by Christopher Rabbitt A thesis submitted in ... - TSpace

FINITE ELEMENT VIBRATION MODELLING AND EXPERIMENTAL

VALIDATION FOR AN AIRCRAFT ENGINE CASING

by

Christopher Rabbitt

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science

Department of Mechanical and Industrial EngineeringUniversity of Toronto

© Copyright 2017 by Christopher Rabbitt

Abstract

Finite Element Vibration Modelling and Experimental Validation for an Aircraft Engine

Casing

Christopher Rabbitt

Master of Applied Science

Department of Mechanical and Industrial Engineering

University of Toronto

2017

This thesis presents a procedure for the development and validation of a theoretical vibration

model, applies this procedure to a pair of aircraft engine casings, and compares select

parameters from experimental testing of those casings to those from a theoretical model using

the Modal Assurance Criterion (MAC) and linear regression coefficients. A novel method of

determining the optimal MAC between axisymmetric results is developed and employed.

It is concluded that the dynamic finite element models developed as part of this

research are fully capable of modelling the modal parameters within the frequency range of

interest. Confidence intervals calculated in this research for correlation coefficients provide

important information regarding the reliability of predictions, and it is recommended that

these intervals be calculated for all comparable coefficients. The procedure outlined for

aligning mode shapes around an axis of symmetry proved useful, and the results are

promising for the development of further optimization techniques.

ii

To Navishta,

Never far from my thoughts, forever in my heart, and inspiring me always.

Acknowledgements

I would like to thank my supervisor, Professor Kamran Behdinan, for his guidance and

support in composing this thesis and throughout my Masters program. I have benefited

immensely from his expertise. I would also like to thank Professor Jean Zu, who brought me

to the University of Toronto and co-supervised me for the majority of my time here. Her

guidance, support, and encouragement was invaluable in getting me to where I am today.

This research could not have happened without the support this project received

from Pratt & Whitney Canada and NSERC. I am also grateful for the support of the BC Public

Service and the Ministry of Transportation and Infrastructure, whose support through a

leave-of-absence and the Pacific Leaders Scholarships for Public Servants program allowed

me to pursue this degree.

I am very thankful for my colleagues in both the Vibration, Design, & Mechatronics

Laboratory and the Advanced Research Laboratory for Multifunctional Lightweight

Structures for the help and friendship they extended to me during my time here. I would like

to especially acknowledge the assistance of Dr. Vincent Iacobellis, whose help in navigating

the department bureaucracy and operating the lab’s equipment has been invaluable.

My friends and family have been there for me throughout my studies, and their good

wishes and companionship have kept me going though even the most difficult times.

Lastly, none of this would have happened without the love and support of my

fiancee, Akanksha. Her patience and understanding throughout this long process has been

incredible, and I will be forever grateful.

iii

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Literature Review 5

2.1 Theoretical Analysis of Vibration . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Main Analysis Types . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Finite Element Types . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Experimental Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Measurement Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Correlation Efforts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Orthogonality Check . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.2 Modal Assurance Criterion . . . . . . . . . . . . . . . . . . . . . . . . 10

iv

2.3.3 Reduction Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.4 Approaches to Axisymmetric Structures . . . . . . . . . . . . . . . . . 13

2.4 Reliability Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Modal Parameters 16

3.1 Frequency Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.1 Modelling Considerations . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.2 Correlation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Natural Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18



3.3 Mode Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20



4 Aircraft Engine Casing Case Study 23

4.1 Finite Element Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Results 27

5.1 MAC Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.1.1 Front Compressor Casing . . . . . . . . . . . . . . . . . . . . . . . . 29

5.1.2 Rear Compressor Casing . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.2 Correlation of Natural Frequencies and Mode Shapes . . . . . . . . . . . . . . 30

v

5.2.1 Front Engine Casing . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2.2 Rear Engine Casing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6 Conclusions 36

6.1 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.2 Recommendations for Future Research . . . . . . . . . . . . . . . . . . . . . . 37

References 38

A Modelling 48

A.1 Finite Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

A.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

A.3 Additional Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

B Coding for MAC Optimization in Axisymmetric Structures 52

vi

List of Tables

5.1 Natural Frequencies & MAC for Front Compressor Meshes . . . . . . . . . . . 32

5.2 Natural Frequencies & MAC for Rear Compressor Meshes . . . . . . . . . . . 35

A.1 Front Compressor Casing Mesh Statistics . . . . . . . . . . . . . . . . . . . . 48

A.2 Rear Compressor Casing Mesh Statistics . . . . . . . . . . . . . . . . . . . . . 49

A.3 Additional Analysis Software . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

vii

List of Figures

2.1 Two Element Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Experimental Modal Analysis Example . . . . . . . . . . . . . . . . . . . . . 8

3.1 A Frequency Response Function . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 MAC Optimization Procedure Flowchart . . . . . . . . . . . . . . . . . . . . . 22

4.1 PW300 Engine Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2 Meshes for Front Compressor Casing . . . . . . . . . . . . . . . . . . . . . . 24

4.3 Meshes for Rear Compressor Casing . . . . . . . . . . . . . . . . . . . . . . . 25

4.4 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.1 MAC Value Response to Computational Remodelling for Front Compressor

Casing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.2 Mode 1 Response to Computational Remodelling for Front Compressor Casing 28

5.3 MAC Value Response to Computational Remodelling for Rear Compressor

Casing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.4 Mode 1 Response to Computational Remodelling for Rear Compressor Casing 31

5.5 Natural Frequencies & MAC for Front Compressor Meshes . . . . . . . . . . . 32

5.6 Natural Frequency Error for Front Compressor Casing . . . . . . . . . . . . . 33

5.7 Natural Frequencies & MAC for Rear Compressor Meshes . . . . . . . . . . . 34

5.8 Natural Frequency Error for Rear Compressor Casing . . . . . . . . . . . . . . 34

A.1 Sample FRFs From Each Component . . . . . . . . . . . . . . . . . . . . . . 50

A.2 Experimental Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

viii

List of Symbols

α Type I Error Rate

X Mean of Observed Predictor Variables

β0 Regression Line Intercept

β1 Regression Line Slope

Yi Predicted Response Variable

Yh Point Estimator for Response Variable

Ψ Eigenvector

b0 Least Squares Estimate of β0

b1 Least Squares Estimate of β1

I Identity Matrix

M Mass Matrix

n Number of observed values

tα/2;n−2 Percentile of the t-distribution

Xh Given Value of Predictor Variable

Xi Observed Predictor Variable

Yi Observed Response Variable

ix

Chapter 1

Introduction

This thesis seeks to develop models for vibration in aircraft engine structures, and validate

those models experimentally. These models must be accurate and computationally efficient,

and must be shown capable of accurately reflecting the dynamic properties of interest. To that

end, methods and strategies for modelling components, experimentally measuring dynamic

properties, and correlating theoretical models with experimental data are all explored.

1.1 Background

Modal analysis is the study of vibration in structures. Vibration can occur in a variety of

structures, from large buildings and industrial equipment, to motor vehicles and small

mechanical components, and it can be either beneficial or detrimental. For example, it is

beneficial when induced to mix chemicals or remove surface contaminants in industrial

processes, but is detrimental when created by the operation of an unbalanced engine or the

buffeting of a bridge by wind.

In either case, a thorough understanding of the occurrence of vibration in the

structure is necessary either to exploit its effects or effect its reduction. Typically, modal

analysis is concerned with the determination of two sets of parameters. The first are the

natural frequencies; i.e., the frequencies at which a structure will oscillate in the absence of

1

CHAPTER 1. INTRODUCTION 2

any applied force. The second are the mode shapes, which refer to the parts of a structure that

move with the same phase at a specific frequency of vibration. These parameters indicate the

frequencies at which a structure will vibrate most efficiently and can therefore be exploited or

avoided, depending on the application. They also indicate at which part of the structure an

applied force is both most and least likely to create the largest vibrations. With this

knowledge, engineers can choose a design that optimizes the vibrational response of the

structure and predict what mitigation measures may be necessary.

1.2 Motivation

In the study of vibrations in structures, mathematical models are often employed to assist in

design and analysis prior to the implementation. Numerous models have been developed, with

Finite Element (FE) models the most widely used. Besides being much less expensive to

construct and modify than physical prototypes, they can provide important information about

parts of the structure which are difficult or impossible to measure using experimental

techniques. In addition, analysis can be carried out much more quickly through computer

modelling than physical testing. However, while these techniques have been successfully used

for the analysis of many structures, the extent of variation between structures demands that

newly developed models be validated against either experimental results or previously

validated models for similar structures. While it is beneficial to compare as many parameters

as is practical during validation, it is critical that they cover the full range of expected

applications. Therefore, correlation criteria must be developed and applied in a systematic

way across all the parameters of interest. Only then can the model be used with confidence as

an adequate representation of the real structure.

The objective of this research is to implement a robust FE model validation

procedure capable of validating a wide range of linear vibration models. These procedures

must include specific provisions for determining the correlation of axisymmetric structures,


and should be applied to a real-world case study to validate their feasibility. Where possible,

the reliability of any results should be assessed, and this research should be conducted in such

a way to maximise and simplify reproducibility by future researchers.

1.3 Challenges

Modal analysis can be carried out theoretically or experimentally, with theoretical modal

analysis further divided into analytical and numerical models. Analytical models have

closed-form mathematical solutions, but they exist for a few relatively simple structures. By

contrast, numerical models, predominantly falling within the class of techniques known as

Finite Element Methods (FEM), are iterative techniques approximating the behaviour of a

structure, and they can be applied to a wide range of structures having arbitrary geometry. In

addition, experimental modal analysis can be carried out using a variety of techniques, all

involving the induction and measurement of the vibrational response of a structure. It is

frequently important to correlate modal parameters from two sources, such as using

experimental results to validate a theoretical model, or comparing experimental results

obtained over time to detect the degradation of a structure. A challenge in establishing

correlations between modal analyses is determining which features to compare and when

differences are significant for the intended application of the data. As real-world structures are

continuous and have an effectively infinite number of possible measurements, it is impossible

to compare every possible set of measurements. However, it is important that the correlation

method accurately illustrate the degree of similarity between the two sources within the

measurement range of interest.

