure 4.0.2: Series 1 Problems- Full F.E. Model

SERIES 1 PROBLEMS, PAR Piezoelectric Spherical Shell

GEOMETRY PLOTIIN:; 1~--------------------------------------. ELEMENT PIOTIIN:;

.DIMENSICNS: (ENTIRE REGICN) ~N= -0.1000000 m ~x- 2.1000001 m ZMIN= -0.1000000 m Z~X= 2.1000001 m

. MATERIAL TYPES: 1-3

.ElEMENT TYPES: 6-7-8-13-15

1 8-NOV-89 09: 41

4-2

P153521.PDF [Page: 25 of 124]

Figure 4.0.3 (a): Series 1 Problems· Fluid Node Numbering


GEOMETRY PLOTTI:tl; 1~~nn~~--------------------------------, IDDE NUMBERS

Figure 4.0.3 (b):

.DlMENSICNS: (ENTIRE REGICN)

RMIN= -0.0500000 rn ~X= 2.0500000 rn ZMIN= -0.0500000 rn Z~X= 2.0500000 rn

• MATERIAL TYPES : 1

• ElEMENT TYPES: 6-7-8-13-15

27-0CT-89 11:39

Series 1 Problems -Solid and Connecting Fluid Node Numbering


4-3

IDDE NUMBERS

.DlMENSIONS: (ENTIRE REGION) RMIN- -0.0500000 rn RMAX= 1.0500001 rn ZMIN= -0.0500000 rn Z~X= 1.0500001 rn

.MATERIAL TYPES: 1-3

• ELEMENT TYPES: 6-8-13

24-0CT-89 14:31

P153521.PDF [Page: 26 of 124]

Figure 4.0.4 (a): Series 1 Problems- FLUID and FI'OF Element Numbering


GECMETRY PLOTTIN::; 1~--~------------------------------------, ELEI£NT NLMBERS

Figure 4.0.4 (b):

• DIMENS ICNS: (ENTIRE REGION) RMIN~ -0.0500000 m RMAX= 2.0500000 m ZMIN= -0.0500000 m ZMAX- 2.0500000 m

.MATERIAL TYPES: 1

• ElEMENT TYPES : 6-7-8-13-15

1 27-ocr-89 11:34

Series 1 Problems ·SOLID and SILV Element Numbering


GIDMETRY PLOTTIN::;

4-4

ELEMENT Na.mERS

.DIMENSICNS: (ENTIRE REGICN) RMIN- -0.0500000 m RMAX= 1.0500001 m ZMIN= -0.0500000 m ZMAX= 1.0500001 m

• MATERIAL TYPES : 1-3

• ElEMENT TYPES: 8-13

1 2 4-ocr-89 14 : 26

P153521.PDF [Page: 27 of 124]

4.1 PROBLEM 1

Analysis Type: STATC- pressure load (hydrophone sensitivity, stresses)

Input File: D01.DAT

The F.E. model, a subset of the full model from Figure 4.0.1 , contains PARQ, FTOS and SILV elements only. The nodes used are shown in Figure 4.0.3(b).

A uniform pressure of 1 Pa, applied to the F nodes of the FTOS elements, is shown in input data as a pressure DOF fixity and shown graphically in Figure 4.1.1. The small "pins" pointing to the fixed nodes indicate nonzero fixities. Voltage on the internal surface of the PARQ elements is fixed at V = 0. GRAV and SPIN body forces (Card 13) are set to zero. The voltage node, to which the calculated hydrophone sensitivity is referred, is Node 5 and input pressure is 1 Pa (Card 14). It is important to notice that Node 5 is located on the silvered external surface of the piezoelectric shell.

Plots of selected output data are shown in Figures 4.1.2 through 4.1.4 . Note that the radial stress depicted in Figure 4.1.4 is cylindrically radial and not spherically radial.

The hydrophone receiving sensitivity printed on Figure 4.1.2 is valid at low frequencies, well below any resonances. Simple theory for thin spherical shells leads to the following expression for receiving sensitivity:

(4.1)

where r = 0.9555 metre is the mean radius of the spherical shell, d31 = -123.E-12 miV is the transverse piezoelectric strain constant,

and £33 T = 1.15E-8 F/m is the dielectric permittivity at constant stress.

With these "book" values for PZT-4 [see Ref. 4.1], we calculate a receiving sensitivity of -159.98 dB re 1 V/J.!Pa, in good agreement with the value obtained by MAVART of -160.00 dB.

4-5

P153521.PDF [Page: 28 of 124]

Figure 4.1.1: Problem 1- Fixities

PROBLEM U, PAR Piezcelectric Spherical Shell, l\xisym. Load (m=O)

Figure 4.1.2: Problem 1- Deformations

FIXITIES

• DIMENSICNS: (ENTIRE REGICN)

RMIN= -0.0500000 m RW\X- 1.0500001 m ZMIN- -0.0500000 m ZM&X- 1.0500001 m

24-0CI'-89 14:44

PROBLEM 11, PAR Piezcelectric Spherical Shell, Pressure Load - 1 Pa

1~----------------·--------------------------,~ .PROBLEM: 'STAT'

H/P SENSITIVITY -160. OOdB (volts/uPa)

• DISPlACEMENT SCAIE R

.... •. I 0.1183043E-09 m I

'

1 2 3 5 6

'

7

4-6

I J J I I

.DISPlACEMENT SCAIE Z

I O.ll83043E-09 m I RMIN= -0.0200000 m RMAX= 1.0200000 m

2~~= -~:8~88888 ~

10 28-Dec-SS 11:19

P153521.PDF [Page: 29 of 124]

Figure 4.1.3: Problem 1· Axial Stresses

PROBlEM i1, PAR Piezoelectric Spherical Shell, Pressure Load = 1 Pa

STRESS AND STRAIN PLOTTIN:; 14-------------------------------------------~ Z STRESS

Figure 4.1.4: Problem 1- Radial Stresses

.DIMENSICNS: (ENTIRE REGION) RMIN~ -0. 0500000 m RMAX= 1.0500001 m ZMIN- -0.0500000 m ZMAX= 1.0500001 m


.MAX VALUE:-.1435 AT NOI:E: 166

• VALUE CCNTOORS AT: -0.7300000

-1.930000 -3.130000 -4.330000 -5.530000

1 24-0CT-89 15:09

PROBlEM if1, PAR Piezoelectric Spherical Shell, Pressure Load = 1 Pa

STRESS AND STRAIN PLOTTIN:; 1q-------------------------------------------~R STRESS

4-7

.DIMENSICNS: (ENTIRE REGION)

RMIN= -0.0500000 m RMAX=" 1.0500001 m ZMIN= -0.0500000 m ZMAX= 1.0500001 m


.MAX VALUE:-.1299 AT NOI:E: 2

• VALUE CCNTOORS AT: -0.7199998

-1.940000 -3.160000 -4.380000 -5.600000

1 24-0CT-89 15:11

P153521.PDF [Page: 30 of 124]

4.2 PROBLEM 2

Analysis Type: EIGEN - resonance

Input File: D02.DAT

The F.E. model, a subset of the full model from Figure 4.0.1 , contains PARQ, and SILV elements only.

As the resonance condition requires, a V = 0 fixity is applied to all of the nodes on the inner and outer surfaces of the shell (i.e., a short circuit condition). Due to the presence of SILV elements, it is sufficient to apply V = 0 to Node 4, in order to assure a imiform potential of V = 0 on the entire external surface of the shell.

Lower and upper frequency estimates (Card 6) are set to 300Hz and 1000Hz respectively. The normalizing DOF number was chosen as 83 after a geometry check run to ensure that this is not a fixed DOF. The value of the normalizing DOF is set to l.OE-6. Values ofERVAL and ERVEC (Card 7) were each reduced to 1.0E-6 to ensure convergence to an accurate solution [See Ref. 4.2]

Plots of mode shape for the first four modes are shown in Figures 4.2.1 through 4.2.4. Mode 4, the "breathing" mode, is the only one of these modes that can be excited piezoelectrically wich the (spherically) radial poling condition of Problem 2.

Analytical theory [ 4.3] for a thin walled spherical shell gives the following expression for the resonance frequency (Mode 4):

fr = (2n2 r 2 p (811 + 812 )}-1/2 (4.2)

where r is the mean radius of the sphere, p is the density, and 811 and 812 are elastic compliance moduli of the material. The short circuit elastic moduli are used for resonance calculation, and open circuit moduli for antiresonance calculation, giving values of 94 7 Hz and 1162 Hz, respectively, using Clevite "book values" for the properties ofPZT-4 [4.1].

4-8

P153521.PDF [Page: 31 of 124]

Figure 4.2.1: Problem 2-1st Mode Shape

PRCBIEM i2, PAR Piezoelectric Spherical Shell

0 1 2 3 4 5 6 7 8

Figure 4.2.2: Problem 2 ·2nd Mode Shape

I

',

.NATi.RAL FREQIJEN:Y: 404.37 Hz

.PRELEM: 'EIGE'

.FCURIER M:IE NLMBER : 0

.NATi.RAL M:IE Nt.MBER: 1

.E6~4~~~~ICN:

.NUMBER OF ITEBATICNS 9

.DISPLACEM?NI' SCAlE R

I 0.7523929E-05m I .DISPLACEMENT SCAlE Z

I o. 7523929E-o5 m I RMIN= -o .0200000 m RMAX= 1.1500000 m

~~ ~:~§88888 ~

9 10 28-De::-88 14:34

PROBlEM lf2, PAR Piez~lectric Spherical Shell

1o=~~~~~~~TH==E_S~O~LI==D~B~O~UN~D~AR==Y~------------~ • NATURAL FRmUENCY

583.68 Hz

1 2 3 5 6 7

4-9

\ ,

.PROBlEM: 'EIGE'

.FOORIER MODE NUMBER 0

• NATURAL MODE NUMBER 2

.EIGEN VAL DEVIATION 0.3782679£-06

• NUMBER CF ITERATIONS 14

.DISPLACEMENT SCAlE R

I o.ioo 6718E 04 m I .DISPLACEMENT SCALE Z

! 0.1006718E-04 m ! I RMIN- -0.0200000 m

RMAX= 1.1500000 m ZMIN= -0.0200000 m ZMAX= 1.1500000 m

9 10 28-Dec-88 13:35

P153521.PDF [Page: 32 of 124]

Figure 4.2.3: Problem2- 3rdMode Shape

PROBlEM #2, PAR Piezoelectric Spherical Shell

DISPLl\CEMENT CF THE SOLID BOUNDARY 1~~~~~~~~~----------------~----------, .NATURAL FREQUENCY:

Figure 4.2.4: Problem 2-4th Mode Shape

PRCBIEM #2, PAR Piezoelectric Spherical Shell

0 1 2 3 4 5 6 7 8

4-10

I

J

838.12 Hz

.PROBLEM: 'EIGE'

• FCURIER MODE NUMBER: 0

• NATURAL MODE NUMBER: 3

.EIGEN VAL DEVIATION: 0. 8784 705E-O 6


• DISPLACEMENT SCAIE R

I o.1216131E o4 m I

• DISPLACEMENT SCAIE Z

I 0.1216131E-04 m I RMIN= -0.1500000 m RMhX- 1.1500000 m ZMIN= -0.1500000 m ZMAX- 1.1500000 m

9 10 28-Dec-88 13:18

'•

• NATURAL E'REOUEN:Y: 949.02 Hz

.PR:ELEM: 'EIGE'

.FOURIER IDE Nt:MBER: 0

• NATURAL IDE Nt:MBER: 4

.Efi~z~~~~ICN:

.NlJMSER OF ITERATIONS 11

.DISPLACEMENT SCALE R

I 0.3767792E-05 m l I .DISPLACEMENT SCAlE z

.I

: I I L ~ _f

9

I o.3767792E-o5 m I RMIN= -0.0200000 m ~ 1.1500000 m

~~ -~:~88888 ~

10 28-Dec-88 14 :36

P153521.PDF [Page: 33 of 124]

4.3 PROBLEM 3

Analysis Type: EIGEN- antiresonance

Input File: D03.DAT

The F.E. model, a subset of the full model from Figure 4.0.1 , contains PARQ, and SILV elements only.

For antiresonance analysis an open circuit condition is required; hence, a V = 0 fixity is applied to one surface, in this case, the internal surface of the shell, while the outer silvered surface is free. The only change that is required to Problem 2 input data is to remove the fixity from Node 4.

The modal frequency analysis data (Card 6) are the same as for Problem 2. Two poling conditions, set by parameter POLANG, for the piezoelectric shell have been used in both resonance and antiresonance analyses: (a) Uniform radial (spherical) poling and (b) The upper five of the ten P ARQ elements have their poling reversed. The first four mode shapes are essentially identical in all cases to those shown in Figures 4.2.1 to 4.2.4 for the resonance (a) condition. The frequencies obtained in the MA V ART analysis are given in the following table for all cases:

(a) FULL RADIAL POLING (b) HALF REVERSED POLING

Mode# Resonance Antiresonance Resonance Antiresonance

1 404.37 404.37 404.43 409.28

2 583.68 583.68 582.74 583.02

3 838.12 838.11 844.18 845.02

4 949.02 1146.8 948.54 974.37

Note that for condition (a), the antiresonance frequency differs from the resonance only for Mode 4, which is the mode of most interest acoustically. For condition (b), the antiresonance is always higher in frequency than the resonance, indicating that the other modes can now be excited piezoelectrically.

The resonance and antiresonance frequencies calculated using Eq. 4.2, 94 7 Hz and 1162Hz, respectively, are in reasonable agreement with the MAVART predictions of949 Hz and 1147Hz, above.

4-11

P153521.PDF [Page: 34 of 124]

4.4 PROBLEM 4

Analysis Type: CAPAC - frequency sweep

Input File: D04.DAT

The F .E. model, a subset of the full model from Figure 4.0.1, contains P ARQ and SILV elements only.

A V = 0 fixity is applied to all nodes on the internal surface of the shell. A sinusoidal driving voltage of1 V amplitude is specified as a fixity at Node 4 on the silvered surface.

The same polarity ofPARQ elements as in Problem 3, condition (b) is used, so that Mode 1 can be excited.

A frequency scan was conducted in the range of 400 Hz through 408 Hz, with frequency start points and increments specified according to Card 8 format. Ten element groups are used, one element per group. NSTAV, the 'number of staves' in the transducer, is 2, as a hemisphere is being analyzed; for a full sphere, NSTAV would be 1.

Plots of deformations, obtained at 404 Hz excitation frequency for two different phase angles, are shown in Figures 4.4.1 and 4.4.2. Capacitance is plotted versus frequency in Figure 4.4.3. It should be noted that the capacitance values passed to the output files are the absolute values. It is necessary to use absolute values in the resonance search mode (see Problem 9) since the logarithm of the value is used. However, any negative (i.e. inductance) values should be restored before output. This will be corrected in a future revision of MA V ART. Some of the capacitance values just above the resonance in Figure 4.4.3 may be negative.

4-12

P153521.PDF [Page: 35 of 124]

Figure 4.4.1: Problem 4- Deformations at 404Hz, Phase= 0

POOBLEM lt4, PAR Piezoelectric Spherical Shel:l, Reversed polarity on

r:ISPLACEMENT CF THE SOLID BOUNDARY half of shell

PHASE- OPI/4 lG---------------------------------------------~ ·-----. .

• DRIVEN FREQUEN:Y: 404.00 Hz

.PROBlEM: 'CAPA'

• CAPACIT.II.NCE: 0.1860525E-04 F

• D ISP LACEME'N'I SCALE R

I 0. 2 II 0064£-05 m I • DISPLACEME'N'I SCALE Z

I o.2II0064E o5 m I RMIN- -0.0200000 m RMAX- 1.1500000 m ZMIN- -0.0200000 m ZMAX- 1.1500000 m

10-NOV-89 10:56

Figure 4.4.2: Problem 4- Deformations at 404Hz, Phase= 1t

POOBLEM lt4, PAR Piezoelectric Spherical Shell, Reversed polarity on

DISPLACEMENT CF THE SOLID BOUND.I'RY lq_----------------------------------------~

PHASE= 4 PI/ 4

4-13

half of shell

• DRIVEN FREQUEN::::Y: 404.00 Hz

.PROBlEM: 'CAPA'

• CAPACITANCE: 0.1860525E-04 F

.DISPLACEME'N'I SCALE R

I o.277oo64E os m I • DISPLACEMENT SCALE Z

I 0.2110064£ 05 m l RMIN= -0.0200000 m RMAX= 1.1500000 m ZMIN= -0.0200000 m ZMAX= 1.1500000 m

10-NOV-89 10:58

P153521.PDF [Page: 36 of 124]

Figure 4.4.3: Problem 4- Capacitance versus Frequency

PROBLEM i4, PAR piezoele:::tric Spherical Shell, Capacitance

Xl0-5 ~ with reversed polarity on half of shell elements

2.00

1. 75

c A 1.50 p

A 1.25 c I T 1.00 A N

0.75 c E

0.50 F

0.25

1\

I \ I \ I \

/ \ / \

~ / \

r---- ~

~ 0.00

400 401 4 2 4 3 4 4 4 5 406 4 7 4 8 FREQIENO (Hz) 26-0CT-89 13:14

4-14

P153521.PDF [Page: 37 of 124]

4.5 PROBLEM 5

Analysis Type:

Input File:

Auxiliary Input File:

Auxiliary Output File:

CAPAC - transient

D05.DAT

D05FOUD.DAT

FOUDC.DAT

The F.E. model, a subset of the full model from Figure 4.0.1 , contains PARQ and SIL V elements only.

Voltage fixities, the driving point and the polarity of P ARQ elements are the same as in Problem 4 .

In order to obtain a time dependent solution for an arbitrary forcing function, it is necessary to decompose the function first into the corresponding Fourier series using the Fourier Decomposition Preprocessor invoked by setting variable PRO BID to "FOUD". At the end of the auxiliary input file D05FOUD.DAT, the Fourier decomposition data are given in a format specified by Cards 16 (a,b). In this case there are sixteen equally spaced values in one cycle of a 1-volt amplitude square wave (2 volts peak-to-peak) with a fundamental frequency of134.7 Hz. Note that this drive is not a true "transient", but a repetitive, nonsinusoidal waveform.

MAV ART execution with the file D05FOUD.DAT as input yields output files §MA V.DAT and FOUDC.DAT. The latter, containing Fourier coefficients, is used automatically in subsequent MA VART execution, with file D05.DAT serving as the main source of input data.

In file D05.DAT, the fundamental frequency, total number of harmonic frequencies and number of times at which displacements and stresses are to be computed are given according to Card 9 (a) . After inspection of data in file FOUDC.DAT, harmonic numbers 1, 3 and 5 were chosen and written into D05.DAT, as required by Card 9 (b). Then, six equally spaced time instances, ranging from 7.5 ms through 10 ms, were specified as per Card 9 (c).

Plots representing snapshots of the deformation at the selected times are shown in Figures 4.5.1 through 4.5.6 .

4-15

P153521.PDF [Page: 38 of 124]

Figure 4.5.1: Problem 5- Deformations at 0.0075 s

PROBlEM lt5, PAR Piezoelectric Spherical Shell

DISPLACEMENT OF THE SOLID BOI:NDARY 1~---------------------------------------.

;----

1

TIME = 7.5 ms

.PIDBLEM: 'CAPA' (TRAN)

• CAPACITANCE: 0. 6862930E-05 F

.DISP!J'CEMENT SCAlE R

I o.144481oE o5 m I • DISPI.JICEMENT SCAlE Z

I 0. I4448IOE-05 m I RMIN= -0.0200000 m ~x- 1.1500000 m ZMIN= -0.0200000 m Z~X= 1.1500000 m

1-NOV-89 10:40


POOBIEM if5, PAR Piezoelectric Spherical Shell

THE SOLID BOUNDARY

4-16

TIME= 8.0 ms

.PIDBIEM: 'CAPA' (TRAN)

• CAPACITANCE: 0. 6862930E-05 F

• DISPLACEMENI' SCALE R

I o.14448o6E os m I • DISPLACEMENI' SCALE Z

I 0.1444806E-05 m I RMIN= -0.0200000 m ~X= 1.1500000 m ZMIN= -0.0200000 m Z~X= 1.1500000 m

1 1-NOV-89 10:20

P153521.PDF [Page: 39 of 124]

Figure 4.5.3: Problem 5 ·Deformations at 0.0085 s

PROBlEM if5, PAR Piezoelectric Spherical Shell

DISPLACEMENT OF THE SOLID BOUNDARY 1~----------------------------------------, TIME - B .5 ms

.PROBlEM: 'CAPA'(TRAN)

• CAPACITANCE: 0 .6B62930E-05 F

• DISPLACEMENI' SCAlE R

I o .1444852E o5 m I .DISPLACEMENI' SCAlE Z

I a .1444852£ us m I RMIN= -0.0200000 m RMAX- 1.1500000 m ZMIN= -0.0200000 m ZMAX= 1.1500000 m

1 1-NOV-89 10:45


PROBLEM 115, PAR Piezoelectric Spherical Shell

ISPLACEMENT CF THE SOLID BOUNDARY 1~----------------------------------------~

TIME - 9.0 ms

.PIDBLEM: 'CAPA' (TRAN)

• Cl\Pl\CITANCE: 0. 6862930£-05 F

• D ISPLl\CEMENI' SCAlE R

I o.1444826E os m I • D ISPLl\CEMENI' SCALE Z

I u.r444826E us m I RMIN= -0.0200000 m ~X= 1.1500000 m ZMIN= -0.0200000 m ZM\X= 1.1500000 m

Hi 1-NOV-89 10:4 9

4-17

P153521.PDF [Page: 40 of 124]

Figure 4.5.5: Problem 5 ·Deformations at 0.0095 s

PROBLEM liS, PAR Piezcelectric Spherical Shell

DISPLACEMENT CF THE SOLID BOUNDARY 1~----------------------------------------, TIME - 9.S ms

.P:OOBLEM: 'CAPA' (TRAN)

• CAPACITANCE: 0. 6862930E-OS F

• D ISPLACEMENI' SCALE R

I o.1444869E oS m I • DIS PLACEMENT SCALE Z

I 0.1444869£ OS m I RM[Nz -0.0200000 m ~X= 1.1SOOOOO m ZMIN= -0.0200000 m ZMAX- 1.1SOOOOO rn

1 1-IDV-89 10:53

l?i~e4.5.6: Problem 5- Deformations at 0.0100 s

P:OOBLEM liS, PAR Piezcelectric Spherical Shell

DISPLACEMENT CF THE SOLID BOUNDARY 1~----------------------------------------, TIME = 10.0 ms

.P:OOBLEM: 'CAPA' (TRAN)

• CAPACITANCE: 0. 6862930E-D5 F

• DISPLACEMENT SCALE R

I o.14448o1E-o5 m I .DISPLACEMENT SCALE Z

I 0. 144480IE-05 m I RMIN- -0.0200000 rn R~X- 1.1500000 rn ZMIN- -0.0200000 rn Z~X= 1.1500000 m

1 1-NCN-89 10:56

4-18

P153521.PDF [Page: 41 of 124]

4.6 PROBLEM 6

Analysis Type: DRIVE - frequency sweep

Input File: D06.DAT

The complete F.E. model from Figure 4.0.2 is used.

Voltage fixities, the driving point and the polarity of P ARQ elements are the same as for Problem 4 .

The frequency sweep is conducted in only one range of 325 Hz through 340 Hz, with a rather coarse 5 Hz step. Acoustic radiation loading is small for this mode (Mode 1), and the sweep spans the narrow resonance region.

The output data indicate that, among the frequencies surveyed, 330 Hz is closest to resonance in sea water. Mass loading due to the water lowers the resonance frequency from the in air value of 404.4 Hz. Figures 4.6.1 and 4.6.2 show frequency dependence of admittance and transmitting voltage response, respectively. A cubic spline fit, available as an option in GRAF1, has been applied to these data.

Plots of selected data, obtained for 330 Hz excitation, are shown in Figures 4.6.3 through 4.6.7.

4-19

P153521.PDF [Page: 42 of 124]

Figure 4.6.1: Problem 6- Admittance Components vs. Frequency

PROBLEM lt6, PAR Piezoelectric Spherical Shell; reversed poling on half

c 0 N D u c T A N c E

s

XlO-J 2.2

2.0

1.8

1.6

1.4

1.2

1.0

0.8

-4!1- CCNDU::TANCE S --~-. SUSCEPTANCE S Xl0-3 4.5

/ "" "" . - I '\

.. -- \ / -.. -.. _ .......

1\, ·--

I ., . ~ "~ .,

I ·· . . " . -. "' I . · ... , \ ., .,_ .

-.. ______ ., ...... _____

4.0

3.5

3.0

2.5

2.0

1.5

1.0 3 2::>.0 327.5 330.0 33: • 5 33 .0 33 .5 340 • 0

FREQUENCY (Hz) 27-DCT-89 14:21

Figure 4.6.2: Problem 6- Transmitting Voltage Response

PROBIEM lt6, PAR Piezoelectric Spherical Shell, Reverse poling on half

T R

R E s p 0 N s E

d B

146

145

144

143

142

141

140

3

-e-- TR. RESPONSE dB -*-· AXIAL WR dB (Q: 24 .5~9 )

152

v---- ~ ./· ~-. -. ~

'·

~ '· .......

-/:/__

v ·, .. ~ -- ··. ~~ - '\

- ·/ ..

~ ·~ - .· '· - .• '· -; ·. . -v ·., .,

151

150

149

148

-., 147

'~ '\

. ' 146

2 ). 0 327.5 330.0 332.5 33 ).0 33 .5 340. 0 FREQUENCY {Hz) 10-NOV-89 10:52

4-20

s u s c E p T A N c E

s

A X I A L

T v R

d B

P153521.PDF [Page: 43 of 124]

Figure 4.6.3: Problem 6- Displacement of Solid Boundary

PRCBLEM lt6, PAR Piezoelectric Spherical Sooll

DISPLACEMENT <F THE SOLID BOUNDARY PHASE= 6PI/ 4 14-------------------------, FREQIEN:Y: 330.00 Hz

Figure 4.6.4:

• PROBLEM: 'DRIV'

.RESISTAN::E: 162.9958 ohms

• REACTANCE : -221.6927 ohms

• CONDUCTANCE: 0.2152747E-02 s

• SUSCEPTANCE: 0.2927978E-02 S

EFFICIENCY0.97505

PEAK TR 150.98 dB

• DIS PLACEMENT SCAlE R

I o. 1598390£-ob m I • DISPLACEMENT SCAlE Z

I o.1598390E-o6 m I RMIN= -0.0200000 m RMAX= 1.1500000 m ZMIN= -0.0200000 m ZMAX= 1.1500000 m

8-IDV-89 15:48

Problem 6- Directional Response at 330Hz

PIDBIEM i6, PAR Piezoelectric Spherical Shell

151.00 with reversed polarity on half of shell elements

4-21

DIROCTIONAL RESPCNSE (dB re 1 ~a/volt @ 1 m)

• DRIVE FRE~EN:Y: 330.000 Hz

.DIRECTIVITY INDEX: -6.90594 dB

.PEAK 'IRANSMIT RESPCNSE: 150.979 dB

• RECEIVING RESPONSE 0 lEG -144.80

27-QCT-89 11:43

P153521.PDF [Page: 44 of 124]

Figure 4.6.5: Problem 6- Near-Field Pressure Contours

PPDBIEM i6, PAR Piezcelectric Spherical Shell; reversed polil'XJ on half ENTIRE REGION RADIUS- 2.000 of shell elements ISOAMPLITODE CONTOURS

1<t--====;::::--------------------, FRE:!UENCY: 330. 00 Hz

- .... ' .......