Axisymmetric structures present special challenges. While the structure is

axisymmetric, the vibrational response may not be. Difficulties can then arise as the

orientation of the structure about the axis of symmetry does not assist in determining the

orientation of the results.


1.4 Applications

Modal analysis has been applied to a wide variety of structures that are expected to undergo

vibration, from large civil structures [1]–[6] to tiny piezoelectric sensors [7]–[10]. Many of

these structures are commonly largely axisymmetric, such as turbines, rotor housings,

pressure vessels, rotational bearings, missiles, and piping. The methods presented in this

paper hold promise for advancing analysis of all these structures.

1.5 Thesis Organization

This thesis will present the background and theoretical basis for this work, followed by a case

study and specific results. Chapter 2 presents a review of literature relevant to the topics

under consideration, with Section 2.1 focusing on theoretical vibration models, Section 2.2

discussing past efforts in experimental modal analysis, and Section 2.3 contrasting the various

methods employed to validate and correlate linear vibration models. Chapter 3 discusses the

most common parameters considered in modal analysis, with sections devoted to each of the

main classes of parameter and their respective modelling and correlation considerations.

Chapter 4 presents details of the aircraft engine casing examined in this research as a case

study, with sections devoted to the development of both the theoretical and experimental

models. Chapter 5 comprises a presentation, analysis, and discussion of the results of the

correlation methods presented in Chapter 3 to the case study presented in Chapter 4. Chapter

6 sums up the conclusions drawn from this research, highlights the contributions this work

makes to the broader research field, and gives recommendations for how this research may be

further extended. The Appendices contain additional details of this research to supplement

those in the main body of this work.

Chapter 2

Literature Review

Vibration in structures has been studied by humans for thousands of years [11], first

experimentally, and later theoretically. Both experimental and theoretical techniques started

simply with what could be learned through direct observation, becoming ever more

sophisticated as understanding of mathematics and physics advanced, and technology allowed

for more detailed observations. In this chapter, the literature to date in the areas of theoretical

and experimental analysis of vibration will be examined. In any scientific endeavour,

determining the degree of correlation between results and ensuring repeatability are important

considerations, and the literature related to these areas as they apply to the study of vibration

is also reviewed.

2.1 Theoretical Analysis of Vibration

Modern vibration theory is often traced back to Galileo Galilei [11] and his work on stretched

strings [12]. Similar analytical solutions were later developed for beams [13] and plates [14].

While exact analytical solutions to vibration problems continue to be developed [15], with the

rapid increase in computing power in recent decades, Finite Element (FE) analysis has

emerged as a widely used technique due to its ability to model structures of greater

complexity than is currently possible through analytical techniques [11].

5

CHAPTER 2. LITERATURE REVIEW 6

2.1.1 Main Analysis Types

For linear structures, three main types of analysis are commonly used: real eigenvalue

analysis, linear frequency response analysis, and linear transient response analysis [16]. Even

for applications where modelling requirements exceed the limits of these analysis methods, it

is common practice for analysts to begin with a real eigenvalue analysis [16]–[18] as a

relatively quick and efficient way of validating the basic behaviour of the model prior to

moving on to more complex analysis. Most commonly used commercial FE packages include

this capability [16], [19]–[21], presumably for this reason.

2.1.2 Finite Element Types

The main solid element types used in FE analysis of structural vibration are tetrahedral (tet)

hexagonal (hex) elements [22]. Both are commonly used, and each offers advantages and

disadvantages. Tet meshes are more amenable for use with a variety of robust algorithms [23],

while hex meshes are considerably more time-consuming to formulate. While Cifuentes et al

[24] found no appreciable advantage to using hex elements over second order tet elements in

terms of solution accuracy, other studies [23], [25]–[27] have cited the superior stability of

hex elements as a reason to continue their use.

G4

G10

G3

G6

G9

G8

G1

G5G2

G7

(a) TET Element

G18

G6

G17

G4 G12

G1

G11

G8

G19

G7

G2

G9

G5

G20

G10G3

G15

G16

G13

G14

(b) HEX Element

Figure 2.1: Two Solid Element Types[28]


2.2 Experimental Modal Analysis

2.2.1 Measurement Techniques

Modal testing has existed for decades [29], advancing quickly as more sophisticated

computing and signal processing technology has developed. Measurement of the vibrational

response has traditionally been done through a sensor in contact with the structure, though

Marudachalam et al. [30] found that even relatively small sensor masses can impact the modal

response in some situations. To avoid this, efforts started in the 1970’s to measure vibration

without contact [31]–[33], with the most successful efforts focusing on the use of lasers.

Another advantage sought by those developing laser vibrometers was the ability to measure

rotational degrees of freedom, which most traditional contact sensors are unable to measure

[34]. While laser vibrometry has continued to advance and find new applications in recent

decades [35], traditional contact-based sensors have continued to find use due to their lower

cost, smaller footprint, and ability to measure surfaces to which a clear line of sight is not

available. Some techniques have been developed to minimize the effect of sensor masses,

though these are primarily aimed at small structures where the sensor mass is significant in

proportion to the structure mass [36]–[38]. Additionally, the ability of a laser vibrometer to

detect vibration is strongly dependent on the surface properties of the structure. For example,

non-retroreflective surfaces may not return a strong signal to the accelerometer, although in

some situations this can be mitigated by surface treatments.

A third technique which has seen some activity is digital image correlation [39].

This technique relies on high speed cameras to record vibration, and digital signal processing

to extract the modal parameters. In some ways, this method uses the same techniques used in

direct observation of vibration for centuries, using computer technology to capture the image

in more detail than the human eye can see. This method shares with laser vibrometry the

advantage of non-contact with the vibrating structure, as well as the ability to record the

response of a large area of the part. Further, digital image correlation offers the potential to do


Structure

Force Transducer

Exciter (shaker)

Signal Conditioner

(power amplifier)

Computer (acquires data and

stores frequency-response

functions)

Spectrum

(FFT) analyser

Accelerometer

Elastic cord

(to simulate free-free condition)

Figure 2.2: Experimental modal analysis example[11]

this without the sophisticated and expensive optics associated with laser vibrometers.

However, the data processing requirements for this type of analysis are much higher than

those for laser or traditional vibrometry, so digital image correlation has not yet become

popular among analysts [40].

2.2.2 Boundary Conditions

When performing modal analysis, a key concern is knowledge of the boundary conditions.

While some modal analysis is carried out in-situ, especially on very large structures where

laboratory testing would be difficult or impossible, it is desirable to control the boundary

conditions as much as possible in order to induce conditions which simplify theoretical

analysis. Ewins [29] recommends an unsupported (or free-free) case be tested initially, as

experience has shown that this condition is easier to achieve in practice than any perfectly

described constraint conditions. Although completely unsupported conditions are impossible

to achieve for any significant length of time under normal circumstances, numerous authors

have presented a variety of methods to approximate these conditions. One commonly applied


method is to suspend the structure from above in such a manner that the structure can freely

oscillate in all directions, both linear and rotational, at a frequency an order of magnitude

lower than the lowest natural frequency of the unsupported structure. In [29], it is

recommended that the analyst attach such supports as close to the nodal points as possible in

order to minimize their influence, although as Avitabile points out in [41], it is rare that the

analyst knows the exact location of all nodal points with certainty a priori.

2.3 Correlation Efforts

From the first theoretical vibration models, it has been necessary to correlate these models

with other results for validation purposes. Galileo [12] validated his work on what would later

be termed lumped mass models by comparing the predominant natural frequency predicted by

his model to that observed in experiments. The simplicity and effectiveness of this technique

means it is still in use today. Correlation of mode shapes has also been an important goal.

While the mode shapes for some simple lumped mass models such as pendulums can be

correlated through direct observation and measurement with simple tools, most structures

require indirect measurement and correlation methods. Early mode shape visualization

techniques were pioneered by Chladni [42] and Hooke [43] and relied on visually comparing

mode shapes in a qualitative way. Visual comparison is still done today, as it is a quick check

not relying on any sophisticated mathematics but which can immediately detect large

discrepancies. Nevertheless, as modal analysis became more sophisticated, quantitative

methods for mode shape correlation were sought.

2.3.1 Orthogonality Check

An early example, arising directly from theory, was the orthogonality check [44]. As normal

modes are orthogonal to each other, it was thought that the eigenvectors from one source

would be orthogonal to the eigenvectors from another source except for those representing the


same mode which would be identical. Targoff [44] represented this with the following

equation:

ΨT1MΨ2 = I (2.1)

When normalized, this approach would ideally return an identity vector, with values

approaching unity along the diagonal showing strong correlation between mode pairs and

values approaching zero everywhere else. While this approach is theoretically sound, it has

several weaknesses. The first is that it requires a mass matrix, which is of course theoretical

and subject to error. This means that if one wanted to correlate two experimental results, for

example to test for changes in the boundary conditions or autocorrelation, an theoretical

model would need to be developed in order to perform this check. Also, the addition of a

theoretical mass matrix causes errors in the mass matrix to propagate through the system as

the degrees-of-freedom (dofs) best able to represent the mode shape may not correspond to

those most representative of the structure’s mass distribution.

2.3.2 Modal Assurance Criterion

An alternative metric is the Modal Assurance Criterion (MAC) [29], [45], [46], given by the

following equation:

MAC =

∣∣ΨT1 Ψ2

∣∣2ΨT

1 Ψ1ΨT2 Ψ2

(2.2)

This metric works the same way as the orthogonality check, in that values near unity indicate

strong correlation between those pairs of modes, while values near zero indicate poor

correlation. Unlike the orthogonality check, it can be performed on experimental results

without reference to a mass matrix. It has proven very popular, and the original paper [45]

continues to be cited frequently in the peer-reviewed literature. The MAC and orthogonality

check, together with their variants, are the most popular quantitative correlation metrics for

mode shapes in use today, although they are not without their drawbacks. Both are indicators

of global correlation, and do not give any information about correlation at a particular location


on the structure which may be of greater importance in practice. They also both struggle with

axisymmetric structures; the matrix algebra assumes a common coordinate system, whereas in

axisymmetric structures any arbitrary rotation about the axis of symmetry could be described

by the same mass matrix but yields different eigenvectors. There has been some research

suggesting approaches to resolve this issue [47]–[49], but no clear consensus has emerged as

to the best approach.

It is also important to note that neither the orthogonality check nor the MAC are

linear metrics, and are simply trigonometric functions of the angle between the eigenvectors.

This means that a MAC value of 0.8 is not necessarily “10%” closer in any absolute sense to

perfect correlation than a MAC value of 0.7. This is not intuitive, and some authors such as

[48] continue to cite MAC values as percentages. In [50], Morales discussed a variety of

methods for linearizing the MAC value. On the basis that simply linearizing the MAC yields a

profile insufficiently sensitive to changes as the value approaches 1, he proposes linearizing

the Normalized MAC (NCO) presented in [29]. This would yield the following equation:

LNCO = 1 − π

2acos

( ∣∣ΨT1MΨ2

∣∣2(ΨT

1MΨ1)(ΨT2MΨ2)

)(2.3)

One can see from this that the mass matrix has been reintroduced, bringing with it the

problems observed for the orthogonality check. Indeed, this metric could be referred to as a

linearized cross orthogonality check. While the advantages of the LNCO cited in [50] are real,

a high degree of confidence in the mass matrix would be required for this to be a useful

metric. If an accurate mass matrix is available, the LNCO would be most valuable in model

updating and optimization applications, where small differences between correlation

coefficients are most important.

A further extension of the MAC is the FMAC[51]. This method plots the MAC

values on a graph, and a linear regression line along the matched mode pairs. Presenting the

data this way provides information regarding correlation of both the mode shapes and the


natural frequencies on the same plot. The autocorrelation values of the mode shapes from one

set of results (the AutoMAC) can also be included in the plot to convey further information.

2.3.3 Reduction Methods

The mode shape correlation metrics identified above rely on the eigenvectors under

comparison being the same length. In many applications this is impractical, the most severe

being when comparing experimental results against FE models which may have thousands or

even millions of dofs. Therefore, a method must be used to either represent the larger model

with fewer dofs, or extrapolate the smaller model to additional dofs. Wang and Luo [52]

present a good overview of the methods in use.

Static reduction, also called Guyan reduction after its author [53], is performed by

creating a transformation matrix from the stiffness matrices and neglecting any inertia effects.

This method provides good results at lower frequencies, but degrades as the frequency

increases and inertia effects become more significant. It is also very sensitive to the choice of

location for the reduced dofs, as the process of creating an equivalent mass and stiffness

matrix can become problematic in locations where the replaced dofs are highly

non-homogeneous. Nevertheless, it is the simplest of the commonly used methods and the

original paper continues to be cited frequently. An extension of this method is dynamic

reduction, first presented in [54]. It works the same way, but creates the transformation matrix

based on the mass matrix, stiffness matrix, and an arbitrary frequency, thereby incorporating

some inertial effects. This can provide good results near a frequency of interest, but results

again degrade away from the selected frequency. Therefore, the advantage offered over Guyan

reduction is limited, and dynamic reduction is less frequently cited.

The next important improvement attempt for model reduction was the Improved

Reduction System (IRS) presented in [55]. This method introduces a procedure to refine the

Guyan technique to approximate the inertial effects over wide range of frequencies. This

method relies on minimization of both the stain and potential energies to approximate the


inertial forces within the system, and renders the reduction relatively less sensitive to the

location of the reduced degrees of freedom. An iterative technique was presented in [56],

which showed an improvement over Guyan reduction over a wide range of frequencies.

System Equivalent Reduction Expansion Process (SEREP) [57] works in reverse, by

first reducing the results of the larger model, then using the resulting transformation matrix to

extrapolate the results of the smaller model to the larger dofs. Consequently, the quality of the

reduced model is much less influenced by the choice of dof location, although the number and

placement of dof must still be sufficient to fully describe the mode shapes. Several

comparative studies [52], [58], [59] have found it to be the most accurate of the reduction

techniques, though it is also the most complicated. However, as Ewins notes [29], this

technique creates a situation where the results of an experimental model are transformed

using results from a theoretical model. This can be seen as a biased process, as it attempts to

fit the experimental results to the theoretical results. Unlike the previous techniques, SEREP

preserves eigenvalues and eigenvector values for both models at the original measurement

points in the comparison, without suffering data loss or corruption in the transformation

process.

2.3.4 Approaches to Axisymmetric Structures

As was noted in the discussion of correlation metrics, the existing quantitative methods suffer

from complications when applied to axisymmetric structures. Several approaches have been

investigated to solve this problem. In [47], Chen et al. present methods for simply rotating the

mode shapes about the axis of symmetry, using an analytical solution to the angle that

maximizes MAC value for a given mode, then averages angle based on a weighting formula to

produce an optimal angle for the modes under measurement. Schedlinski et al. [48] present a

similar but iterative rather than analytical approach. Neither paper presented a comparison of

their approach to a proven global optimum, or investigated the sensitivity of the MAC value to

the angle parameter.


An alternative approach is offered through digital pattern recognition. This work,

beginning in the 1990s [60], uses image recognition methods applicable to a wide range of

applications to identify matching shapes regardless of orientation. Its use is vital for the

digital image correlation of modal data collection discussed earlier, as pattern recognition is

necessary to extract the mode shapes of the structure from the broader image. However, Wang

et al. [61] make a convincing argument for its adoption as a complement to the more

traditional MAC as well. As it does not depend on orientation, it is well suited for

axisymmetric structures. In addition, it could even be used in in cases where the coordinate

system is unknown, although in this case it would introduce additional uncertainty into the

modal parameters due to approximations inherent in the pattern matching.

2.4 Reliability Estimation

Modelling invariably involves some degree of error, and while it should be the goal of the

analyst to reduce it as much as feasible, in practice, the amount will never decrease entirely to

zero. Two types of error are generally recognized: systematic and random. Systematic error

consists of errors pervasive throughout the data, resulting in a consistent difference between

the true value and its modelled counterpart. Causes of systematic error would include a

miscalibrated measurement device, or an incorrect stiffness matrix in a FE model. These

errors are repeatable, in the sense that twice measuring a value with a miscalibrated device

would return the same error each time. Determining and reducing this error is the primary

goal of model updating.

Conversely, random errors are those where previous measurements of a value do not

guarantee the same value for future measurements. These can be particularly important in

experimental settings where a large number of unpredictable variables may be present: signal

noise, analyst performance, and environmental conditions among others. There is a large body

of research dedicated to analysis of these random errors [62], mostly involved with methods


of quantifying the uncertainty.

With respect to modal analysis, reliability of the parameter estimations is rarely

considered, and few of the references cited so far make any reference to it. A notable

exception is digital image correlation, where the heuristic nature of the analysis necessitates a

calculated degree of confidence in the solution [63]. For more traditional experimental modal

analysis approaches, this is often not done. However, there has been some research in this

area. Any analysis of random error involves some degree of repeated measurement, to

establish the amount of random variation present. [64] used the bootstrap, a simulated

sampling technique, to estimate the error distribution of experimental measurements and

calculate confidence intervals for the modal parameters. As the bootstrap depends on

independent measurements, all individual responses must be recorded as opposed to the

averaged responses more commonly collected by software. To see if this requirement could be

relaxed, [65] conducted a comparison study on the bootstrap and the Monte Carlo technique

which can be used with averaged data. Unfortunately, they were unable to conclude that the

two techniques yielded equivalent results.

Other approaches have been attempted, including the use of fuzzy numbers [66], but

variations of the traditional least-squares technique [67], [68] appear to have gained the most

acceptance. These techniques work by assuming a model, and positing that the best

parameters for that model are those which minimize the sum of the squared residuals, and

assume that the error follows some known distribution. If the distribution is known, the

confidence interval can then be calculated based on the existing sample measurements.

Chapter 3

Modal Parameters

Modal analysis consists of the study of vibration in structures through the measurement of a

few well-defined parameters. These are the frequency response functions, the natural

frequencies, and mode shapes, which together provide a fairly complete description of the

vibrational response of the structure, at least within the frequency domain. Analysis of

behaviour in the time domain, such as the transient response analysis referenced in the

literature review, is would not normally be referred to as modal analysis and falls outside of

the scope of this thesis.

3.1 Frequency Response Functions

As the name implies, frequency response functions (FRFs) are functions describing the

frequency-domain response of a structure. An example of a FRF plot is provided in Figure

3.1. They more completely describe the vibrational response of a structure than the other

parameters, as they provide information at regarding the response at all frequencies, and can

be transformed through a Fourier transform to show the time-domain response as well.

Natural frequencies and mode shapes can both be extracted from FRFs, with natural

frequencies being represented as peaks on an FRF graph, and mode shapes being the

represented by a vector of the values of the FRFs at the natural frequencies at different points

16

CHAPTER 3. MODAL PARAMETERS 17

Frequency

Am

plitu

de

Figure 3.1: A Frequency Response Function

on the structure.

FRFs are particularly important in the study of non-linear response, as

non-linearities will appear as distortions of the natural frequency peaks within the FRF. As the

models being developed for this thesis are linear, non-linear FRFs will not be discussed

further at this time, though their use may be necessary if this work is extended to more

advanced useage cases at a later time.

3.1.1 Modelling Considerations

In spite of their advantages, FE modelling of the frequency response function is often avoided

when possible. As the full FRF is a continuous infinite function, full numerical computation is

impossible, and some amount of truncation is required. This inherently produces error, and

cannot be avoided. Additionally, this degree of detail is often not required. Resonant

frequencies will dominate any response, and the range of frequencies at which a structure will

typically be excited is likewise limited. Therefore, modelling techniques which focus on these

parameters to the exclusion of others are often used in place of full FRF modelling.

Nevertheless, FRF modelling remains in use for applications where more complete modelling

is required, such when non-linearities are present.