' \ \ \ I "-, \

" \ I \

- ..... ..... ............,

\ \

POSITIVE - - - ZEPD

NEG1\.TIVE

ISOAMPLITO'DE COOTCXJRS • MAXIMUM AMPLITUDE:

77.66270 Pa • MINIMUM AMPLITUDE:

0.3761682 Pa .REFERENCE AMPLITUDE:

5.405019 Pa .AMPLITUDE CONTOURS:

20.30000 dB 17.40000 dB 14.50000 dB 11.60000 dB B. 700001 dB 5.800000 dB 2.900000 dB

O.OOOOOOOE+OO dB -2.900000 dB -5.800000 dB -8.700001 dB -11.60000 dB -14 .50000 dB -17.40000 dB -20 .30000 dB

1 27-0:::T-89 13:06

Figure 4.6.6: Problem 6 ·Detail of Near-Field Pressure

PPDBIEM i6, PAR Piezcelectric Spherical Shell; reversed polil'XJ on half Rl- 0.9000 R2- 1.5000 Zl- 0. 7000 Z2- 1.3000 of shell elements

1ISOAMPLITUDE CONTOURS 4---------,--,f---:1,...--1.,--/---,---,--, FRE;;!UENCY: 330. 0 0 Hz

I I

I

I

I 1

/ _ _ _ ~~MTIVE

I / / - .. - . - . - -- NEG1\.TIVE

I I /

/

4-22

ISOAMPLITUDE COOTCXJRS • MAXIMUM AMPLITUDE:

11.83618 Pa • MINIMUM AMPLITUDE:

0.3761682 Pa .REFERENCE AMPLITUDE:

2.110070 Pa .AMPLITUDE CONTOURS:

13.30000 dB 11.40000 dB 9.500000 dB 7. 600000 dB 5. 700000 dB 3. 800000 dB 1. 900000 dB

0. OOOOOOOE+OO dB -1.900000 dB -3.800000 dB -5.700000 dB -7.600000 dB -9.500000 dB -11.40000 dB -13.30000 dB

1 27-0:::T-89 14:14

P153521.PDF [Page: 45 of 124]

Figure 4.6. 7: Problem 6 ·Near-Field Pressure Gradient Contours .

PPDBIEM t6, PAR Piezcelectric Spherical Shell; reversed poling on half of shell elements.

~--==~=---------------------------------, PRESSURE GRADIENT

4-23

FRroUENCY: 340.00 Hz

• DIMENSICNS: (ENTIRE REGION) RMIN- 0. 0000000 m ~x~ 2.0000000 m ZMI:N~ 0. 0000000 m z~x~ 2.ooooooo m

• MATERIAL TYPES: 1

.MAX VALUE: 100.8 AT NODE: 172

• VALUE C<NTaJRS AT: 91.80000 73.40000 55.00000 36.60000 18.20000

27-0CT-89 14:27

P153521.PDF [Page: 46 of 124]

P153521.PDF [Page: 47 of 124]

5 SERIES 2: Pr PIEZOELECTRIC RING

A radial, cross sectional diagram of the piezoceramic ring, immersed in a sphere of sea water, is given in Figure 5.0.1. The inner radius of the ring is 0.833 m, the outer radius is 1.0 m, and its half height is 0.333 m. It is made -from 48 staves of tangentially poled Channel 5400. The outer radius of the sea water sphere is 2 m.

The mesh of the F.E. model is shown in Figure 5.0.2, together with the element numbers. As in the case of the Series 1 model, only the upper hemisphere needs to be modelled because of symmetry. Figures 5.0.3 (a,b) give details of the node numbering of the F.E; model.

The ring is modelled by two PTQ elements. The fluid model contains both quadrilateral and triangular elements. The fluid and PT solid are connected by FTOS elements. FTOF elements are grouped along the outer boundary of the fluid. Node 303, the only node ofE type, is positioned at the origin of the system of coordinates and coincides with F type Node 236, at the geometrical centre of the model. Coordinates of E nodes are immaterial in the analysis.

The horizontal symmetry plane requires proper boundary conditions for solid elements; crossplane (Z) displacements for Nodes 98, 99 and 132 are fixed to zero.

5-1

P153521.PDF [Page: 48 of 124]

Figure 5.0.1:

z

Figure 5.0.2:

Series 2 Problems- Cross-Sectional Diagram of Free-Flooded Ring in Sea Water

{m)

Series 2 Problems- Element Numbering

Series 2 Problem~ PT Piezoelectric Ring

GEOMETRY PLOTTIN:;

2

5-2

ELEM!!NT NUMBERS


RMIN- -0.1000000 m RMAX= 2.1000001 m ZMIN= -0.1000000 m ZMAX= 2.1000001 m


.ELEMENT TYPES: 4-6-7-14-15

10 23-NCN-89 13:39

P153521.PDF [Page: 49 of 124]

Figure 5.0.3 (a): Series 2 Problems- Outer Node Numbering

Series 2 Problems, PT Piezoelectric Ring

GEO~TRY PLOTTIN:; 14-----------------------------------------~ IDDE Nl:MBERS 3

• DIMENSICNS: (ENURE REGION)

RM[N= -0.1000000 m RMAX= 2.1000001 m ZMIN= -0.1000000 m ZMAX= 2.1000001 m

.MATERIAL TYPES: 1

• EIE~NT TYPES: 4-6-7-14-15

23-IDV-89 13:41

Figure 5.0.3 (b): Series 2 Problems- Inner Node Numbering Detail

Series 2 Problem!!. PT Piezoelectric Ring

GEO~TRY PLOTTIN:> 1 1

5-3

IDDE Nl:MBERS

• DIMENSICNS: RM[N= -0.0560000 m RMAX= 1.0440001 m ZMIN= -0.0560000 m ZMAX= 1.0440001 m


9 • EIE~NT TYPES: 4-6-7-14-15

23-IDV-89 13:45

P153521.PDF [Page: 50 of 124]

5.1 PROBLEM 7

Analysis Type:

Input Files:

EIGEN - resonance ( m = 2,0 )

D07A.DAT - m = 2 D07B.DAT - m = 0

The F.E. model is a subset of the full model from Figure 5.0.2 and contains the PTQ elements only.

As the resonance condition requires, a V = 0 fixity is applied to the only electrical DOF at Node 303 . The Fourier mode number m = 2 is written into the problem parameters data of Card 2 (!MODE).

Although the lower and upper bounds on the frequency estimate were set to 100 Hz and 500 Hz respectively, the predicted resonance frequency was found at 72.9 Hz, outside of the 100 to 500Hz band. Figure 5.1.1 shows the corresponding mode shape in the Z-R plane at¢ = 0. Tangentially, the displacement dependence is proportional to cos m¢ (=cos 2¢ in this case).

The same input data file was run with Fourier mode number m set to zero, giving a resonance frequency of 516.06 Hz. The mode shape is shown in Figure 5.1.2.

We see that the cross sectional area of the ring remains essentially constant under vibration when m = 2, indicating the pure bending nature of the mode. On the other hand, when m = 0, the modes are extensional and the area varies in vibration, as seen in Figure 5.1.2.

5-4

P153521.PDF [Page: 51 of 124]

Figure 5.1.1: Problere. 7 -1st Mode Shape ( m = 2 ) at 72.9 Hz

PRCBIEM i7, P'I Piezoelectric Ring (m=2)

DISPLACEMENT CF THE SOLID BOUNDARY l~-----------------------------------------,

.---~------,. -- •• I

I

• NATURAL FREQUENCY: 72.90 Hz

.PROBLEM: 'EIGE'

.FOURIER MODE NUMBER: 2

.NATURAL MODE NUMBER: 1

.EIGEN VAL DEVIATION: 0. 8029532£-07


.DISPLACEMENT SCAlE R

I 0.3110809£ OS m I .DISPLACEMENT SCAlE Z

I o.377o8o9E-o5 m I RMIN= 0.7000050 m RMA.X= 1. 1332 94 9 m ZMIN= -0.0499950 m ZMAX= 0. 3832950 m

23-IDV-89 11:00

Figure 5.1.2: Problem 7- 1st Mode Shape ( m = 0) at 516Hz

PROBLEM ll7, PT Piezoelectric Ring (m=O)

ISPLACEMENT CF THE SOLID BOUNDARY 14-----------.:....-------------, . NATURAL FREQUENCY:

----------·----· •t ' I

5-5

516.06 Hz

.PROBlEM: 'EIGE'

• FCXJRIER MODE NUMBER: 0


.EIGEN VAL DEVIATION: 0.3946554E-06


.DISPLACEMENT SCALE R

I 0.316II94E 0:> m I .DISPLACEMENT SCALE Z

I o.3761194E os m I RM!Nm 0.7000050 m RMA.X- 1.1332 94 9 m ZMIN= -0.0499950 m ZMAX= 0.3832950 m

1 23-NOV-89 11: 01

P153521.PDF [Page: 52 of 124]

5.2 PROBLEM 8

Analysis Type:

Input File:

CAPAC- frequency sweep (m = 0)

D08.DAT (frequency sweep)

The F.E. model is a subset ufthe full model from Figure 5.0.2 and contains the two PTQ elements only.

A sinusoidal driving voltage of1 V amplitude is specified as a fixity at Node 303.

Similar to Problem 4, three frequency ranges are specified for scanning. They cover the band from 508 Hz through 524 Hz. One element group composed of both PTQ elements is used (Card 11 ). It is not necessary to specify the coordinates of an integration line for PT elements; i.e., RR1, ZZ1, RR2, and ZZ2 are not used.

In the output data the maximum computed capacitance was found at a frequency of 516 Hz. As this is very close to the actual resonance of 516.06 Hz predicted in Problem 7, the capacitance is very high. Being an undamped driven problem, the capacitance and displacement become infinite at the resonance. A plot of the deformation at 516 Hz is shown in Figure 5.2.1 and a plot of absolute value of the capacitance versus frequency is given in Figure 5.2.2.

5-6

P153521.PDF [Page: 53 of 124]

Figure 5.2.1: Problem 8- Deformations at 516Hz

POOBLEM if8, PT Piezoelectric Ring (m=O)

ITISPLACEMENT CF THE SOLID BOUNDARY 1~--------------------------------------~ • DRIVEN FREQUEN::Y :

516.00 Hz

Figure 5.2.2:

t• ·-------- ·------ • PROBlEM: 'CAPA'

• CAPACITANCE: 0.9407460E-03 F

• DISPIACEMEN'I SCALE R

I o. 3384816E-o4 m I • DISPIACEMEN'I SCAlE Z

I o. 33B4BI6E o4 m I RMIN= 0. 7000050 m R!¥.X= 1.1332949 m ZMIN= -0.0499950 m ZMAX= 0. 3832950 m

23-NOV-89 11:19

Problem 8- Capacitance versus Frequency

PROBLEM #8, P'I Piezoelectric Ring (m=O)

c A p

A c I T A N c E

F

X10- 3

1.0

0. 8

0. c

o. 4

0. 2

0. 0

--e-- Capacitance (F)

I ~ ~ ~ ~ 508 .:-io 512 514 516 5 8

FREQl:ENCY (Hz)

5-7

520 522 52 4

23-NOV-89 11:16

P153521.PDF [Page: 54 of 124]

5.3 PROBLEM 9

Analysis Type:

Input Files:

DRIVE- resonance search

D09A.DAT - extra DOF dropped D09B.DAT - extra DOF kept


Two versions of the input file differ in the extra (w, fij displacement) degrees of freedom (DOF) either being dropped (IFOU=O), or retained (IFOU=1), in the analysis. The extra DOF's are carried automatically when the Fourier mode number m > 0, and is only required for m = 0 if data from different Fourier modes are to be combined.

The voltage fixity is the same as in Problem 8, theE node (#303) is fixed at 1 volt.

To locate the resonance, instead of frequency sweep as in Problem 2 , another strategy is applied for the case without the extra DOF's. Card 8 in file D09A.DAT specifies only one frequency range within which the search is to be performed. The negative value of parameter F2 in Card 8 activates the resonance search mode, which continues until the maximum of a specified parameter is located with peak tolerance specified by F2I in decibels, or until I2 iteration cycles are reached. If the maximum is at one or the other end of the specified search range, the search range will be extended until a lower value is seen or until the I2 iteration cycles are completed. Various parameters, including a DOF at a specified node, may be chosen as the search parameter by specifying a nonzero value for NFUN in Card 12. Allowable values for NFUN are discussed in Reference [5.1]. In a DRIVE analysis, radiated power (i. e., conductance, NFUN = 2) is the default parameter on which the resonance search is based; in CAPAC, it is capacitance (NFUN = 3).

The parameter NSEL in Card 12 specifies a node whose DOF values are written to the plot file for GRAF1 processing; one of these DOF's may be used in the resonance search [5.1]. IfNSEL is zero, the default node used is the first in the node list.

The output data indicate that 375Hz is the frequency closest to resonance in sea water. Plots of selected output data are shown in Figures 5.3.1 through 5.3.6. The cubic spline fit option has been selected in GRAF1 to plot the frequency response data in Figures 5.3.3 and 5.3.4.

The case with the extra DOF's carried (D09B.DAT) was run only at 375Hz excitation frequency. The results are identical to the previous run for that frequency.

5-8

P153521.PDF [Page: 55 of 124]

Figure 5.3.1: Problem 9 ·Deformation at375 Hz

PIDBLEM i9, PT Piezoelectric Ring (m=O)

SPLACEMENT CF 'IHE SOLID BOUNDARY 14-----------------------, FREQUENCY: 375.00 Hz

.POOBIEM: 'DRIV' .- -- .. -- ------- - .. --I .RESISTAOCE:

203.8722 • REACTI\NCE :

-64.77062

ohms

ohms

• CCNDOCTANCE: 0.4455336£-02 s

• SUSCEPTJI..NCE: 0.1415470E-02 s

EFFICiENCY 1.0000

PEAK m. 14 9. 94 dB

.DISPLACEMEN~ SCALE R

I 0. /003491E 0 I m I .DISPLACEMENT SCAlE Z

I a. 7oo3491E-o7 m I RMIN~ 0. 7000050 m RMI\X= 1.1332949 m ZMIN= -0.0499950 m ZMAX= 0.3832950 m 23-NOV-89 13:47

Figure 5.3.2: Problem 9 ·Tangential Stress Contours

POOBLEM ll9, PT Piezoelectric Ring (m=O)

S~SS AND Sm.AIN PLOTTIN:; 14-----------------------, TAN;. STBESS

I \

I \

I \ \ \

\ \ \

I \ \ \

\ \

\ \

I !

I \ I I

\ \ \ \

\ \.

\

\ \ \

\ \

\

\ \

5-9

FREQUENCY: 375.00 Hz

.DIMENSICNS: RMIN= 0. 7333350 m RMAX- 1.0999650 m ZMIN= -0.0166650 m ZMAX= 0.3499650 m

.MATERIAl TIPES: 4

.MAX ~LUE: 1351. AT NOCE: 131

• VALUE CCNTOURS AT:

B~~:~88 1226.800 1178.400 1130.000

29-NOV-89 14:16

P153521.PDF [Page: 56 of 124]

J?i~e 5.3.3: Problem 9 ·Admittance Components vs. :Frequency

POOBIEM i9, PT Piezoelectric Ring

X10

_ 3 ~ CONDUCTANCE S

(m=O)

--*·· SUSCEPTANCE S

4.5

c 4.0 0 N D 3.5 u c T A 3.0 N c E 2.5 s

2.0

1.5

: I r\ :

: I \ --: l \ :

- -~·f. \ , ·.

I -~ \ l

I •·. ~ ·-

1.0

/ ·,. ~ 'x ~ .. -----········~ 'K -· , ... --------· ...

~- .. _ ... _ 0.5

350 375 400 425 4 0 FREQlEOCY (Hz) 23-NOV-89 13:31

Problem 9 ·Transmitting Voltage Response

POOBIEM i9, PT Piezoelectric Ring (m=O)

~ TR. RESPONSE dB -*-- AXIAL TVR dB

0.007

0 .006S u s c 0 .005E p T

0 .004A N c

0 .003E s

0.002

0.001

0.000

-0.001

~ 134

T A R X

~0 132 I A

R L E s T p

148 v 0 130 R N s d E B

d B

146 128

425 4 0 FREQlEOCY (Hz) 23-NOV-8 9 13 :2 9

5-10

P153521.PDF [Page: 57 of 124]

~i~5.3.5: Problem 9 ·Directivity at375 Hz

PROBLE~ !9, PT Piezoelectric Ring {m=O)

145.0

140.0

135.0

130.0

125.0

DIREX::TIONAL RESPONSE (dB re 1 uPa/volt @ 1 m)

.DRIVE FRECPEOCY: 375.000 Hz


.PEAK 'IRANSMI'I RESPCNSE: 149.943 dB

.RECEIVING RESPCNSE 0 lEG: -166.87

23-NOV-89 13:33

~igure 5.3.6: Problem 9- Near-llield Pressure Contours at 375Hz

PROBLEM i9, PT Piezoelectric Ring (m=O) E'NIIRE REGION RADIUS= 2 .000

1 ISCAMPLITUDE CONTOURS--------------------------.

'

. -. \.

... '

' ,_

' .. .. ·,

·. ' ' \.

'.

'~ '\

-, l

\

' .. I

',

5-11

FREQUENCY: 375.00 Hz POSITIVE ZERO NEGATIVE

ISOAMI?LITUCE CCNTCXJRS .MAXIMJM AMPLITUDE:

95.63044 Pa • MINIMJM AMPLITUDE:

8. 430649 Pa • REFERENCE AM'LITUDE:

28.39413 Pa • AMPLITUDE CCNTOURS:

10.40000 dB 9.100000 dB 7. 800001 dB 6. 500000 dB 5.200000 dB 3.900000 dB 2. 600000 dB 1. 300000 dB

O.OOOOOOOE+OO dB -1.300000 dB -2.600000 dB -3.900000 dB -5.200000 dB -6.500000 dB -7.800001 dB -9 • 10 0 00 0 dB -10.40000 dB

23-NCV-89 13:35

P153521.PDF [Page: 58 of 124]

P153521.PDF [Page: 59 of 124]

6 SERIES 3: TRJLAMINAR BENDER DISK

A cross sectional diagram of the trilaminar disk is given in Figure 6.0.1. It is made of two outer layers ofPZT-4 ceramic 1 em thick with a 2-cm aluminum layer between. The piezoelectric material on both sides of the disk is poled in the same +Z direction. Although the structure is symmetrical geometrically with respect to the R axis, it cannot be considered electrically symmetrical due to the chosen piezoelectric polarity, and as a result the complete structure has to be analyzed.

SLIDER elements are inserted between the PZT-4 and the aluminum. Physically, they can be considered as representing the adhesive joints bonding the materials together, and are only necessary if the adhesive contributes a significant compliance to the structure. Otherwise, the model can be simplified by eliminating the SLIDERs and allowing the aluminum to share the adjoining A nodes of the PZT-4. The SLIDER elements can serve thr~e purposes:

• they can "soften" the junction of two structural components; • they provide for connection of two dissimilar nodes, such as H and A types,

where both must exist because of the elements that they serve; and • they can be used to apply damping in complex driven analyses (damping

applications are covered in Problem 18)

The mesh of the F.E. model is shown in Figure 6.0.2. There was no necessity of using triangular elements to model the structure; they were chosen in order to demonstrate their use.

Figures 6.0.3 and 6.0.4 (a,b) give details of element and node numbering, respectively. The axial scale has been expanded to allow the numbers to be read more easily.

Figure 6.0.1: Series 3 Problems- Cross-Sectional Diagram

Z (em)

PZT-4

0 ~~~~~77~~~~~~~77~~~~--~

-2 R {em)

0

6-1

P153521.PDF [Page: 60 of 124]

Figure 6.0.2: Series 3 Problems- F.E. Model

Series 3 Problems, PAR Trilaminar Bender Disk

GECMETRY PLOTTIN:; 1.~~--------------------------------------. ELEMENT PlOTTING


RMIN= -0.0034000 m RMAX= 0.2034000 m ZMIN= -0.1034000 m ZMAX- 0.1034000 m

.MATERIAL TYPES: 2-3-9

.ELEMENT TYPES: 1-2-9

1 2 3 4 5 6 7 B 9 IO 2B-NOV-B9 OB:59

Figure 6.0.3: Series 3 Problems- Element Numbering

Series 3 Problem•, PAR Trilaminar Bender Disk

GEOMETRY PLOTTIN:; 1*---------------------------------------~ ELEMENT NUMBERS

.DIMENSICNS: (ENURE REGION)

RMIN= -0.0034000 m RMAX= 0.2034000 m ZMIN= -0.0413600 m ZMAX= 0.0413600 m


• ELE!v.ENT TYPES: 1-2-9

1 2B-IDV-B9 09:02

6-2

P153521.PDF [Page: 61 of 124]

Figure 6.0.4(a): Series 3 Problems- Node Numbering, Aluminum Layer

Series 3 Problems, PAR Trilaminar Bender Disk

NOIE NU.JBERS

.DIMENSICNS: (EN'IIRE REGION)


.MATERIAL TYPES: 2

• EIEM;:NT TYPES: 1-2-9

28-NOV-89 09:07

Figure 6.0.4 (b): Series 3 Problems· Node Numbering, Piezoelectric Layers

Series 3 Problems, Pl\R Trilaminar Bender Disk

GEOMETRY PLOTTII'G l.q-----------------------------~~----~

~~~~~~~~~~~~n-n~~~~~~~~

6-3

NOIE Nl:MBERS

• D IMENSICNS : (ENTIRE REGION)


.MATERIAL TYPES: 3

• EIEM;:NT TYPES: 1-2-9

28-NOV-89 11:08

P153521.PDF [Page: 62 of 124]

6.1 PROBLEM 10

Analysis Type: STATC- electric stresses

Input File: D10.DAT


A 1-volt fixity is applied to all A nodes on the top and the bottom surfaces of the disk. A zero volt fixity is applied to all A nodes attached to the SLIDER elements. The SLIDER stiffness is set at 1.0E+15 Pain all three moduli. Node 77 is restrained in the Z direction.

A plot of the deformation of the disk is shown in Figure 6.1.1. In Figure 6.1.2, an expanded axial scale is used to show radial strain contours in the materials; the solid contour line is the line of zero strain, long dashed lines are positive strain contours, and the short dashed lines are negative contours. The waviness in the contours is an artifice caused by the use of the triangular elements.

6-4

P153521.PDF [Page: 63 of 124]

Figure 6.1.1: Problem 10 ·Electrical Deformation with a Hard SLIDER

PAR Trilaminar Bender Disk, Electrically stressed, SLIDER 1.0E15 Pa

1 CISPLACEMENT CF THE SOLID BOUNDARY

. - .. _ ... --

-"---- .. ----_..,._ --""' -- ~'-

~--- ..... -

~---... -

I

'

.PBOBIEM: 'SIAT'

.DISPLACEMEN! SCALE R

I o.s3o6226E-o7 m I .DISPLACEMEN! SCAlE Z

I 0.5306226£-0 I m I RMIN= -0.0019200 m RMAX= 0.2060800 m ZMIN= -0.1019200 m ZMAX= 0.1060800 m

1 :2 3 5 b 7 B 9 1 23-NCW-89 15:09

Figure 6.1.2: Problen110 · Radial Strains

PROBLEM 10 - PAR Trilaminar Bender Disk, Electrically stressed

STRESS AID SmAIN PLOTriN:; 'STATC' 1~----------------------------------------~ R STRAIN

b

--·- ----.-..... ... _ .. ......... ·- , .......... -- ~-....... -- , .........

--- .. - .. - __ .. _, __

__ , --- -~ ---~·~-~,~---~----~--,~-----~~---~-~-~---~-~

• DIMENSICNS: (ENTIRE REGION)

RMIN- -0.0034000 m RMAX= 0.2034000 m ZMIN= -0.0440000 m ZMAX= 0.0440000 m


.MAX ~LUE:0.1340E-07 AT NODE: 171

• VALUE CCNTCURS AT: 0.1200000E-07 o. 8oooogoE-08 0.40000 OE-08 O.OOOOOOOE+OO

-0. 4000000E-08 -0.8000000E-08 -0.1200000E-07

2 4 5 '6 '7 '8 9 10 27-NCW-89 13:07

6-5

P153521.PDF [Page: 64 of 124]

6.2 PROBLEM 11

Analysis Type: STATC- thermal stress

Input File: D11.DA'r

The complete model from Figure 6.0.2 is used.

A zero volt fixity is applied to all A nodes on the top and bottom surfaces of the piezoelectric layers. The SLIDER stiffness is set at 1.0E+15 and Node 77 is restrained in the Z direction, as in Problem 10.

The element data contains extension Cards 5 (b) specifying element temperature differences from zero strain: +20° C for elements of the top piezoelectric layer, -20° C for the bottom piezoelectric layer. The temperature of the aluminum disk is left unchanged from its zero strain state.

Figures 6.2.1 and 6.2.2 show output data for the thermally stressed disk that corresponds to Figures 6.1 J. and 6.1.2 for the electrically stressed disk. We note that the form of the deformation is essentially the same in both cases but the strain is very different: there are three lines of zero strain (solid lines) with thermal stresses but only o:ae with electrical. Adjacent parts of the materials undergo opposite strains. The hydrophone sensitivity printed OI). Figure 6.2.1 is meaningless in this situation.

6-6

P153521.PDF [Page: 65 of 124]

Figure 6.2.1: Problem 11- Thermal Deformation with a Hard SLIDER

POOBLEM 11 - PAR Trilaminar Bender Disk, thermal stresses

DISPLl\CEMENT CF THE SOLID BOUNDARY 1~----------------------------------------.

--~- ....... _ - .. - -- ---- ... -... - .. ----

-- ... ---------'--, ... - .......

, ... ,'

.PROBlEM: 'STAT'

H/P SENSITIVITY: -542. 02dB (volts/uPa)

.DISPlACEMENT SCALE R

I o.33BI992E o3 m I • DISPlACEMENT SCALE Z

I o.3381992E o3 m I RMIN= -0.0019200 m RMAX= 0.2060800 m

~~~= -s:l8t6~88 ~

D 1 "2 3 4 7 '8 9 10 27-NC:W-89 11:46

Figure 6.2.2: Problem 11- Radial Strains

ProBLEM 11 - Pl\R TrL"'minar Bender Disk, the:onal stresses

S'IRESS AID S'JRAIN PLOTTIN:; lq-------------------------------------------,R STRAIN

-- ... -- - ..... -... -- • --- ... --- .... -- .. - -- • - # ... ... _, - ...

::--- ~- ~.:'- -- _........ -- _,:--. -- _::--- __ / . ........... _ ~- ........... _ ......... _ '""""-

-"'- - --'-' - -~ - --...__. - ---- ..-- -~.' ... ------. - --- ... -... -·- . -... -· ~ . :-::-. -~

- ......... - ',:: -, :.: ... ••• ·_. # - •• -.. - , -- - -...... - ~,. -- - - ..........

... .. - , ...... , • - .. # ... , - - .. -- - , ...... , .. - ....

,-- ---· - -- -- / f------------------- --


RMIN= -0.0034000 m RMAX= 0.2034000 m ZMIN- -0.0440000 m ZMAX= 0.0440000 m

.MATERIAL 'IYPES: 2-3-9

.MAX ~LUE:0.3957E-04 AT NODE: 189

• VALUE CCN'IOURS AT: o. 3000000E-04 o.zooogoOE-04 0.1000 OOE-04 0. OOOOOOOE+OO

-O.lOOOOOOE-04 -0.2000000E-04 -0. 3000000E-04

1 2 3 4 5 6 7 !l 9 1 J 27-NC:W-89 13:00

6-7

P153521.PDF [Page: 66 of 124]

6.3 PROBLEM12

Analysis Type:

Input Files:

CAPAC - without slider softening (a) - with full slider softening (b) -with partial slider softening (c)

D12A.DAT -hard slider D12B.DAT - soft slider D12C.DAT - softened slider D12D.DAT -hard slider resonance


A sinusoidal 100-Hz drive with an amplitude of1 volt is applied to all A nodes on the top and bottom surfaces of the disk. All A nodes attached to the SLIDER elements share a 0 volt fixity. No structural nodes are restrained in the Z direction.

Two element groups, one for each piezoelectric layer, are used. Three SLIDER conditions are analysed. The material properties of the SLIDER elements in Case (a) make the aluminium-piezoelectric connection stiff (en = c22 = c33 = 1.0E+15), as in Problem 10. In Case (b) the connection remains stiffin the . Z direction while allowing free sliding in the R direction Cc22 = C33 = 0.). Case (c) is an intermediate condition Cc22 = c33 = 1.0E+11), providing some shear compliance in the R-~ plan~.

Plots of deformations at 100Hz are shown in Figures 6.3.1 to 6.3.3 for the three cases (a), (b), and (c), respectively. The clamping effect of the aluminum plate on the piezoelectric layers is reflected in the reduced capacitance as the sliders become stiffer.

An 'EIGE' run was carried out for the hard slider, Case (a), predicting a resonance of1647.8 Hz. Deformations are shown in Figure 5.3.4. Since no node is fixed, the first mode will always be at zero frequency, hence the data indicate a NATURAL MODE NUMBER of '2'.