3.1.2 Correlation Methods

Due to the inherent errors present in FRF modelling techniques, successful correlation has

often been found to present difficulties[69]. Nevertheless, it is important to visually inspect

the FRFs from experimental testing to get a sense of if the modelling assumptions are correct;

primarily the presence or absence of non-linearity. As the FE models developed assume a

linear response, the FRFs from experimental testing should be consistent with this and show

straight vertical peaks with roughly symmetrical responses surrounding them. Aside from

this, for the reasons discussed in the preceding section, the FE models developed through the

research discussed here have not been used to generate FRFs; therefore, no direct correlation

procedure has been presented.

3.2 Natural Frequencies

Natural frequencies are commonly measured parameters, as they are scalar values, easily

compared. They can be defined as the frequencies at which a system vibrates when subjected

to an impulse from rest[70], and are sometimes referred to as the eigenvalues. The number of

possible natural frequencies is equal to the number of dofs for the structure. While there are

theoretically an infinite number of these for any continuous system, only a limited number

will be relevant for a given usage case.


Calculating natural frequencies within a given range from a FE model is comparatively quick.

As with any FE simulation, mesh density and quality can have a significant impact on the

results. For natural frequency calculations, the finite element model tends to exhibit higher

natural frequencies than the physical object and increasing the mesh density can reduce this

error. This effect can be reversed in poor quality meshes or with geometries prone to

phenomena such a shear-locking; in these cases, the use of higher-order elements can help


eliminate these effects [71]. A number of solution methods exist, but MSC Nastran

recommends the use of the Lanczos method for most problems due to its efficency and

reliability[16].


Two primary methods exist for correlating natural frequencies [29], and both were used in this

research. The first is simply comparing the percent difference between two sets of results,

often in tabular or bar graph form. The second is to use simple linear regression to model the

relationship between the two sets of values. This is the practice of formulating an equation

that describes a linear relationship between the experimental and theoretical frequencies. The

method used in this work is the method of least squares; a procedure that seeks to minimize

the difference between the experimental value predicted from the equation, and that found in

practice. This requires the minimization of the following equation [72]:

Q =n∑i=1

(Yi − β0 − β1Xi) (3.1)

The regression equation then takes the form [72]:

Yi = b0 + b1Xi (3.2)

The ideal case for model validation would be the experimental and theoretical results having

a 1-to-1 relationship, resulting in slope of 1 when plotted in a graph. This second method has

the advantage of accounting for all the natural frequencies within the range of interest, thereby

revealing systematic errors in the model. Points showing little scatter but resulting in a line

with a slope other than 1 or an intercept other than zero can indicate an error in the material

properties of the model. The significance of these deviations can be assessed using standard


statistical techniques, such as the regression confidence interval[72], [73]:

Yh ± tα/2;n−2

√MSE

(1

n+

(Xh − X)2∑(Xi − X)2

)(3.3)

This equation calculates the upper and lower bounds of a predicted value based on the

assumption that either the error or the estimator follow a normal (Gaussian) distribution. The

requirement for the theoretical models used in this research is that the frequency deviations

fall within ± 10% of the experimental value for the natural frequencies of interest.

3.3 Mode Shapes

Mode shapes are critical parameters in the study of vibration, as they provide information as

to which parts of a structure are most excited at a given frequency. In addition, they also

provide a metric for validating the overall vibration model. In the context of the equations of

motion, mode shapes are represented by (and sometimes referred to as) the eigenvectors.


The typical limitations of FE modelling apply to the extraction of mode shapes as well.

However, the experimental measurement of mode shapes involves additional considerations

not present in the measurement of natural frequencies or FRFs. While natural frequencies and

FRFs can be measured and evaluated at discrete points, mode shapes require measurement at

a range of locations. Points must be of sufficient number and placement to capture all points

of inflection on the mode shape. The number and location of these points can be predicted

through use of a pre-test FE simulation; however, additional points should be included to

account for possible errors in the FE model [41]. It must be noted that the number of

additional points should be kept to a minimum, as collecting data for additional sampling

points is a time-consuming procedure. It is also important to consider the eventual model


reduction when choosing sampling locations, as the points should be chosen such that a dof at

that location can be assigned properties that adaquately account for the larger number of

nearby dofs replaced in the reduction procedure.


MSC.ProCOR is used to calculate the MAC between the experimental and theoretical results,

using Guyan reduction to equivalence the dofs. In addition, a new procedure has been

developed to maximize its effectiveness for axisymmetric structures. As noted in the literature

review, for an axisymmetric structure, any arbitrary rotation of the results about the axis of

symmetry results in an equally valid solution. However, the MAC treats the angle of rotation

as significant; experimental and theoretical mode shapes with poorly matched rotations will

result a poor MAC value, as seen in [47] . It is important to note, that although the rotation of

the results can be arbitrary, the rotation of individual mode shapes is not independent but

connected to that of the other shapes. This is apparent when one considers the response of a

single point on a structure; it can be a node for one mode, and antinode for another, and

somewhere in-between for a third. By the principle of superposition, the response is the sum

of all these modes, meaning that while the overall rotation may be arbitrary, each mode shape

must maintain its rotation relative to the other modes. To account for this, this work sought to

maximize the average MAC across the modes of interest as opposed to maximising any

individual MAC number. To accomplish this, a script was written in the Patran Command

Language (PCL) which automatically calculated the MAC between experimental and

theoretical results. This script can be used to either obtain the results for the rotation angle

that optimizes the averaged MAC or generate a table for the MAC values for a set of

iterations. The requirement for models used in this research is that the MAC be >0.9 for all

correlated modes of interest, and <0.1 for all uncorrelated modes.


Read FE model Calculate model CG Run full FE model

Run reduced FE modelRead experimental

results

Rotate experimental

results by arbitrary

angle θ

Calculate MAC for

each mode

Average MAC across

modes of interest

Figure 3.2: MAC Optimization Procedure Flowchart

Chapter 4

Aircraft Engine Casing Case Study

To demonstrate and validate the procedures discussed in the proceeding chapter, front and rear

engine casing components from the Pratt & Whitney PW300 turbofan engine were subjected

to modelling and testing. The components in question and their associated 3D CAD files were

provided by Pratt & Whitney Canada (PWC) and are depicted in Figure 4.1. These models

were made as part of a larger project to model and analyse the vibration throughout the entire

engine. This case study serves as a validation for the modelling techniques used for this part

of the project.

(a) Front Compressor Casing

(b) Rear Compressor Casing

Figure 4.1: PW300 Engine Components

23

CHAPTER 4. AIRCRAFT ENGINE CASING CASE STUDY 24

(a) Fine Mesh (b) Coarse Mesh

Figure 4.2: Meshes for Front Compressor Casing

4.1 Finite Element Modelling

The FE models for this case studies were developed for use with the MSC.NASTRAN normal

modes solver. Each component was meshed twice, resulting in a fine, high-fidelity mesh, and

a coarse, simplified mesh. A coarse mesh was desirable for computational efficiency, and as a

base from which to build a larger, more complex model with which to carry on future research.

The fine, high-fidelity mesh was required as an additional validation step for the coarse mesh

pre-experiment, also serving as a second model to be validated through experimental testing.

The components modelled are not fully axisymmetric. The fine meshes, shown in

Figures 4.2a and 4.3a, modelled all features to a high degree of precision, and therefore

preserved these asymmetries. The coarse meshes, shown in Figures 4.2b and 4.3b, feature the

component modelled as a revolved solid, and therefore are fully axisymmetric. Statistics and

properties for all depicted meshes can be found in Appendix A.


(a) Fine Mesh (b) Coarse Mesh

Figure 4.3: Meshes for Rear Compressor Casing

4.2 Experimental Setup

For this case study, experimental modal analysis was carried out using the test setup shown in

Figure 4.4. A test piece of high stiffness is placed on a softly sprung support. As the stiffness

of the support is orders of magnitude lower than the stiffness of the test piece, the vibrational

response approximates that of an unsupported (free-floating) piece. A piezoelectric triaxle

accelerometer is attached to the piece by a thin layer of wax, and both it and an impact

hammer containing an integral accelerometer are attached to a signal analyser. The signal

analyser digitizes the signal received from both pieces of equipment and sends them to the

computer for further processing and analysis. The accelerometer measures the acceleration

along three orthogonal axes at a point. When the impact hammer strikes the test piece, it

sends a signal to the computer through the signal analyser, which allows measurements from

the accelerometer to be linked to that specific strike. Sophisticated signal processing by the

computer then extracts modal properties, such as the frequency transfer function, natural

frequencies, and mode shapes for the test piece.

To collect the experimental data used in this case study, the shapes of the engine

casing components were entered into the analysis software (LMS Test.Lab), along with the

strike and measurement points. The engine casing components were then placed on a thick

soft foam pad. Since the components are composed of steel or titanium alloys, their stiffness

far exceeds that of the foam, so the vibrational response should approximate that of an

unsupported object. The software was then turned to data acquisition mode, and the test piece


Test Piece

Signal Conditioner

Soft

Spring

Support

Triaxle

Accelerometer

Figure 4.4: Experimental Setup

struck at the selected point with the hammer. Upon impact, the software registered the strike

and the resulting response from the accelerometer. The newly generated data was analysed,

after which the software prompted the experimenter to strike the object again. The process

was repeated until adequate data had been collected at the measurement point, at which time

the accelerometer was moved to the next point. This procedure is known as a roving

accelerometer test. The adequacy of the data was determined by the absence of high signal

noise and by compatibility between repeated strikes according to predetermined limits set in

the software. Generally, this required 3-5 strikes at each measurement point, but additional

strikes were required in cases of missed strikes, loss of accelerometer signal, or other signal

errors. Once adequate data was collected, the software was switched to data analysis mode,

the frequency transfer function examined, and the natural frequencies and corresponding

mode shapes extracted. Sampling points were determined based on those sufficient to detect

and discriminate between those predicted by the corresponding theoretical model.

Potential sources of error in this test setup include those resulting from the

approximations used in formulating the boundary conditions, calibration error of the

instruments, signal noise, and manufacturing discrepancies in the test pieces. Attempts were

made to minimse these errors to the extent possible. All sensors used were professionally

calibrated.