6-8

P153521.PDF [Page: 67 of 124]

Figure 6.3.1: Problem.12 ·Deformations without Slider Softening

ProBLEM 12 - PAR Trilaminar Bender Disk, hard SLIDER, 1. OE+15 Pa

L:ISPLACEMENT CF THE SOLID BOUNDARY 1~----------------------------------------,

~-. ~- ----- .. '. ---~--- .... -----.... _"" ....... _.,. -.. -

- , . ... ... - f

-~

• DRIVEN FREQOEN:Y: 100.00 Hz

.ProBlEM: 'C.APA'

• Cl\PACITJl.NCE: 0.2749600E-06 F

• DISPlACEMENT SCALE R

I o.267122dE-o7 m I • DISPlACEMENT SCALE Z

I 0.261122UE 0 I m I RMIN- -0.0040000 m RMAX= 0.2274000 m ZMIN= -0.1040000 m ZMAX= 0.1274000 m

1-DB:-89 13:01

Figure 6.3.2: Problem 12 ·Deformations with Slider Softening

PROBlEM 12 - PAR Trilaminar Bender Disk, soft SLIDER

DISPLACEMENT CF THE SOLID BOUNDARY 1~----------------------------------------~

, r = = :: __ :--- c::f::" .. ·- - -- -- -- -- _____ j

• DRIVEN EREQUEN::Y : 100 .00 Hz

.ProBLEM: 'CAPA'

.CAPACITANCE: 0.2892971E-06 F

.DISPlACEMENT SCAlE R

I o. 9255468E os m I .DISPlACEMENT SCALE Z

I 0. 9255468£ OS m I RMIN= -0.0040000 m RMAX= 0.2274000 m ZMIN= -0.1040000 m ZMAX= 0.1274000 m

1 2 3 4 5 6 7 B 9 lb 1-DB:-89 11:53

6-9

P153521.PDF [Page: 68 of 124]

Figure 6.3.3: Problem 12 ·Deformations with Partial Slider Softening

PROBLEM 12 - PAR Trilaminar Bender Disk, softened SLIDER, 1. OE+ll Pa

1~ISPLACEMENT CF THE SOLID BOUNDARY

;···------- ------~-------------- .. -- ... ,-- --.

- .... __

• DRIVEN EREQUEJ:.r:Y : 100 .00 Hz

.PROBIEM: 'CAPA'

• CAPACITP.NCE: 0.27B6057E-06 F

.DISPIACEMEN'I SCALE R

f o.2492o63E-o7 m I .DISPIACEMEN'I SCALE Z

I 0.2492063£-0 I m I RMIN- -0.0040000 m RMAX= 0.2274000 m ZMIN= -0.1040000 m ZMAX- 0.1274000 m

1 2 3 11 5 6 7 8 9 10 1-DEC-89 13: 03

Fi~e6.3.4: Problem 12 ·First Mode Shape

ProBLEM 12 - PAR Trilaminar Bender Disk, hard SLIDER, Resonance

DISPLACEMENT OF THE SOLID BOUNDARY PHASE= 4PI/4 1¥-----------------------------------------,

.. -- ... __ '9-t __ ,. __ _

.... . - .... ~

• NATURAL FRECUENCY: 1647.81 Hz

.PROBLEM: 'EIGE'

• FCURIER MODE NUMBER: 0


• EIGEN VAL DEVIATION: 0.1611546E-Q6

.NUMBER OF ITERATIONS 7


I 0.1~5:5!!23E-O~ Ill I • DISPlACEMENT SCAIE Z

I o. rs65923E=o3 m I RMIN= -0.0040000 m RMAX= 0.2066000 m ZMIN= -0.1040000 m ZMAX= 0.1066000 m

D ·.1. ·;t. :::s ·q ·:;, o ., <! 9 lU 29-NOV-89 13:30

6-10

P153521.PDF [Page: 69 of 124]

7 SERIES 4: SHELL THJNDISKJTUBE

This series of problems demonstrates the application of SHELL (Type 10) elements in modelling a thin aluminum disk and a thin walled aluminum tube. A cross sectional diagram of the disk is given in Figure 7 .0.1 and of the tube in Figure 7.0.2. The radius of the disk is 10 em, and the tube is 10 em long and 5 em in radius. In both cases the shell thickness is 2 mm, and is manifest in the material properties as discussed in the Theoretical Manual [1.2]. The properties specified for aluminum in program MATER (See [1.6], [1.7]) are Young's modulus= 7.16E10 Pa, density= 2700 kgfm3, Poisson's ratio= 0.344 and coefficient of thermal expansion = 23.4E-6 /°C.

The F.E. model of the disk with node and element numbering, is shown in Figures 7.0.3 and 7.0.4, respectively. The corresponding F.E. model for the tube is given in Figures 7.0.5 and 7.0.6. GRAF1 has no provision for indicating the thickness of a shell.

7-1

P153521.PDF [Page: 70 of 124]

Figure 7.0.1: Series 4 Problems- Disk Radial Cross-Section

~z I Aluminum Disk r 2 mm \ I ~ \ R (em) ~ssssssssssssssss ssssssssssssssssssss1ssssssssssssssssssssssssssssssssssssss;--- -~

0 5 10 '

Figure 7 .0.2: Series 4 Problems- Tube Radial Cross-Section

,tz (em) Sf------------...

I

Aluminum Tube

1 R (em) 0 , _____________ ~-------------1 ->

0 5 10

~ < 2mm

7-2

P153521.PDF [Page: 71 of 124]

Figure 7 .0.3: Series 4 Problems- Disk Node Numbering

Series 4, SHELL, thin disk

GEOfJETRY PLOTTIN:; 1~--------------------------------------~ NOCE NUMBERS

Figure 7.0.4:

.DIMENSI<NS: (ENURE REGION)


.MATERIAL TYPES: 10

.EIEfiENT TYPES: 10

123456 8 9 IOIII213141516111819202

1 2 3 5 6 7 8 9 1 J 6-I:EC-89 15: 4 7

Series 4 Problems· Disk Element Numbering

Series 4, SHELL, thin disk

GEOfJETRY PLOTTIN:; 1~--------------------------------------, ELEMENT NUMBERS

~ --~1 ___ 2~~3~--4~~5~~6~~7--~8--~9--~1~0~

.

• DIMENSICNS: (EN'IIRE REGION)


.MATERIAL TYPES: 10

• EIEfJENT TYPES : 10

1 2 3 ~ 5 t; 7 8 9 1 J 6-I:EC-89 15: 4 6

7-3

P153521.PDF [Page: 72 of 124]

Figure 7 .0.5: Series 4 Problems· Tube Node Numbering

Series 4, SHELL, thin-walled tube

GEOMETRY PLOTTIN:; 1~----------------------------------------, NJDE NOMBERS

DIMENSICNS:

Figure 7.0.6:

1 2 4 5

1

6 7

8 9 0 1 2

3 4 5 6 7 8

9 0 1

(ENTIRE REX;ION) RMIN= -0.0050000 m RMAX= 0.0830000 m ZMIN- -0.0550000 m ZMAX- 0.0550000 m

.MATERIAL TYPES: 10

.ELEMENT TYPES: 10

10 7-DEC-89 15 :27

Series 4 Problems- Tube Element Numbering

Series 4, SHELL, thin-walled tube

GEOMETRY PLOTTIN:; 1~----------------------------------------. ELEI£NT NUMBERS

1

3

4

6

7

8

9

0

4 5

7-4

.DIMENSICNS: (ENTIRE REX;ION)



.ELEMENT TYPES: 10

10 7-DEC-89 15:26

P153521.PDF [Page: 73 of 124]

7.1 PROBLEM13

Analysis Type:

Input Files:

EIGEN- flexural (disk) ·and hoop (tube) modes

D13A.DAT -disk D13B.DAT - tube, ends restrained (infinite tube) D13C.DAT -tube, centre restrained (finite tube) D13D.DAT -tube, hoop mode

All entries into the material property table of the aluminum SHELL elements, according to the data requirements [1.1a], [1.7], depend on the shell thickness.

For the disk, the search for resonances is confined to the frequency band between 350 Hz and 450 Hz. As in Problem 2, ERV AL and ERVEC (Card 7) were each reduced to 1.0E--6 to ensure accurate convergence [ 4.2]. The resonance frequency was located at 458.35 Hz. A plot of the corresponding mode shape is shown in Figure 7 .1.1, where the vibration is displayed at five phase angles, 7t/4 radians apart. (The 27t/4 vibration curve lies along the disk, so is not seen.) Further testir.g of flexural and extensional modes in the thin disk are reported in [4.2]

For the tube, since hoop type modes are expected to fall within a much higher frequency range, a 15,000 Hz through 16,500 Hz band is used. Also, a number of modes are expected in close proximity, so the tolerance for convergence of eigenvalues (ERVAL on Card 7) is lowered to 1.0E--8 and the tolerance for convergence of eigenvectors, ERVEC is l.OE--6. Similarly, in anticipation of an increase in the number of necessary iteration cycles, the "stop iteration number", ITSP is increased to 50.

When the ends of the tube are fixed in Z and 13 (rotations), two modes were found in the search range. The first mode is antisymmetrical at 15,839 Hz and the second is symmetrical at 16,403 Hz. A plot of the first mode shape is shown in Figure 7 .1.2. Extending the search range, we find a pure hoop mode at 17,456 Hz as Mode 4. The -:lnd fixities simulate the constraints in an infinite tube, and calculation of this resonance from simple elastic theory gives 17,457 Hz.

When there are no fixities, three modes occur in the search range, two symmetrical modes at 15,838 and 16,396 Hz, and one antisymmetrical mode at 16,314 Hz. The first mode shape is shown in Figure 7.1.3.

In the free tube, no real mode displays pure hoop vibration, but when rotations are suppressed by artificially applying 13 fixities of zero to all nodes, a hoop mode is found at 15,924 Hz. The mode shape, shown in Figure 7.1.4, displays both radial and axial vibration. If, in addition to the 13 fixities, the axial motion is suppressed by Z fixities applied to the ends of the tube, the pure hoop mode at 17,456 is again observed, now as Mode 1.

7-5

P153521.PDF [Page: 74 of 124]

lfi~~ 7.1.1: Problem 13 ·Disk, First Flexural Mode Shape

PIDBLEM U3A, SHELL, thin disk

DISPLACEMENT OF TilE SOLID BOUNDARY 1¥-----------------------------------------, • NATURAL FREQUENCY:

458.35 Hz

------ .. ----------- ---~:-~ .. --.:.:.:. :..,,..._,_

_______ .,._______ ,_ .... ----

___ ,...., --~··;;-:------ -

'l '2 3 '4 '!:) '6 7

----------:""~----~-.. ____ .. , ...... --

........ _-_ ...... : ..

'8 9

• PFDBIEM: 'EIGE'

• FOORIER MOOE NUMBER: 0

• NATURAL MOOE NUMBER: 2

• EIGEN VAL DEVIATION: 0. 7052880E-09

• NUMBER OF ITERATIONS 2


:3. 7!'51194 m I • DISPlACEMENT SCAIE Z

3. 16!194 m I RMIN- -0.0020000 m ~x- 0.1020000 m ZMIN= -0.0520000 m ZMAX= 0.0520000 m

0 6-DEC-89 15:42

Figure 7 .1.2: Problem 13- Tube, First Mode Shape, Fixed Ends

PIDBLEM 13B, SHELL, thin-walle:i tube, ends fixed in Z and Jj

DISPLACEMENT OF TilE SOLID BOUNDARY 1~----------------------------------------, 1 • NATURAL FREQUENCY:

I

1

' I

' I

'

I I

I I

I

I I

I

.,

7-6

• 15838 .53 Hz

• PIDBIEM: 'EIGE'

• FOORIER MOOE NUMBER: 0

• NATURAL MOOE NUMBER: 1

.EIGEN VAL DEVIATION: 0. 7501590E-09


.DISPlACEMENT SCAIE R

m I • DISPlACEMENT SCAlE Z

3. 16!194 m I RMIN- -0.0150000 m ~X= 0.0890000 m ZMIN- -0.0520000 m ZMAX- 0.0520000 m

'8 9 . 0 6-DEC-89 16:10

P153521.PDF [Page: 75 of 124]

Iri~ 7.1.3: Problem 13- Tube, First Mode Shape, Free Ends

PRJBLEM 13C, SHELL, thin-walled tube, centre node fixed in Z

ISPIJ\CEMENT OF THE SOLID BOUNDARY

1¥---------------------------------------~

4 5

I

' ' I

: , I ,

I

'

• NATURAL FRECUENCY: 15838.00 Hz

.PROBlEM: 'EIGE'

• FOORIER MOOE NUMIER: 0

• NATURAL MOOE NUMIER: 1

• EIGEN VAL DEVIATION: 0.1208875E-Q9


.DISPlACEMENT SCAlE R

15:3. "78432 m I • DISPlACEMENT SCAlE Z

63.831/6 m I RMIN= -o. 0150000 m RMAX= 0.0890000 m ZMIN= -0.0533000 m ZMI\X= 0. 0533000 m

6-DEC-89 16:07

Figure 7.1.4: Problem 13- Tube, Hoop Mode Shape, Fixed Rotations

PRJBIEM 13D, SHELL, t:J.in-walled tube, all rotations fixed

1 ISPIJ\CEMENT OF THE SOLID BOUNDARY

-----------------------. . NATURAL FREQUENCY: 15924.41 Hz

.PROBlEM: 1 EIGE'

• FOORIER MODE NUMBER: 0


.EIGEN VAL DEVIATION: 0.5831946E-07

.NUMBER OF ITERATIONS 8

.DISPlACEMENT SCALE R

3. 84 13:3"72 m I .DISPlACEMENT SCALE Z

3.848312 m I RMIN= -0.0150000 m RMI\X= 0.0890000 m ZMIN= -0. 052 0000 m ZMAX= 0.0520000 m

0 '1 "L ·..; ·.:, ·a 'I ·a ·g ro 7-DEC-89 15:15

7-7

P153521.PDF [Page: 76 of 124]

7.2 PROBLEM14

Analysis Type:

Input Files:

STATC

P14A.DAT P14B.DAT

- transverse temperature differential (a), - and nodal load (b)

Only the disk structure is used in these problems.

In Case (a) a 20° C transverse temperature difference is assumed across the shell thickness, with the higher temperature on the top surface of the disk. The data format requires that, for the right side of the element (as seen from the first node on the element) being hot, the temperature difference is positive. In this case the left side is hotter, so a negative value of DELDT is used on the element extension data, Card 5b.

A plot of the thermally deformed disk is shown in Figure 7.2.1.

In Case (b) the disk is clamped around its periphery by specifying zero u,v,w, and 13 fixities at Node 21 . A 10-N force is applied as a load fixity at Node 1 in the +Z direction.

A plot of the deformed disk is shown in Figure 7 .2.2. In both Figures 7 .2.1 and 7 .2.2, the printed hydroph0ne sensitivity is meaningless.