Chapter 5

Results

The FRF depicted in Figure 3.1 is representative of those found experimentally from this case

study. It exhibits two natural frequency peaks, and appears to be linear due to its smooth and

symmetric shape.The two closely spaced natural frequencies are characteristic of

axisymmetric structures, and this pattern can be seen in the tables below as well for both FE

and experimental results. Sample FRFs for each component can be found in Appendix A. The

frequencies presented have been normalized between zero and the largest measured natural

frequency to preserve the proprietory nature of the underlying data.

5.1 MAC Optimization

The PCL algorithm presented in Section 3.3.2 was used to optimze the MAC values between

the axisymmetric FE models and the experimental results. A large number of values were

calculated for each component, increasing the probability of the maximum values closely

approximating the global optimum.

27

CHAPTER 5. RESULTS 28

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0 45 90 135 180 225 270 315 360Iteration #

MA

C V

alue

Mode #●

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12345678910

Figure 5.1: MAC Value Response to Computational Remodelling for Front Compressor Casing

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0 45 90 135 180 225 270 315 360Iteration #

MA

C V

alue

(a) MAC Value with Rotation

0

1

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0.5 0.6 0.7 0.8 0.9MAC Value

Den

sity

(b) MAC Value Density

Figure 5.2: Mode 1 Response to Computational Remodelling for Front Compressor Casing


5.1.1 Front Compressor Casing

Figure 5.1 presents the response of the MAC value of the front compressor casing to

computational remodelling as a result of rotating the experimental results about the axis of

symmetry. A large amount of scatter is present, with the largest amount of scatter

accompanying the best-correlated modes. Figure 5.2 isolates the response of the mode shape

to computational remodelling for the first mode. This is one of the most widely dispersed

modes, with peak values of >0.9 and low values of <0.5. No clear pattern is apparent from

these plots, limiting the potential for development of an optimization algorithm to determine

optimal rotation angles.This is likely the result of the multiple reductions of the FE model

resulting in unpredictable mode shape orientation about the axis of symmetry. However,

Figure 5.2b shows a strong concentration of values at the upper end of the range, indicating

that a close approximation of the optimal MAC value has likely been achieved. Therefore,

these results are sufficient for determination of an approximate optimal MAC for each pair of

mode shapes; in this case, those values are found at iteration numbers 185, 267, 189, 272, and

241. With this information, the experimental results were rotated to produce Figure 5.5a and

Table 5.1a.

5.1.2 Rear Compressor Casing

Figure 5.3 presents the response of the MAC value of the rear compressor casing to

computational remodelling as a result of rotating the experimental results about the axis of

symmetry. The amount and pattern of scatter is similar to that observed for the front

compressor casing. This further supports the hypothesis that the repeated calculation of the

reduced FE model by the algorithm is leading to unpredictable changes in the mode shape

orientation about the axis of symmetry. In this case, the bias of MAC value concentration

shown in Figure 5.4b is even more pronouced than previously, increasing confidence that an

approximate optimal MAC value can be obtained; in this case, at iteration numbers 49, 162,

68, 310, and 196. As with the front compressor casing, the experimental results for the rear


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1.0

0 45 90 135 180 225 270 315 360Iteration #

MA

C V

alue

Mode #●

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123456789

Figure 5.3: MAC Value Response to Computational Remodelling for Rear Compressor Casing

compressor casing were rotated by the specified angles to produce Figure 5.7a and Table 5.2a.

5.2 Correlation of Natural Frequencies and Mode Shapes

The results of the described experiments are shown in Figures 5.5 and 5.7, as well as Tables

5.1 and 5.2. These figures follow a modified form of the FMAC introduced by Fotsch and

Ewins [51]. The diameter of the green circles represents the AutoMAC value for each

experimental frequency, while the diameter of the hollow blue circles represents the MAC

values for each paring of experimental and theoretical modes. The red line represents the

simple linear regression line between correlated mode pairs. 1-to-1 correlation is desired,

therefore slopes other than 1 are considered representative of error within the model. The

dashed lines represent ±10% error limits on the regression line, and the linear regression

equation is displayed on the plot. Grey shading around the regression line indicates the 95%

confidence band for that line, and can indicate the probability that the results of further

experimentation will show acceptable correlation.


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1.0

0 45 90 135 180 225 270 315 360Iteration #

MA

C V

alue

(a) MAC Value with Rotation

0

1

2

3

4

5

0.5 0.6 0.7 0.8 0.9 1.0MAC Value

Den

sity

(b) MAC Value Density

Figure 5.4: Mode 1 Response to Computational Remodelling for Rear Compressor Casing

5.2.1 Front Engine Casing

From 5.5, it is apparent that the relationship between correlated mode pairs for the front

compressor casing appears to be linear and no consistent bias is apparent for error relative to

the regression line. Both MAC and AutoMAC values which fall on the regression line are

significantly larger than those off it, with no significant MAC values appearing off the

diagonal for either mesh. Due to discretization error, the intercept of the regression line for

the coarse mesh is higher than that for the fine mesh. This indicates consistently higher

natural frequencies resulting from the coarse mesh as predicted by theory [74], and is

therefore expected. Of note is the fact that the regression line for the coarse mesh falls outside

the 10% deviation limit. While this can indicate a problem with the mesh properties [29], the

fact that the regression line for the fine mesh, which uses the same mesh properties as the

coarse mesh, falls well within acceptable limits would appear to indicate an issue with the

geometric fidelity of the coarse mesh instead.

Table 5.1 presents the numerical MAC values for the front compressor casing. The

cells are colour-coded on a gradient scale, with high MAC values having green shading and


● ●● ●● ●● ●●

● ●● ●● ●● ●●

●● ● ●● ●● ●●

●● ● ●● ●● ●●

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● ●● ●● ●●

●● ●● ●●

●● ●● ●●

● ●● ●●

y = 0.01917 + 0.8473x

0.2

0.4

0.6

0.8

0.2 0.4 0.6 0.8Normalized Frequency from FEM

Nor

mal

ized

Fre

quen

cy fr

om E

xper

imen

tal R

esul

ts

(a) Coarse

● ●● ●● ● ● ● ●

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y = 0.0122 + 0.941x

0.2

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0.6

0.8

0.3 0.5 0.7Normalized Frequency from FEM

Nor

mal

ized

Fre

quen

cy fr

om E

xper

imen

tal R

esul

ts

(b) Fine

Figure 5.5: Natural Frequencies & MAC for Front Compressor Meshes

Table 5.1: Natural Frequencies & MAC for Front Compressor Meshes

(a) Coarse

0.93

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0.98

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0.91

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0.79

0.04

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0.02

0.27

0.04

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0.16

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0.11

0.16

0

0.57

0.02

0.195

0.2

0.231

0.234

0.523

0.592

0.614

0.767

0.81

0.9714

0.207

0.207

0.266

0.266

0.583

0.583

0.677

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0.9208

0.9208

Normalized Frequency from FEM

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(b) Fine

0.93

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0.231

0.234

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0.592

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0.767

0.81

0.9714

0.192

0.196

0.239

0.24 0.542

0.543

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0.633 0.8

0.8563Normalized Frequency from FEM

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987654321

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20% Difference between FE and Experimental

Exp

erim

enta

l Mod

e #

(a) Coarse

10987654321

0 1 2 3 4 5 6% Difference between FE and Experimental

Exp

erim

enta

l Mod

e #

(b) Fine

Figure 5.6: Natural Frequency Error for Front Compressor Casing

low MAC values having no shading. A clear diagonal line of high MAC values is apparent in

both Table 5.1a and Table 5.1b, indicating that the MAC has proven effective in selecting

correlated mode pairs. Of particular interest is the 5th experimental frequency; the tables

clearly indicate that a second natural frequency is predicted nearby which was not detected by

the experiment. Additional signal processing of the experimental data may be able to yield the

missing mode. Also of note is the poor MAC values for the higher frequency modes in Table

5.1. While these values are higher than the surrounding values, they are well below optimal.

Since this issue arises in both the fine and coarse meshes, it may indicate a more fundamental

problem with the model parameters rather than an issue with the meshing strategy.

5.2.2 Rear Engine Casing

As for the previous part, the intercept of the regression line is higher for the coarse mesh than

for the fine. Again, this is expected. Both regression lines fall well within the acceptable

bounds, indicating a high degree of correlation between both models and experimental results.

In Figure 5.7a, the scatter of high MAC and AutoMAC is small, indicating a strong linear

relationship between the model and experimental results. In particular, there are no large

MAC or AutoMAC values off the diagonal, indicating a low degree of autocorrelation

between results and a high degree of certainty in selecting correlated mode pairs. Figure 5.7b

shows a near ideal result for a plot of this type: it is readily apparent that no significant MAC

or AutoMAC values appear off the regression line, the MAC values for correlated mode pairs


● ●● ● ●● ● ●

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●● ● ●● ● ● ●

y = 0.01375 + 0.9589x0.25

0.50

0.75

1.00

0.25 0.50 0.75 1.00Normalized Frequency from FEM

Nor

mal

ized

Fre

quen

cy fr

om E

xper

imen

tal R

esul

ts

(a) Coarse

● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ●

● ● ● ●

●

● ● ●

● ● ●

●

●

●

y = 0.002835 + 0.976x0.25

0.50

0.75

0.25 0.50 0.75Normalized Frequency from FEM

Nor

mal

ized

Fre

quen

cy fr

om E

xper

imen

tal R

esul

ts

(b) Fine

Figure 5.7: Natural Frequencies & MAC for Rear Compressor Meshes

987654321

0 1 2 3 4 5 6 7 8 9% Difference between FE and Experimental

Exp

erim

enta

l Mod

e #

(a) Coarse

987654321

0 1 2% Difference between FE and Experimental

Exp

erim

enta

l Mod

e #

(b) Fine

Figure 5.8: Natural Frequency Error for Rear Compressor Casing

are all close to 1, the regression line intercept is low, and the slope is close to 1.

Table 5.2 shows additional opportunities to improve the experiment. While the

paring of the 1st & 2nd, 3rd & 4th, and 9th & 10th FE modes appear to be reversed, the

frequencies for these modes are identical, so the order of those columns can be reversed to

create a diagonal trend with no change to the substance of the table. Of note is the difference

in paring for the 8th and 9th experimental frequencies.