7-8

P153521.PDF [Page: 77 of 124]

~i~ 7.2.1: Problem 14- Disk Thermal Deformations

POOBLEM U4A, SHELL, thin disk, thermal stresses

1! ISPLACEMENT CF THE SOLID BOUNDARY

-· ----------------. J ------------------------~~~---------

.POOBIEM: 'S'IAT'

H/P SENSITIVITY: -194.27dB (vclts/uPa)


I o. 958I64UE oJ m I • DISPIACEMEN'I SCALE Z

l o.958164CE o3 m I RMIN= -0.0020000 m RMAX= 0.1020000 m ZMIN- -0.0520000 m ZMAX= 0.0520000 m

1 :2 3 ll 5 6 7 !l 9 1tl 8-DEI::-89 14:4 4

~i~e 7.2.2: Problem.14- Disk ~orced Deformations

POOBLEM if14E, SHELL thin disk, fixed edge, centre load

1 DISPLACEMENT CF THE SOLID BOUNDARY

'

.POOBIEM: 'S'IAT'

H/P SENSITIVITY: -208. 62dB (vclts/uPa)


I o.n9D26YED3m I • DISPIACEMEN'I SCALE Z

I o.l39o269E-o3 m I RMIN= -0.0020000 m RMAX= 0.1020000 m ZMIN= -8.0520000 m ZMA.X- • 0520000 m

1 :2 3 5 6 7 B 9 lJ 6-DEC-89 16:56

7-9

P153521.PDF [Page: 78 of 124]

P153521.PDF [Page: 79 of 124]

8 SERIES 5: ANALYTICAL FLUID PROBLEM

A cross sectional diagram of the structure modelled in this series is given in Figure 8.0.1. It is a 10 em diameter piston that vibrates axially in an infinite hard baffle. The half space over the piston and baffle is sea water. The outer radius of the sea water sphere used in the analysis is 0.165 m.

The mesh of the F.E. model with node numbers is shown in Figure 8.0.2, while Figure 8.0.3 gives the element numbering. The piston is modelled with six SHELL (Type 10) elements.

The model shown in Figures 8.0.2 and 8.0.3 has been changed from that in the original Examples Manual [1.1c]. The main change has been to reduce the number and size of the fluid elements to increase accuracy and to reduce the run time. Testing that resulted in these changes is reported in Ref. [4.2]

Figure 8.0.1: Series 5 Problem- Diagram of Piston in Infinite Bame

8-1

P153521.PDF [Page: 80 of 124]

Figure 8.0.2: Series 5 Problems- Node Numbering

PBOBLEM US - Piston in Haro Baffle

l GEOMETRY PLOTTIN:> KOCE NrMBERS

• DIMENSICNS: (EN'IIRE REGION)



.ELE~NT TYPES: 4-6-7-10-15

11-I::EC-89 13:04

Fi~e8.0.3: Series 5 Problems -Element Numbering

PROBLEM illS - Pist6r in Hard Baffle, thin FLUID elerrents, small sphere

GEOMETRY PLOTTING 1¥-----------------------------------------. ELEMENT NUMBERS



• MATERIAL TYPES: 1-10

• ELEMENT TYPES : 4-6-7-10-15

IO 19-DEC-89 13:32

8-2

P153521.PDF [Page: 81 of 124]

8.1 PROBLEM 15

Analysis Type: DRIVE- piston in infinite hard baffle

Input File: D15.DAT

The piston is driven at a vibration amplitude of 1 mm by applying fixities in the Z direction on all H nodes of the SHELL elements. Only one excitation frequency ofl2,000 Hz is used in the analysis.

Test results reported in [ 4.2] show that this model is acoustically accurate at frequencies up to 18,000 Hz. It was shown that FLUID element dimensions should not exceed 0.4 wavelength at the highest frequency where accurate results are sought.

A plot of far-field directivity at 12,000 Hz is shown in Figure 8.1.1. The peak pressure of269.28 dB re1~a @1m is the same as predicted by analytical theory [ 4.2]. When an acoustic radiation problem is analysed as a hemisphere, a mirror of symmetry is assumed, hence the directivity pattern is repeated for the lower hemisphere. Under these conditions, a hard baffle condition on the symmetry plane is automatic. A soft baffle could be created by imposing pressure fixities of zero on all fluid nodes on the plane, but this soft baffle would extend only to the spherica:. boundary of the near-field region analysed.

Figure 8.1.2 shows near-fie:d pressure contours at 12,000 Hz.

8-3

P153521.PDF [Page: 82 of 124]

Figure 8.1.1: Problem 15- Piston Directivity at 12,000 Hz

PROBLE!{ U5 - Piston in Hard Baffle

269.30

DIRB::TIONAL RESPONSE (dB re 1 JI.Pa/volt B 1 rn)

.DRIVE FRE<:UEOCY: 12000.0 Hz

.DIRECTIVIIT INDEX: -11.0764 dB

.PEAK 'IRANSMI'I RESPCNSE: 269.277 dB

.RECEIVING RESPONSE 0 IEG: 0.00

11-DEC-89 12:54

Figure 8.1.2: Problem 15- Near-Field Pressure Contours at12,000Hz

PROBLEM U5 - Piston in Hard Baffle EN'IIRE REGION RADIUS-D.1650 ISCAMPLITl:DE OON'IOUFS 11.J---;==~---------------, FRmUENCY: 12,000.00 Hz

......... \ '-

\ \ \ \

I -, \ \

\ I

\

' J I ' . . I I I

/ I / / \ ·- .. ,

.,

' ' " \ ' ' '

\ ' '

',

·, ·, t

8-4

t

POSITIVE - - -ZERO

_ ••••••. _ _ NEGATIVE

ISOAM?LITUCE CCNTOURS .MAXIMUM A!{PIITUDE:

0.2286280E+09 Pa • MINIMUM A!{PliTUDE:

0.2084398E+08 Pa . REFERENCE A!{P LITUDE:

0. 6903272E+08 Pa • AMPLITUDE CCN'IOURS:

10.40000 dB 9.100000 dB 7. 800001 dB 6.500000 dB 5. 200000 dB 3.900000 dB 2. 600000 dB 1. 300000 dB

O.OOOOOOOE+OO dB -1.300000 dB -2.600000 dB -3.900000 dB -5.200000 dB -6.500000 dB -7.800001 dB -9.100000 dB -10.40000 dB

11-DEC-89 13:07

P153521.PDF [Page: 83 of 124]

9 SERIES 6: COMPLIANT FLUID

This series demonstrates the use of Compliant Fluid modelled with SOLID elements to simulate a 20-cm diameter spherical air bubble pulsating in sea water. Figure 9.0.1 is a cross sectional diagram of the structure analysed.

The mesh of the F.E. model is shown in Figure 9.0.2. The material properties of the air within the bubble are determined assuming that the bubble is placed at 10 metres depth, where the absolute hydrostatic pressure in the water is two atmospheres. Values for density and bulk modulus are 2.585 kgfm3 and 0.285E+6 Pa, respectively. The adiabatic bulk modulus of the air is used, as this is a dynamic problem; for static problems, the isothermal bulk modulus would be used.

Figures 9.0.2 (a,b,c) and 9.0.3 (a, b) give details of node and element numbering.

9-1

P153521.PDF [Page: 84 of 124]

Figure 9.0.1: Series 6 Problems- Air Bubble in Water

~z

0 10

R (em) 10~

Figure 9.0.2 (a): Series 6 Problems- Outer Fluid Node Numbering

Series 6, Spherical air bubble GEG1E'1RY PLO'ITIN3

9-2

NCDE NU1BERS

.DTI£N3I<NS:

~m::o~~om RMAX= 1. 01700h m

~ -£:8H8~£~ ~TYPES:

~TYIES:

0 15-DEC-89 15: 35

P153521.PDF [Page: 85 of 124]

Figure 9.0.2 (b): Series 6 Problems- Middle Fluid Node Numbering

Series 6, Spherical air bubble

1 GEI:M:TRf PI.OIT!ffi

N:lE NtMBERS

·:~81~~888 ~ ~ -s:g§~~888 ~

iMM'ERIAL TYPES:

.EIEM::Nl' TYEE$ : 4-15

0 15-IEC-89 15:40

Figure 9.0.2 (c): Series 6 Problems- Compliant FluidAir Node Numbering ·


GEI:M:TRf PI.OIT!ffi

61

9-3

N:lE NtMBERS

.DlMENSiaiS: ~ -R:£~~RRR ~ ZMIN= -0. oo3sooo m ZMlJ{= 0.1084000 m

gMM'ERIAL TYPES:

i~TYEES:

lO 15-IEC-89 15:23

P153521.PDF [Page: 86 of 124]

Figure 9.0.3 (a): Series 6 Problems ·Outer Element Numbering


1 GEI:M:TR'l PIDTI'IN3

78

ELEMENT NlMBERS

.DJMENSICNS:

~o~~om ~ 1: o17oo(h m

~ i:8B888~~ .MATERIAL TYPES: 1

.EI.EM!:Nl' TYfES : 4-7-15

ro 15-IEC-89 15:30

Figure 9.0.3 (b): Series 6 Problems ·Inner Element Numbering

Series 6, Sphe~ical air bubble

1 GE!:l£TR'l PID'ITIN3 -------r--------"71 47

35

ELEMENT NU1BERS

.DJMENSICNS:

~ -R: ~8g~~R ~ ZMIN= -0.0008000 m ~ 0.1084000 m


• EIEl'£Nl' TYEES: 1-4-6-12-15

ro 15-IEC-89 15:27

9-4

P153521.PDF [Page: 87 of 124]

9.1 PROBLEM 16

Analysis Type:

Input File:

DRIVE - spherical air bubble

D16.DA':'

Since only the upper hemisphere of the complete structure is analyzed, all "air" nodes on the symmetry plane must be restrained in the Z direction.

A uniform volumetric harmonic body force was applied to the SOLID elements comprising the air by placing thermal loads on the elements.

Frequency response plots of radiated far-field pressure (dB re 1f..LPa@ 1m) and radial displacement of the surface of the bubble at Node 69 are given in Figure 9.1.1. The resonance frequency is 45.934 Hz with a Q of 51.57. The resonance frequency predicted by analytical theory [ 4.2] is 45.9 Hz with a Q of 52.0.

The pressure in the air bubble, given as the pressure in the stress printout for each element, is constant at 11000 Pa. This is equal to the nodal pressure in the water at the surface of the bubble.

9-5

P153521.PDF [Page: 88 of 124]

Figure 9.1.1: Problem 16- Far-Field Pressure and Radial Displacement of a Resonant Air Bubble in Water

PRJB 16 - Spherical Air Bubble

T R

~ s p 0 N s E

d B

-e- TR. FESEONS:: d3 ·-M·- Padial Dif:Pl. m X1f3 (Q= 51.572) 1B L

17 1.2

17 1.0

16 0.8

~ 16 . 0.6 ) ..

{

'X ·,,, J

0.4

·~.,rr.;-.,.,-r.,-rr+..-.,,_ .. .,-r.,,-r+-0.2

9-6

.0 09:27

R a Q. ~

! II ~

5 ~·

m

P153521.PDF [Page: 89 of 124]

10 SERIES 7: RING

To demonstrate the use of RING elements, the same tubular shell structure is used as in Problem 13, with RING stiffeners added. A representative cross sectional diagram is given in Figure 10.0.1. The node numbering is the same as in Figure 7.0.2. Ten RING elements (#11 to #20) are added to the structure of Figure 7.0.3, and are located at the odd numbered H nodes of the shell. The cross section of a single ring is 5 mm high and 2 mm thick. The end nodes, 1 and 21 are stiffened with single ring ribs, while the rest of the appropriate nodes receive double reinforcing.

Figure 10.0.1: Series 7 Problems- SHELL Tube with RING Stiffening

Z (em)

5 f I . .

-,-RING Ribs I

I ~ I . . , . I ~

I ~ Aluminum Tube

I . I E

. -I R (em) ,_------------{ -0 --------------:;,..... I 0 10

~ I

I f ---I

.... - 2mm I ~

I I E I

-r I -5

10-1

P153521.PDF [Page: 90 of 124]

10.1 PROBLEM 17

Analysis Type:

Input Files:

EIGEN - RING-stiffened SHELL tube, (a) with no additional inertia1

(b) with nodal masses.

D17A.DAT D17B.DAT

In Case (a), the original structure of Figure 7.0.1 (Problem 13) is stiffened with aluminum RING elements added to odd numbered H nodes of the shell. No inertia term (density) is specified in the material properties data for the ring, so that the stiffness alone may be modelled. The end nodes, 1 and 21 are free to move only in the R direction, corresponding to the condition of Figure 7 .1.2. The parameter POLANG in the element data for RING elements defines the number of elementary rings assigned to any given RING element; thus, PO LANG is unity for elements 1 and 10, and 2 for the remaining RING elements.

In Case (b), nodal masses corresponding to the masses of each respective RING element are specified at the appropriate nodes. This essentially takes the place of the density term that could have been included in the material data for the RING. Note that the total mass of the ring is added to each DOF on which it i.s effectively acting. If the inertial moments of the RING elements are to be modelled, the density must be included in the material data, as the added masses cannot represent inertial moments.

In both cases the radial displacement of Node 1 is used as the normalizing DOF and is set to 1. The values of ERV AL and ERVEC are set to 1.0E-8 and 1.0E-6, respectively.

The results show the first natural frequency of the RING-stiffened shell to be 21,756 Hz without the nodal masses (Case (a)). After adding nodal masses, the frequency drops to 15,418 Hz (Case (b)). The mode shapes are both essentially the same, and are the same as shown in Figure 7 .1.2 for Problem 13. As in Problem 13, the pure hoop mode is again Mode 4, but its frequency is lowered to 16,933 Hz.

10-2

P153521.PDF [Page: 91 of 124]

11 SERIES 8: DAMP1NG APPLICATIONS

The model used in Problem 9 (Figure 5.0.1) was chosen for demonstration of damping employment in solids and fluids. Node numbering is the same as in Figure 5.0.3 (a,b). Element numbering for Problem 18A is the same as in Figure 5.0.2. For Problem 18B, an additional five VISCous fluid elements (#81 to #85) are added to the structure, using existing fluid nodes.