For the coarse mesh, they appear to be paired with the 10th and 9th FE modes

respectively, while for the fine mesh, they are paired much more strongly with the 9th and 12th

FE modes. This may indicate a problem with the coarse mesh, and should be investigated


Table 5.2: Natural Frequencies & MAC for Rear Compressor Meshes

(a) Coarse

0

0.97

0

0

0

0

0

0

0

0.99

0

0

0

0

0

0

0

0

0

0

0

0.97

0

0

0

0

0

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0

0.97

0

0

0

0

0

0

0

0

0

0

0.98

0

0

0

0

0

0

0

0

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0

0

0

0

0

0

0

0.96

0.02

0

0

0

0

0

0

0

0.01

0.89

0

0

0

0

0

0

0

0

0

0

0.82

0

0

0

0

0

0

0

0.72

0

0

0

0

0

0

0

0

0.2

0.07

0

0

0

0

0

0

0

0.04

0.06

0.1

0.11

0.298

0.304

0.376

0.578

0.591

0.927

0.967

0.1 0.1

0.292

0.292

0.405

0.405

0.578

0.578

0.959

0.959 1 1


Nor

mal

ized

Fre

quen

cy fr

om E

xper

imen

tal R

esul

ts(b) Fine

0.98

0.03

0

0

0

0

0

0

0

0.01

0.95

0

0

0

0

0

0

0

0

0

0.98

0

0

0

0

0

0

0

0

0

0.97

0

0

0

0

0

0

0

0

0

0.98

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.98

0.06

0

0

0

0

0

0

0

0

0.85

0

0

0

0

0

0

0

0

0

0.96

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.02

0

0

0

0

0

0

0

0

0

0.96

0.09

0.1

0.275

0.28

0.347

0.533

0.545

0.855

0.892

0.09

0.1 0.279

0.285

0.352

0.359

0.541

0.554

0.873

0.887

0.903

0.913


Nor

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ized

Fre

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further if those higher mode shapes are of importance. Table 5.2b also shows 3 possible

missing modes from the experimental results corresponding to FE modes 6, 10, and 11.

Further signal analysis of the experimental data may be able to extract those missing modes if

they exist.

Chapter 6

Conclusions

This work has presented a comprehensive procedure for the development and validation of a

finite element vibration model, culminating in a demonstration of this procedure to validate a

pair of aircraft engine casing components. The MAC and FMAC show good correlation

between the models and the parts tested, and highlight issues and avenues of further

investigation not apparent through more basic correlation methods. For the rear compressor

casing, Table 5.2b indicates the likely absence of three modes from the experimental results.

For the front compressor casing, Table 5.1 indicates a missing mode near the 5th experimental

mode. These absences are not apparent from the FMAC alone, which show strong correlation

between the matched modes and frequencies. The use of both MAC tables and the FMAC in

concert is therefore shown to be an efficient and effective method of correlating normal modes

in structures.

The PCL algorithm was able to successfully determine a reduced model resulting in

high MAC values for two axisymmetric structures. In both cases, a high degree of scatter in

the MAC values was observed. However, the scatter appears biased, showing a distinct

distribution, leaving open the possibility that a near optimal results may be possible with a

smaller number of iterations.

36

CHAPTER 6. CONCLUSIONS 37

6.1 Contribution

The algorithm described in this work was able to locate the optimal MAC values for each set

of mode pairs in two axisymmetric structures. In addition, the FMAC has been extended to

show the confidence interval for the regression line, providing further information to the

analyst as to the reliability of their results. Coding for the PCL algorithm is provided in

Appendix B. This document was written in LATEX with knitr[75], the code for which is

available upon request for easy reproduction of Figures 5.1 to 5.8, Tables 5.1 and 5.2, and

associated analysis. An experimental setup and testing procedure were developed which

proved capable of measuring the natural frequencies and mode shapes of interest.

6.2 Recommendations for Future Research

Plotting of the MAC value for an axisymmetric part against the iteration number revealed a

large amount of scatter in the values. While some amount of scatter is unavoidable due to the

discretized nature of both the experimental sampling points and the FE model, more precise

modal expansion methods such as SEREP may have some beneficial impact. Their effect on

the scatter should be explored. Additionally, for many applications, obtaining the optimal

interation while minimizing the computation time may be of interest; the datasets presented

here should serve as a good basis from which to test various optimization techniques requiring

a smaller number of computations. Approaches for evaluating and limiting the effects of

heteroscedasticity in the fitted models should be considered.

References

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rational finite-element model selection for ten high-rise buildings based on on-site

ambient vibration measurements”, Journal of Performance of Constructed Facilities,

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Appendix A

Modelling

A.1 Finite Element

The FE modelling for this project required the meshing of 15 components, two of which are

referenced in this thesis. The geometry was determined from solid models provided by Pratt

& Whitney Canada. The actual material properties were not provided, as they were

considered proprietary, so the properties of similar generic materials were used instead. The

2016.1 version was used for all MSC products referenced in this thesis (MSC Nastran, MSC

Patran, MSC ProCOR).

Tables A.1 and A.2 provide the properties and number of elements used for the two

parts examined in this thesis. The two parts were also weighed, and their weights compared

against those predicted by their respective solid models; the percent discrepancy is also

Table A.1: Front Compressor Casing Mesh Statistics

Front Compressor CasingYoung’s Modulus 113.8 GPaPoisson’s Ratio 0.342Density 4.43 g/ccWeight Discrepancy from CAD 11.3%

Fine Mesh Coarse Mesh1220644 2ndorder TET 12760 1storder HEX

48

APPENDIX A. MODELLING 49

Table A.2: Rear Compressor Casing Mesh Statistics

Rear Compressor CasingYoung’s Modulus 204.8 GPaPoisson’s Ratio 0.291Density 8.22 g/ccWeight Discrepancy from CAD 3.12%

Fine Mesh Coarse Mesh1020642 2ndorder TET 45588 1storder HEX

348 1storder PENTA

provided in these tables.

The fine meshes were created using the Altair Hypermesh v13.0 solid automesher,

creating 2ndorder tet elements which closely followed the supplied CAD geometry. These

meshes included all the asymmetries of the actual part. The coarse meshes were created by

sweeping an approximate cross-section of the component about the central axis, resulting in a

fully axi-symmetric part. The coarse meshes are made using 1storder hex elements for

increased efficiency.

A.2 Experimental

Figure A.2 shows the sampling grid used to collect the experimental data used in this thesis.

The front compressor casing was modelled with 90 dofs, and the rear compressor casing was

modelled with 140 dofs. LMS Test Lab R16A was used for all analysis, processing data from

a LMS SCADAS Mobile 02 Mainframe. A PCB Piezotronics Model 356A31 triaxial

accelerometer and Model 086C03 general purpose modal analysis impact hammer with a

nylon tip were used for data collection. Figure A.1 shows sample FRFs from the components

tested in as part of this work.


Frequency

Am

plitu

de

(a) Front Compressor Casing

FrequencyA

mpl

itude

(b) Rear Compressor Casing

Figure A.1: Sample FRFs From Each Component

(a) Front Compressor Casing(b) Rear Compressor Casing

Figure A.2: Experimental Models


A.3 Additional Software

In addition to the modelling software discussed above, Table A.3 presents software used for

the analysis presented in this thesis. All listed software is open source, and available on

multiple platforms.

Table A.3: Additional Analysis Software

Software VersionR 3.3.2ggplot2 2.1.0ggExtra 0.5reshape2 1.4.1knitr 1.14

Appendix B

Coding for MAC Optimization in

Axisymmetric Structures

uil_file_new.go( "C:\MSC.Software\Patran_x64\20160\mscnastran_template.db", "C

:\Users\crabbitt\Desktop\PCL Practice\Part2\rot_practice1.db" )

set_current_dir( "C:\Users\crabbitt\Desktop\PCL Practice\Part2" )

nastran_input_import( "C:\Users\crabbitt\Desktop\PCL Practice\Part2\" // @

"preg.bdf", "default_group", 11, [TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, @

TRUE, FALSE, TRUE, TRUE], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, @

0, 0, 0, 0, 0, 0, 0], [-2000000000, -2000000000, -2000000000, @

-2000000000, -2000000000, -2000000000, -2000000000, -2000000000, 0, 0, 0] )

ga_view_aa_set( -67., 0., -34. )

uil_file_report.open_file( "mass_prop.rpt", 80 )

REAL mass_prop_mass_props_array(VIRTUAL)

STRING mass_prop_entity_list[VIRTUAL]

STRING mass_prop_reject_list[VIRTUAL]

mass_prop.cleanup_detailed( mass_prop_mass_props_array, mass_prop_entity_list,

@

mass_prop_reject_list )

INTEGER mass_prop_noutputs

STRING mass_prop_gnames_out[32](VIRTUAL)

INTEGER mass_prop_entities(VIRTUAL)

INTEGER mass_prop_entity_types(VIRTUAL)

INTEGER mass_prop_len_elist

INTEGER mass_prop_num_ents

INTEGER mass_prop_len_rlist

INTEGER mass_prop_num_rejects

INTEGER mass_prop_segment_id

INTEGER mass_prop_cord_label

INTEGER mass_prop_user_cord_label

52

APPENDIX B. CODING FOR MAC OPTIMIZATION IN AXISYMMETRIC STRUCTURES 53

mass_prop_create_detailed( "3D", "Group", "FEM", "Summary", 1, [ @

"default_group"], "", "Coord 0", "Use Element Properties", @

"Use Element Properties", "Coord 0", 3, 1, FALSE, FALSE, mass_prop_noutputs, @

mass_prop_gnames_out, mass_prop_entities, mass_prop_entity_types, @

mass_prop_mass_props_array, mass_prop_entity_list, mass_prop_len_elist, @

mass_prop_num_ents, mass_prop_reject_list, mass_prop_len_rlist, @

mass_prop_num_rejects, mass_prop_segment_id, mass_prop_cord_label, @

mass_prop_user_cord_label )

mass_prop.gen_report_detailed( 80, mass_prop_noutputs, "Summary", "FEM", @

mass_prop_gnames_out, mass_prop_entities, mass_prop_entity_types, @

mass_prop_mass_props_array, "Use Element Properties", @

mass_prop_user_cord_label, mass_prop_entity_list, mass_prop_num_ents, @

mass_prop_len_elist, mass_prop_reject_list, mass_prop_num_rejects, @

mass_prop_len_rlist )

ui_exec_function( "mesh_seed_display_mgr", "init" )

mesh_seed_display_mgr.erase( )