11.1 PROBLEM 18

Analysis Type:

Input Files:

DRIVE -damping in the PT solid ring (a), -damping in a VISCous fluid layer (b)

D18A(1,2).DAT D18B(1 ,2,3,4).DAT

In Case (a) the stiffness and mass distributed damping coefficients, j.lk and j.lm

equal to 0.03 (3%), were arbitrarily chosen. Otherwise the data did not differ from those used in Problem 9. The two damping coefficients were applied separately, and resonance searches were applied in both cases in the range from 250 to 450 Hz. Transmitting voltage responses for the two types of damping are plotted in Figure 11.1.1, together with the result of Problem 9 for no damping. Corresponding efficiency plots are given in Figure 11.1.2 for the two damped cases. (The efficiency for the undamped structure is 100 percent.)

In Case (b), the material damping coefficients for the PT ring were set to zero, and a viscous dissipating fluid layer was inserted on the ring. This comprises five VISC elements, laid on F nodes of the FTOS elements.

The appropriate material properties matrix for the VISC elements was added to the data. Viscous damping is invoked by setting the "density" to 1.0 for pressure varying damping and/or setting the stiffness coefficient en to 1.0 for pressure gradient varying damping; these are flags and have no relevance to their normal properties. The magnitude of the damping is applied in the same way as for solid material damping: j.lk for stiffness distributed damping and j.lm for mass distributed damping, although these parameters are no longer fractions between zero and one. The four types of viscous damping were applied separately to the model and a resonance search was done in the same range as for material damping. Values of j.lk and j.lm were selected so as to yield efficiencies near 80% at the 375Hz resonance.

Plots of the transmitting voltage responses with viscous damping are compared to those for the undamped structure in Figures 11.1.3 to 11.1.6. The corresponding efficiencies are plotted in Figure 11.1. 7.

11-1

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Figure 11.1.1: Problem 18(a) -Transmitting Voltage Responses of a Free-Flooding Ring with Different Material Damping

PEOBLEM lflBA, PT Piezoelectric Ring with Material Damping

Figure 11.1.2:

---a-- Stiffness damping ·----- Mass damping .. .x.. No damping

;n 15

R 14 a

c

d i a .fl. 14

1

T '"' v 13

R

d

-

j - I

!j

/ _1_

-v .: ' - _/ - /

.c-

B 13

12 . 2'50 1.75

If\ .~- :~:x_ ll(,•

J.' .......... -· · ... ;_-~- ........ .. '...' .... -:-,

< ··.:-... "

/ ~/ -~-· ... ~~

/ "-,

~ .l

I r------

./ ... .. · ,...

t-

00 325 3'50 3 5 4'00 4:25 450 FREQIENCY (Hz) 2-JAN-90 15:40

Problem 18(a)- Efficiencies with Material Damping

PEOBLEM UBA, PT Piezoelectric Ring with M:lterial Damping

--Stiffness damping, 3% ·------·Mass damping, 3%

E F F I c I E N c y

1.

0.

0.

0.

0.

o.

n

c

..,

n.

-

-

.-.. ..

-

v -

250

____ ..,--~ -·- ..... --- ..... -~ _ .......

~ .r- __ , ...

-.. - v --.il.-"'"' ---- / .. . -

-. / ..

/ _.-I

/ v

L'7 5 00 -325 350 3'75 400 4 5 45 0 FREQIENCY {Hz) 2-JAN-90 15:53

11-2

P153521.PDF [Page: 93 of 124]

Figure 11.1.3: Problem 18(b) ·Transmitting Voltage Response for a Free-Flooding Ring with Viscous Pressure Damping, Jlk = 2.0E+8

P.EOBLEM USB, PT Piezoelectric Ring, VISC Fluid Layer, Press.K

Figure 11.1.4:

15

R 14 a d i a 14 l

T

v 13 R

d

B 13

12

-e-- Press. K damping --K-- No damping

n

- ,x K f-- _.// f-- f-

~- ~ f-

- ~~ ~ 1----, •'

,j I I

I

. v c: .)

( 1-1-

n ,i

- / f-- f-- f-

"-1-.

250 <75 bo 325 3SO 3 5 400 4 5 450

FREQUENCY (Hz) 3-JAN-90 13:05

Problem 18(b) -Transmitting Voltage Response for a Free-FIQOdi.ng Ring with Viscous Pressure Damping, Jlm= 30

PROBLEM Jfl8B, PT Piezoelectric Ring, VISC Fluid Layer, Press -M

15

R 14 a d i a 14 l

T v 13 R

d B 13

12

n

"

n

~

"'

-- Press_ M damping ---- No damping

- ., ,t ~ 1-

~-I/ ~

~ •'

I v J

1/ v

250 J 275 300 325 350 375 400 425 450

FREQUENCY (Hz) 3-JAN-90 13 :17

11-3

P153521.PDF [Page: 94 of 124]

Figure 11.1.5: Problem 18(b) -Transmitting Voltage Response for a Free-Flooding Ring with Pressure Gradient Damping, Ilk= 1.5E+7

ProBLEM USB, PT Piezoelectric Ring, VISC Fluid Layer, P .grad.K

Figure 11.1.6:

~ Pr.grad. K damping

'" 15 -

-"'-R 14

a d i

()..

---

I

I I '

-4E-- No damping

.J<

~ 1-

.7 r-1-

~ r--· v "'-; ~

r-~

I a 14 1

T v 13 R

d B 13

12

" ---,-

---

.,-

250

I

v 1-r-

J r-v r-r-1-1-

275 300 325 350 375 400 4 5 450 PREQOENCY (Hz) 3-JAN-90 13:21

Problem 18(b) ·Transmitting Voltage Response for a Free-Flooding Ring with Pressure Gradient Damping, J..Lm= 3

ProBLEM USB, PT Piezoelectric Ring, VISC Fluid Layer, P.grad.M

----- No damping -- Pr. grad. M damping

15"~-----.----.----,-----.--~~---.-----.----~ - #~ ........

·~ R 14~"----~----+---~----~;r-~--+---~~~··~·~~-r----+-

j' ~~---~ a -d -i

(\-

a 14 1 -

-T .. -v 13 R -

--

d B 13

,-

---

12 c::-

2 0

v /

v 00 325

FREQOENCY (Hz)

11-4

3'50 375 400 4 5

rr---

---

450 3-JAN-90 14:35

P153521.PDF [Page: 95 of 124]

Figure 11.1. 7: Problem 18(b) -Efficiencies for a Free-Flooding Ring with Different Types of VISCOus Fluid Damping

PBOBLEM ff18B, PT Piezoelectric Ring, VISCous fluid layer

-- Pr. K ··········· Pr. M --- P.Grad K ······ P.Grad M

E F F I c I E N c '[

1.0~----~---r----~---r----r----r----~--~

- - -~ , ,:-_;...;. P-<"··········· ·-1-----il----1---+--:::=+ ·-········ -~ ~~·:-... :. . ..... .

0.8 ............ ~ -.---:·:·. ~:·:·~·:·: :·;;? . 0.6 . ,. i_,-?

- -· v 0.4 -~i-----1----+----+----+----+----+--~

-0. 2:-1-----+----l-----+----l-----+----1-----+----1

0.0-1-~~~~~~~~~~~~~~~~~~~M

250 2 5 3bO 3~ 350 3 5 400 4 5 4 0 FREQrENCY (Hz) 3-JAN-90 13:08

We note that the solid material stiffness damping has a profound effect on the resonance frequency of a free flooding ring, while the other forms of damping have virtually no effect on the frequency. In all cases, the dependence of efficiency on frequency is greater for stiffness damping than for mass damping. Further investigation of damping in a free flooding ring transducer will be reported in a future DREA Note.

11-5

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P153521.PDF [Page: 97 of 124]

12 SERIES 9: RING ARRAY, RIGID NODE

Two PT rings, each made of 48 staves of Channel 5400 ceramic, are immersed in sea water and separated by an aluminum ring as shown in Figure 12.0.1. The mesh of the F.E. model is given in Figure 12.0.2. Figures 12.0.3 (a-d) and 12.0.4 (a, b) give details of node and element numbering, with the radial_ dimensions expanded relative to the axial dimensions. TheE type Nodes 975 and 987 coincide with F type Node 376, and are placed at the origin of the system of coordinates.

·For this edition of the Examples Manual, the model was changed by deleting the four outer layers of fluid elements from the original model. This reduces the size of the problem with a possible improvement in accuracy as noted under Problem 15. Also, the eigh~ elements representing the two PT rings have been placed in two element groups, one for each ring (NELG = 2 on Card 10). When all eight elements are plac~d in one group, the radiated power, admittance and efficiency are in error.

12-1

P153521.PDF [Page: 98 of 124]

Figure 12.0.1: Series 9 Problems- Cross-Sectional Diagram of Model

12-2

P153521.PDF [Page: 99 of 124]

Figure 12.0.2: Series 9 Problems · F.E. Mesh

Series 9, Two Free PT Rings with SOLID Spacer

GECMETRY PLOlTIN:i CCMPIETE GRID

.DIMENSICNS RMIN= -0.1024000M RMAX= 0.3024001M ZMIN= -0.2024000M ZMAX= 0 .2024000M


.ElEMENT TYPES: 4-6-7-12-14-15

Figure 12.0.3 (a): Series 9 Problems· Outer Node Numbering

Series 9, Two F".::ze PT Rings with SOLID Spacer

12-3

NJDE NUMBERS

.DIMENSICNS: {ENTIRE RffiiON) RMIN= -0.0100000 m RMAX= 0.2100000 m ZMIN= -0.2027520 m Z~X= 0.2112000 m


• ElEMENT TYPES: 4-6-7-12-14-15

5-FEB-90 13:38

P153521.PDF [Page: 100 of 124]

Figure 12.0.3 (b): Series 9 Problems- Middle Node Numbering

Series 9, Two Free PT Rings with SOLie Spacer

GEO?£TRY PLOT.riN:;

IDDE NUMBERS

• DIMENSIONS: RMIN= -Q. 0023760 m RMI\X= 0.1163765 m ZMIN- -0 .10 97 627 m zw.x- 0.1137714 m

.MATERIAL 'IYPES: ~~~~~~~~ 1

~~~1~~~~~±~~tit*~~~i~-11~ .ElEMENT TYPES: p· 4-6-7-12-14-15

5-FEB-90 14:03

Figure 12.0.3 (c): Series 9 Problems -Inner Fluid Node Numbering

Series 9, Two Free PT Rings with SOLID Spacer

GEOMETRY PLOTTIN:;

12-4

NJDE NUMBERS

DIMENSICNS: RMIN= 0.0332498 m RMAX= 0.0688755 m ZMIN= -0.0561145 m ZMAX= 0.0578879 m

• MATERIAl TYPES : 1

• ELEMENT TYPES: 4-6-7-12-14-15

5-FEB-90 14:05

P153521.PDF [Page: 101 of 124]

Figure 12.0.3 (d): Series 9 Problems- Solid Node Numbering

Series 9, 'Iwo Free P'I Rings with SOLID si>acer

GECMETRY PLOTI'INS 1~----------------------------------------~

12-5

N<DE NUMBERS

• DlMENSIONS RMIN= 0.0235839M RMAX= 0.0801373M ZMIN= -0.0607200M ZMAX= 0.0607200M


.ElEMENT TYPES: 4-6-7-12-14-15

P153521.PDF [Page: 102 of 124]

Figure 12.0.4 (a): Series 9 Problems· Outer Element Numbering

Series 9, Two Free PT Rings with SOLIC Spacer

GEO!'ETRY PLOTTI:t-n

ELE~NT NU:!!BERS

DIMENSICNS: (ENTIRE RffiiON) ~N= -0.0100000 m RMAX= 0.2100000 m

~~~= -g:~i88888 ~ .MATERIAL TYPES: 1-2-4 .

• ELEMENT TYPES: 4-6-7-12-14-15

5-FEB-90 14 :10

Figure 12.0.4 (b): Series 9 Problems -Inner Element Numbering

Series 9, Two Free PT Rings with SOLIC Spacer

1 GEOM::TRY PLOTTIT:\G

42 43 44 45 46 l .,0

29 30 ...... jlJ 5o /

41 3-'22 33 35 -,10 !

15 16 n l.l ,a~ ~l ~~0 28 ........ _.:::::,

7 .L'l .U ~ - 1n 1 1

1 2 ....., _E._ <2 27

.} ~ 5 6 ~.L.

l

133 134 135 P"n~ MQ. t.L" 159 I ........ ~·- ~~~ ..::::--

146 145 ~ '"' 147 148 -~ [54 l55 E- 160

14..~:p.H58 .L:J I .L P'b"' . 161 162 lbl3 ~us i~~; !197 173 1 6

/ .::::, 174 :'.75 176 177 178

--0 1 2 3 4 5 6 7 8 9 lb

12-6

ELE~NT NUMBERS

DIMENSICNS: RMIN= -0.0012000 m RMAX= 0.0780000 m ZMIN- -0.0714000 m ZMAX= 0.0714000 m


ELEMENT IYPES: 4-6-7-12- 4-15

5-FEB-90 14:15

P153521.PDF [Page: 103 of 124]

12.1 PROBLEM 19

Analysis Type: DRIVE- two free flooding PT rings with SOLID spacer

Input Files: D19.DAT

The model shown in Figures 12.0.3 and 12.0.4 is analyzed. A 6000Hz, 1-V amplitude excitation is applied to both E type nodes. The inner midplane nodes

·of each PT ring and the spacer are restrained in the Z direction, i.e., Nodes 185, 458, and 324.

Plots of displacement, far-field directivity, and near-field pressure contours are shown in Figures 12.1.1 through 12.1.3, respectively.