INTEGER chan1, integer_holder(8)

REAL real_holder(8)

STRING string_holder[80](8)

text_open("mass_prop.rpt", "OR", 0, 0, chan1)

While( text_read(chan1, "%OF%%A%%F15%", integer_holder(1), real_holder(1),

string_holder(1)) != -1)

IF(string_holder(1) == "X") THEN real_holder(2)=real_holder(1)

IF(string_holder(1) == "Y") THEN real_holder(3)=real_holder(1)

IF(string_holder(1) == "Z") THEN real_holder(4)=real_holder(1)

END While

STRING fem_create_nodes__nodes_created[VIRTUAL]

fem_create_nodes_1( "Coord 0", "Coord 0", 2, "999999", @

" [" // str_from_real(real_holder(2)) // " " // str_from_real(real_holder(3))

// @

" " // str_from_real(real_holder(4)) // "]", fem_create_nodes__nodes_created

)

$? NO 1

$? YES 6022095

mesh_seed_display_mgr.erase( )

$? NO 1

sys_free_string(fem_create_nodes__nodes_created)

jobfile.open( "pre-g", "ANALYZE NO JOBFILE" )

$? YES 6016027

msc_delete_old_files( "pre-g", ".bdf", ".op2" )

jobfile.write_spl( "/* Jobfile for PATNAS created %A% at %A% */", ["25-Apr-17"

@

, "17:33:19"] )

jobfile.writec( "", "TRANSLATOR = pat3nas" )

jobfile.writec( "DATABASE", "C:\Users\crabbitt\Desktop\PCL Practice\Part2\" //

@

"rot_practice1.db" )


jobfile.writec( "JOBNAME", "pre-g" )

jobfile.writec( "ANALYSIS TITLE", "" )

jobfile.writec( "ANALYSIS SUBTITLE", "" )

jobfile.writec( "ANALYSIS LABEL", "" )

jobfile.writec( "", "" )

jobfile.writec( "OBJECT", "Entire Model" )

jobfile.writec( "METHOD", "Full Run" )


jobfile.writec( "MODEL SUFFIX", ".bdf" )

jobfile.writec( "RESULTS SUFFIX", ".op2" )


jobfile.writec( "", "/*" )

jobfile.writec( "", " * File Search Path Declaration" )

jobfile.writec( "", " */" )


jobfile.writec( "File Search Path", "C:\Users\crabbitt" )

jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160" )

jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\help" //

@

"files" )

jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\alters"

)

jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\icons" )

jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\icon" //

@

"s\RibbonIcons" )

jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\bin" )

jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\bin\exe"

)

jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\msce" //

@

"xplore_files\" )

jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\mscp" //

@

"rocor_files\dmap" )


@

"rocor_files\plb" )


@

"rocor_files\lib" )


@

"rocor_files\icons" )

jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\shar" //

@

"eware\msc\unsupported\utilities\icons" )



@

"eware\msc\unsupported\utilities\plb" )


@

"eware\msc\unsupported\utilities\extra_files" )



jobfile.writec( "", " * Translation Parameters" )



jobfile.writec( "DATA OUTPUT", "OP2" )

jobfile.writec( "OUTPUT2 REQUESTS", "P3 Built In" )

jobfile.writec( "OUTPUT2 FORMAT", "Binary" )

jobfile.writec( "DIVISION TOLERANCE", "1.0e-08" )

jobfile.writec( "NUMERICAL TOLERANCE", "1.0e-04" )

jobfile.writec( "WRITING TOLERANCE", "1.0e-20" )

jobfile.writec( "GEOM CHECK", "INFORM" )

jobfile.writec( "SORTED BULK", "NO" )

jobfile.writec( "CARD FORMAT", "small" )

jobfile.writec( "NODE COORDINATES", "reference frame" )

jobfile.writec( "COORD COORDINATES", "global" )

jobfile.writec( "MSC.Nastran VERSION", "2016." )

jobfile.writec( "WRITE STORED PRECISION", "TRUE" )

jobfile.writec( "PROPS ON ELEM ENTRY", "FALSE" )

jobfile.writec( "CONTINUATION ENTRY", "FALSE" )

jobfile.writec( "PCOMPG ENTRY", "FALSE" )

jobfile.writec( "CONVERT CBAR CBEAM", "FALSE" )

jobfile.writec( "ITERATIVE SOLVER", "FALSE" )

jobfile.writei( "SUPER ELEMENT 0", 0 )

jobfile.writec( "SEALL WRITE", "FALSE" )

jobfile.writec( "PART SUPERELEMENT CREATE", "TRUE" )

jobfile.writec( "AUTOQSET", "FALSE" )

jobfile.writec( "FIXEDB", "FALSE" )

jobfile.writei( "SUPER TREE COUNT", 0 )

jobfile.writec( "MODEL TOLERANCE", "0.0049999999" )

jobfile.writec( "ELEMENT PROPERTY OFFSET", "0" )

jobfile.writec( "MATERIAL PROPERTY OFFSET", "0" )

jobfile.writec( "TABLE OFFSET", "0" )

jobfile.writec( "LOAD SET OFFSET", "0" )

jobfile.writec( "LOAD CASE OFFSET", "1" )

jobfile.writec( "CONTROL SET OFFSET", "0" )

jobfile.writec( "RIGID ELEMENT OFFSET", "0" )

jobfile.writec( "SCALAR POINT OFFSET", "0" )

jobfile.writec( "BEGINNING CONTINUATION MARKER", "+ A" )

jobfile.writec( "NUMBER ONLY", "ON" )

jobfile.writec( "BEGINNING NUMBER", "OFF" )


jobfile.writec( "TRAILING NUMBER", "OFF" )

jobfile.writec( "SYNTAX NUMBER", "ON" )

jobfile.writec( "SYNTAX MARKER", "." )

jobfile.writec( "EXTERNAL SUPERELEMENT METHOD", "NONE" )



jobfile.writec( "", " * Solution Parameters" )



jobfile.writec( "SOLUTION TYPE", "NORMAL MODES" )

jobfile.writei( "SOLUTION SEQUENCE", 103 )

jobfile.writec( "DATABASE RUN", "ON" )

jobfile.writec( "INTERACTIVE MODES", "OFF" )

jobfile.writec( "CYCLIC SYMMETRY", "OFF" )

jobfile.writec( "AUTOMATIC CONSTRAINTS", "ON" )

jobfile.writec( "SHELL NORMAL TOLERANCE", "" )

jobfile.writec( "MASS CALCULATION", "Lumped" )

jobfile.writec( "DATA DECK ECHO", "None" )

jobfile.writec( "PLATE RZ STIFFNESS FACTOR", "0.0" )

jobfile.writec( "MAXIMUM PRINTED LINES", "999999999" )

jobfile.writec( "MAXIMUM RUN TIME", "600" )

jobfile.writec( "WT-MASS CONVERSION", "1.0" )

jobfile.writec( "NODE ID FOR WT-GENER", "999999" )

jobfile.writec( "RIGID ELEMENT TYPE", "LINEAR" )

jobfile.writec( "USE SOL600", "OFF" )

jobfile.writec( "RESIDUAL VECTOR", "Nastran Default" )

jobfile.writec( "NEF STRESS UNITS", "MPA" )

jobfile.writec( "NEF STRESS COMBINATION", "ABSMAXPR" )

jobfile.writec( "NEF RESULT LOCATION", "NODA" )

jobfile.writec( "NEF RECOVERY LOCATION", "CORNER" )

jobfile.writec( "NEF LAYER LOCATION", "WORST" )

jobfile.writec( "NEF INTERPOLATION LIMIT", "0" )

jobfile.writec( "NEF LEVEL OF MESSAGE", "0" )

jobfile.writec( "NEF NUMBER OF THREADS", "1" )

jobfile.writer( "NEF CERTAINTY OF SURVIVAL", 50. )

jobfile.writer( "NEF OVERALL SCALE FACTOR", 1. )

jobfile.writer( "NEF TOP STRESS PERCENTAGE FILTER", 100. )

jobfile.writer( "NEF TOP DAMAGE PERCENTAGE FILTER", 100. )

jobfile.writec( "NEF TIME HISTORY DIRECTORY", "C:\Users\crabbitt" )

jobfile.writei( "NEF MAX LOAD PER EVENT", 1 )

jobfile.writec( "NEF EVENT DATA", "0" )

jobfile.writec( "NEF LOAD SEQUENCE DATA", "0" )

jobfile.writec( "USE CONTACT TABLE", "OFF" )

jobfile.writec( "INITIAL CONTACTPAIR LOADCASE NAME", "" )

jobfile.writei( "MDOF DATA", 0 )

jobfile.writec( "CELL WRITE", "ON" )

jobfile.writei( "CELL INPUT 0", 1 )


jobfile.writec( "CELL INPUT 1", "NASTRAN PARALLEL=20" )

jobfile.writec( "FMS WRITE", "ON" )

jobfile.writei( "FMS INPUT 0", 7 )

jobfile.writec( "FMS INPUT 1", "init master(s) $ do not save database" )

jobfile.writec( "FMS INPUT 2", "ASSIGN OUTPUT2='pregke.op2' ,UNIT=23,DELETE" )

jobfile.writec( "FMS INPUT 3", "ASSIGN OUTPUT2='pregdpr.op2',UNIT=41,DELETE" )

jobfile.writec( "FMS INPUT 4", "ASSIGN OUTPUT4='pre_grun.op4'," )

jobfile.writec( "FMS INPUT 5", " UNIT=31," )

jobfile.writec( "FMS INPUT 6", " FORM='FORMATTED', DELETE" )

jobfile.writec( "FMS INPUT 7", "$ Modes output to .pch file" )