Figure 12.1.1: Problem 19 ·Displacement at 6000Hz with SOLID Spacer

PIDBIEM !1~ Two Free PT Rings with SOLID Spacer

1 prsPLACEMENT CF THE SOLID BOUNDARY

.l.lf---------, --------~ -----

1--~-------, FREQUENCY: 6000. 00 Hz

~ : : ' PROBlEM: I DRIV' : ! , • • RESISTANCE:

..E ; : 5.428192 ohms , ___ • • RFACTAN:E:

D 'OJ ~ ~ . . . . ' I

-----~---

-273.1754 ohms

• CCNDO:::TAN:E: 0.7271110E-04 S

• SOSCEPTAN:E: 0. 3659207E-02 S

EFFICIEN:Y 1.0000

PEAK 'lR 132.75 dB

• DISPlACEMENT S:::AIE R

I o. HgoBgl: O"l m I • DISPlACEMENT S:::ALE Z

I 0. I690199E=o I m I RMIN = -o .0150000 m R-!AX= 0 .0910800 m ZMIN= -0.0530400 m ZMAX= 0.0530400 m

'1 '2 '3 4 '5 b ., '8 " 10 25-Jl\N-90 13:38

12-7

P153521.PDF [Page: 104 of 124]

Figure 12.L2: Problem 19 ·Far-Field Directivity at 6000Hz with. SOLID Spacer

PRCBLEM U 9, Two Free PT Rings with SJLID Spacer

132 .so

Figure 12.1.3:

DIROCTIONAL RESPCNSE (dB re 1 uPa/vo1t @ 1 m)

.DRIVE FRE~N':Y: 6000.00 Hz


.PEAK TRANSMIT RESPCNSE: 132.750 dB

.RECEIVING RESPONSE 0 Dffi: -204.23

25-JAN-90 13:45

Problem 19 ·Near-Field Pressure Contours with SOLID Spacer

PROBLEM -lt19, Two Free PT Rings with SOLID Spacer Rl= 0.0000 R2- 0.2000 Z1= 0.0000 Z2- 0.2000 ISOl\MPLITUDE CDNIDUF.3

.. ' '

.... ·•

· ...

'

.

.

..

\ ~

~

-~

~.

··~

'4

.,

. . ..

,

'5

', I .

\

' .

'6 '7

12-8

FREQUENCY: 6000.00 Hz

POSITIVE ZERO NEXiATIVE

ISJAMPLITUDE CCNTOURS • MAXIMJM AMPLIWDE:

187.2069 Pa • MINIMJM AMPLIWDE:

8. 788887 Pa • REFERENCE AMPLITUDE:

40.56279 Pa .AMPLITUDE CDNIDURS:

11.90000 dB 10.20000 dB 8.500000 dB 6. 800000 dB 5.100000 dB

r:188888 ~ 0. OOOOOOOE+OO dB -1.700000 dB -3. 400000 dB -5.100000 dB -6.800000 dB -8.500000 dB -10.20000 dB -11.90000 dB

13:44

P153521.PDF [Page: 105 of 124]

12.2 PROBLEM 20

Analysis Type:

Input File:

DRNE - two free flooding PT rings with a RIGID node replacing the spacer

D20.DAT

Using the model from Problem 19, the aluminum spacer ring is replaced by a single R type (rigid) node. Of all the S nodes of the spacer, only the central Node 282 is retained and changed to R type. The R node becomes a common solid node for all the FTOS elements surrounding the spacer, while the F nodes remain the same as in Problem 19. Again, the same nodes of the PT rings as well as the R node are restrained in the Z direction. The same excitation as in Problem 19 is used.

Results of the analysis are practically the same as in Problem 19. Figure 12.2.1 shows a plot of near field pressure contours for comparison to Figure 12.1.3.

Figure 12.2.1: Problem 20- Pressure Contours with a Rigid Node as the Spacer Ring

POOBIEM !20, Tv.u Free PT Rings with RIGID Node Replacing Spacer Rl= 0.0000 R2= 0.2000 Z1- 0.0000 Z2= 0.2000

01\MPLITtDE a:lNTOllliS 11f--"""'=:::::::::::---------------, FREOUENCY: 6000.00 Hz

. ~ -' '

', . " ' .

' ' " \

' ' -. " . '

.,

- l ·- ...

- .. -' -- ... _ --

..

12-9

-- -- -- POSITIVE ----ZERO -·····----NEGATIVE ISOAM!?LITUE:E CCNTOORS .MAXIMUM AMPLITUDE:

183.7204 Pa .MINIMUM AMPLITUDE:

8.526333 Pa .REFEREN:E AMPLITUDE:

39.57855 Pa .AM?LITUDE CCN'IDURS:

11.90000 dB 10.20000 dB 8.500000 dB 6.800000 dB 5.100000 dB 3. 400000 dB 1. 700000 dB

0. OOOOOOOE+OO dB -1.700000 dB -3.400000 dB -5.100000 dB -6.800000 dB -8.500000 dB -10.20000 dB -11.90000 dB

25-JAN-90 15:24

P153521.PDF [Page: 106 of 124]

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13 SERIES10: TORSIONALPROBLEMS

An aluminum cylinder 20 em in diameter and 40 em long is used to demonstrate torsional problems. A diagram of the structure is given in Figure 13.0.1. The Finite Element mesh is shown in Figure 13.0.2 with element numbers, and node numbers are given in Figure 13.0.3.

Since real torsional vibration modes cannot couple to a fluid acoustically, fluids will never be used meaningfully in a torsional problem.

Driving of torsional deformations piezoelectrically has not yet been demonstrated with MAVART. This will require axial/radial poling with a tangential driving field or tangential poling with axial/radial driving fields, implying that some modification to PAR and/or PT elements may be necessary.

Figure 13.0.1: Series 10 Problems- Solid Cylinder Model for Torsional Problems

20-1 I

Fixed End

_ ,...>R (em)

1~~ 20 ~

13-1

P153521.PDF [Page: 108 of 124]

Figure 13.0.2: Series 10 Problems ·Element Numbering

Series 10, Torsional m=O Examples

GECME:J:RY PLOTriNS 1rr-----------------------------------------,

3 4

ElEMENT NlMBERS

.DIMENSIONS (ENTIRE REGION) RMI~ -0 .1700000M RMAX= 0.2700000M ZMIN= -0 .0200000M ZMAX~ 0.4200000M

.MATERIAL TYPES: 2

• ElEMENT TYPES : 12

7-FEB-90 17:00:42

Figure 13.0.3: Series 10 Problems· Node Numbering

Series 10, Torsional m=O Examples

GE<l1E:J:RY PLO'ITING 1rr-----------------------------------------,

11 a l.;j 21,1 5

6 17 8 19 0

f.L ..LL .~ ..... 5

7 ~ 9 0

L .j q

13-2

NffiE NOMBERS

• DIMENSIONS (ENTIRE REGION)

RMIN= -0.1700000M RMAX= 0.2700000M ZMIN= -0.020DOOOM ZMAX= 0.42DOOOOM

.MATERIAL TYPES: 2

• ElEMENT TYPES : 12

7-FEB-90 16:59:02

P153521.PDF [Page: 109 of 124]

13.1 PROBLEM 21

Analysis Type: STATC- Torsional load

Input File: D21.DAT

Torsional analysis is invoked by setting MSYM = -1 in Card 2, with the Fourier mode number IMODE = 0. The nodes on one end of the cylinder are fixed in R, Z, and¢, and a 100 N torsional load is applied at Node 25 on the other end of the cylinder. As this load is at a radius of0.1 m, the torque is 10 Nm.

From simple elastic theory, the displacement in radians for a circular cylinder of radius a and length L, subjected to a torque Tis given by

~ = 2TL I (Gm4) (13.1)

where G = 0.2664E11 Pais the shear modulus of the aluminum. Eq. 13.1 predicts a tangential displacement of 0.9559E-6 radian at the end of the cylinder for a uniform torque of10 Nm. MAVART predicts a displacement of 0.9927E-6 rad at Node 25, and 0.9152E-6 rad at Node 23, which is at midradius. This variation is due to the load being applied on the outer radius, and not uniformly across the end of the cylinder.

The graphics program GRAF1 does not support plotting of torsional displacements, hence no graphic output is displayed.

· 13.2 PROBLEM 22

Analysis Type: EIGEN- Torsional modes

Input File: D22.DAT

A modal analysis was conducted on the model depicted in Figure 13.0.1. The same zero fixities were applied and the 100 N load was removed, so the vibrating structure is a fixed-free cylinder. Torsional modes will occur when the cylinder length is an odd multiple of one quarter wavelength. The shear

velocity is given by c5 = .V(G I p ), where p = 2700 kg!m3 is the density of the aluminum. The calculated fundamental frequency for the 0.4 m long cylinder is 1963.1 Hz. MAVART predicts 1963.7 Hz, in very close agreement.

13-3

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P153521.PDF [Page: 111 of 124]

14 SERIES ll PROBLEMS· NONLINEAREFFEC'IS

A cylinder as shown in Figure 14.0.1 is used to demonstrate effect of geometrical nonlinearity on longitudinal vibrations of a uniform bar. The finite element mesh consists of 16 quadrilateral solid elements (Type 12) and 85 nodes.

Figure 14.0.1 Series 11 Problems - Solid Cylinder Model for Nonlinear Effects

40cm

I~ .~ 10 em

14-1

P153521.PDF [Page: 112 of 124]

The free end of the circular cylinder is subjected to uniform external compression of magnitude p. The other end of the cylinder is fixed in R, Z and <j>.

14.1 Problem 23

Analysis Type: Static-nonlinear

Input File: d23.dat

A nonlinear effect was analyzed applying several pressure values to the free end of the cylinder. Resulting displacement of the cylinder end at the centre (R=O) is shown in Table 14.01.

Table 14.01 Cylinder Displacement

p (Pa) Linear Nonlinear Linear Deformation Deformation Theory

Compression/Tension

106 -5.4830.10-6 -5.4863.1 o-6!-5.4 797.1 o-6 -5.5811.1 o-6

107 -5.4830.1o-5 -5.5160.1 o-5 /-5.4505.1 o-5 -5.5811.1 o-5

108 -5.4830.1 o-4 -5.8379.1 o-4!-5.1 783.1 o-4 -5.5811.1 o-4

14.2 Problem 24

Analysis Type: EIGEN - Nonlinear effects

Input File: d24.dat

A modal analysis was conducted on the model shown in Figure 14.01. The same zero fixities were applied and the pressure load was removed. Additional stiffness matrices resulting from the pressure load were generated by running the STATIC problem with the input NLGEO flag set to 1, prior to running a corresponding EIGEN problem. For the additional stiffness to be included in the EIGEn analysis, the input NGLEO flag was set to 2. Results of the nonlinear EIGEN analysis are summarized in Table 14.02.

14-2

P153521.PDF [Page: 113 of 124]

Table 14.02 Effect of additional stiffness on Natural Frequency of the Cylinder (Hz)

p Linear Nonlinear Theory Problem Problem

Compression/Tension

0 3266.4 3266.4 3220.1

106 3266.4 3263.2/3269.6

107 3266.4 3234.6/3297.9

108 3266.4 2929.7/3565.9

The wave longitudinal velocity is C0../ El p where p=2700 kgfm3 is the density of

aluminium and E=7.167.1o10 is the Young's modulus. The calculated fundamental frequency of a cylinder of length L=0.4 m, is f = Co/4L = 3220.1 Hz.

14-3

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15 SUMMARY

1. With few exceptions, all example problems from the Examples Manual [1.1c] have been run successfully at DREA, using one or the other of the latest two versions ofMAVART (MAVART 11.0 and 12.0). Some of the problems' input data required modification or minor correction to yield satisfactory results.

2. The FTOS Reversibility deficiency in MA V ART has been resolved, and Problem 16 has been reformulated with Compliant Fluid modelling in keeping with the findings of[4.4].

3. A deficiency in MAV ART remains:

• It is not clear whether piezoelectric drive of torsional deformations can be acommodated without some modification to the PAR and/or PT elements (Series 10 problems).

4. The postprocessor program GRAF1 has been used to display a broad range of analysis data from all example problems except the torsional problems. As well, GRAFl has been used to display model geometry for all examples.

15-1

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16 RECOMMENDATIONS

1. An example problem that demonstrates the piezoelectric drive of torsional deformations should be added to the Series 10 Problems. If necessary, MA V ART should be modified to allow such a driving mode.

2. The DREA postprocessor GRAFl should be updated to support plotting of tangential displacements for both torsional problems and for problems where the Fourier symmetry m is greater than zero.

16-1

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17 REFERENCES

[1.1] (a) User's Manual for Program MAVART, (b) Theoretical Manual for Program MAVART, (c) Examples Manual for Program MAVART Vol. I & II, DREA Contractor Report CR/87/442, E. L. Skiba, June, 1987.

[1.2] User's Manual for Program MAVART, E. L. Skiba and G. W. McMahon, DREA Technical Memorandum TM 91/201, January, 1991.

[1.3] Examples Manual for Program MAVART, G. W. McMahon and E. L. Skiba, DREA Technical Memorandum TM 90/204, August, 1990.

[4.1] Piezoelectric Technology, Data for Designers, Clevite Corporation, Piezoelectric Division (1965).

[4.2] MAVART Testing- Part I, G. W. McMahon, DREA Note SP/89/8, July1989.

[ 4.3] Research and Development of Spherical Transducers, Final Development Report, H. Jaffe, F. Rosenthal, and H. Baerwald, Clevite Research Center, ONR Contract Nonr 1825(00), April1957.

[ 4.3] Resolution of FTOS Reversibility Issue, DREA Contractor Report CR/91/467, E.L. Skiba, June 1991.

17-1

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Contractor Report

UNCLASSIFIED SECURITY CLASSIFICATION OF FORM

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DOCUMENT CONTROL DATA

l X ) UnUmited distribution Distribution limited to defence departments and defence contractors; further distribution on! as approved l Distribution limited to defence departments and Canadian defence contractors; further distr~utlon only as approved

l \ Distribution limited to government departments and agencies; further distribution only as approved Distribution limited to defence departments; further distribution only as approved Other (please specify):


DCD03 2/06187 -M

P153521.PDF [Page: 122 of 124]


This document is the third in a set of documents that provide information on the finite element Model for the Analysis of the Vibrations and Acoustic Radiation of Transducers (MA V ART). The other two documents in the set are: (1) User's Manual for Program MAVARTand (2) Theoretical Manual for Program MAVART. The program MAV ART has been developed for the Defence Research Establishment Atlantic (DREA) under research contracts to Canadian industry from 1976 to the present. This manual provides a number of sample problems to aid users in model development and output interpretation. It demonstrates virtually all elements and features of MA V ART. It also provides a set of problems for validating modifications and new installations.

, or ec mea y meanrng u erms or s o p rases a ara enze a ocumen an cou e p u m cataloguing the document They should be selected so that no security classification is required. Identifiers, such as equipment model designation, trade name, military project code nam~graphic location may also be included. If possible keywords should be selected from a pubfished lhesaurus. e.g. Thesaurus of Engineering and Scientific Terms (TEST) and that thesaurus-identified. If it not possible to select indexing terms which are Unclassified, the classificatlon ot each shoufcfbe indicated as with the tiUe).

Underwater Acoustics Transducer Design Analysis Finite Element Model Computer Software Axisymmetric Piezoelectric


P153521.PDF [Page: 124 of 124]

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