jobfile.writec( "EXEC WRITE", "ON" )

jobfile.writei( "EXEC INPUT 0", 2 )

jobfile.writec( "EXEC INPUT 1", "INCLUDE 'C:\MSC.Software\Patran_x64\20160" //

@

"/mscprocor_files/dmap\p" )

jobfile.writec( "EXEC INPUT 2", "rocor.v2016'" )

jobfile.writec( "CASE WRITE", "ON" )

jobfile.writei( "CASE INPUT 0", 3 )

jobfile.writec( "CASE INPUT 1", "PARAM,RUNTYPE,GSET" )

jobfile.writec( "CASE INPUT 2", "PARAM,DMIGPHI,YES" )

jobfile.writec( "CASE INPUT 3", "PARAM,KEFILT,1.000E-8" )

jobfile.writec( "BULK WRITE", "ON" )

jobfile.writei( "BULK INPUT 0", 12 )

jobfile.writec( "BULK INPUT 1", "$ change gset disp, spcf, and ese patran " //

@

"output" )

jobfile.writec( "BULK INPUT 2", "DTI,PREGLAB,1,CR,1,DW,83,84,2" )

jobfile.writec( "BULK INPUT 3", ",PRE-G" )

jobfile.writec( "BULK INPUT 4", "DTI,PREGLAB,2,ER,ENDREC" )

jobfile.writec( "BULK INPUT 5", "$ change gset computed mode ke % label fo" //

@

"r patran output" )

jobfile.writec( "BULK INPUT 6", "DTI,PREGKE,1,CR,1,DW,83,89,7" )

jobfile.writec( "BULK INPUT 7", ",PREGKE,1,KINETIC,ENERGY" )

jobfile.writec( "BULK INPUT 8", "DTI,PREGKE,2,ER,ENDREC" )

jobfile.writec( "BULK INPUT 9", "$ change gset computed mode DPR label for" //

@

" patran output" )

jobfile.writec( "BULK INPUT 10", "DTI,PREGDPR,1,CR,1,DW,83,89,7" )

jobfile.writec( "BULK INPUT 11", ",PREGDPR,1,DPR--RA,W" )

jobfile.writec( "BULK INPUT 12", "DTI,PREGDPR,2,ER,ENDREC" )

jobfile.writec( "CELL DTI POSITION", "START" )

jobfile.writec( "FMS DTI POSITION", "START" )

jobfile.writec( "EXEC DTI POSITION", "END" )

jobfile.writec( "CASE DTI POSITION", "START" )

jobfile.writec( "BULK DTI POSITION", "START" )

jobfile.writec( "", "END" )


jobfile.close( )

$# Create "pre-g" loadcase

loadcase_create2( "pre-g", "Static", "", 1., [""], [0], [0.], "", 0., TRUE )

$# Create "pre-g" subcase

mscnastran_subcase.create( "103", "pre-g", "" )

mscnastran_subcase.create_char_param( "LOAD CASE", "pre-g" )

mscnastran_subcase.create_char_param( "SUBCASE TITLE", "" )

mscnastran_subcase.create_char_param( "SUBCASE SUBTITLE", "pre-g" )

mscnastran_subcase.create_char_param( "SUBCASE LABEL", "" )

mscnastran_subcase.create_char_param( "SUBCASE TITLE FLAG", "OFF" )

mscnastran_subcase.create_char_param( "SUBCASE SUBTITLE FLAG", "ON" )

mscnastran_subcase.create_char_param( "SUBCASE LABEL FLAG", "OFF" )

mscnastran_subcase.create_char_param( "pre-g", "STRUCTURAL" )

mscnastran_subcase.create_char_param( "CONSTRAINT FORCES", "1" )

mscnastran_subcase.create_char_param( "CONSTRAINT FORCES 1", @

"SPCFORCES=0" )

mscnastran_subcase.create_char_param( "ELEMENT STRAIN ENERGIES", "1" )

mscnastran_subcase.create_char_param( "ELEMENT STRAIN ENERGIES 1", "ESE=0" )

mscnastran_subcase.create_char_param( "EIGENVECTORS", "1" )

mscnastran_subcase.create_char_param( "EIGENVECTORS 1", "VECTOR=0" )

mscnastran_subcase.create_char_param( "SUBCASE WRITE", "ON" )

mscnastran_subcase.create_char_param( "SUBCASE DIRECT TEXT POS", "ON" )

mscnastran_subcase.create_int_param( "SUBCASE INPUT 0", 0 )

mscnastran_subcase.create_char_param( "REAL EIGENVALUE EXTRACTION METHOD", @

"Lanczos" )

mscnastran_subcase.create_char_param( "LOWER FREQUENCY RANGE", "10" )

mscnastran_subcase.create_char_param( "UPPER FREQUENCY RANGE", "1200" )

mscnastran_subcase.create_char_param( "REAL EIGNVALUE NUMBER OF DESIRED ROOTS"

@

, "44" )

mscnastran_subcase.create_char_param( "DIAGNOSTIC OUTPUT LEVEL", "0" )

mscnastran_subcase.create_char_param( "REAL EIGENVALUE NORMALIZATION METHOD",

@

"Mass" )

mscnastran_subcase.create_char_param( "ALL EXPLICIT MPCS", "ON" )

$# Start "pre-g" Run

mscnastran_job.associate_subcases( "103", "pre-g", 1, ["pre-g"] )

analysis_submit( "MSC.Nastran", "pre-g", TRUE)

sys_free_array(mass_prop_entities)

sys_free_array(mass_prop_entity_types)

sys_free_array(mass_prop_gnames_out)

sys_free_array(mass_prop_mass_props_array)

sys_free_string(mass_prop_entity_list)

sys_free_string(mass_prop_reject_list)

sys_free_string(fem_create_nodes__nodes_created)

$# Start reading in ASET Nodes & create map

Integer chan1, count, integer_holder(9), lrecl, nodecount, uff_trace(VIRTUAL)


Integer count3, count2

real real_holder(8), uff_coords(VIRTUAL)

string string_holder[80](8), record[1]

sys_allocate_array(uff_coords, 1,300, 1,3)

sys_allocate_array(uff_trace, 1, 900)

text_open("Experiment2_rot.unv", "OR", 0, 0, chan1)

real_holder(1) = 1

integer_holder(1) = 1

count = 0

nodecount = 0

count2 = 0

count3 = 0

while( text_read(chan1, "%OF%%I8%", integer_holder(1), real_holder(1),


count += 1

IF(integer_holder(1)==15) THEN

text_read(chan1, "%OF%%I8%%33W%%3F13%", integer_holder(1:3), real_holder

(1:3), string_holder(1:3))

while(integer_holder(1) != -1)

count += 1

nodecount +=1

uff_coords(nodecount,1:3) = real_holder(1:3)

IF(count > 422000) THEN BREAK

text_read(chan1, "%OF%%I8%%33W%%3F13%", integer_holder(1:3),

real_holder(1:3), string_holder(1:3))

end while

end IF

IF(integer_holder(1)==82) THEN

Integer i

text_read(chan1, "%OF%%8I10%", integer_holder(1:8), real_holder(1:8),

string_holder(1:8))


string_holder(1:8))


string_holder(1:8))

while(integer_holder(1) != -1)

count2 += 1

For(i=1 to 8)

uff_trace(count3*8 + i) = integer_holder(i)

End For

count3 += 1

count += 1



string_holder(1:8))

end while

end IF



end while

text_close(chan1, "")

sys_reallocate_array(uff_trace, 1, count2*8)

count = 0

$# Variables for nodematch

String match_out[VIRTUAL], string1[1], groupnodes[14], string2[14], string3[

VIRTUAL]

String node_coord[40], str_max[12], deleted_nodes[VIRTUAL], not_found_nodes[

VIRTUAL]

Real distance_out(VIRTUAL), real1

Integer length_out, entity_count, node1

Integer node_map(VIRTUAL), max_node, int1(1)

ga_group_create("UFF_nodes")

ga_group_current_set("UFF_nodes")

db_count_nodes_in_group(1,node1)

sys_allocate_array(node_map, 1, node1-1)

groupnodes = "Node 1:" // str_from_integer(node1-1)

while(count < nodecount)

count += 1

string_write("[%3E%]", int1 , uff_coords(count,1:3), string1, node_coord)

fem_create_nodes_1( "Coord 0", "Coord 0", 2, "#", node_coord, string3 )

db_get_max_node_id(max_node)

str_max = "Node "// str_from_integer(max_node)

fem_geom_match_closest(str_max, groupnodes, @

entity_count, match_out, distance_out, length_out)

node_map(count) = str_to_integer(str_substr(match_out, 5, 6))

sys_free_string(match_out)

sys_free_array(distance_out)

String match_out[VIRTUAL]

Real distance_out(VIRTUAL)

fem_delete_free_node(str_max, deleted_nodes, not_found_nodes )

End while

$# Clear variables from nodematch


sys_free_string(deleted_nodes)

sys_free_string(not_found_nodes)


$# Populate ASET elements

Integer uff_trace2(VIRTUAL)

sys_allocate_array(uff_trace2, 1, 2, 1, count2*8)

count3 = 0

for(i=1 to count2*8)

IF(uff_trace(i) == 0) THEN CONTINUE

IF(uff_trace(i+1) == 0) THEN CONTINUE

count3 += 1

uff_trace2(1, count3) = node_map(uff_trace(i))


uff_trace2(2, count3) = node_map(uff_trace(i+1))

End for

String string_holder2[VIRTUAL](2), traceformat[12]

sys_allocate_string(string_holder2, 2000)

traceformat = "Node %" // str_from_integer(count3) // "I%"

string_write(traceformat, uff_trace2(1,1:300), real_holder, string_holder,

string_holder2(1))


string_holder2(2))

fem_create_elems( "Bar ", "Bar2", "#", "Standard", TRUE, string_holder2(1),

string_holder2(2), "", "", "", "", "", "", "" )

$# sys_free_array(uff_trace)

sys_free_array(uff_trace2)

sys_free_string(string3)

sys_free_string(string_holder2)

Integer max_elem, min_elem

db_get_max_elem_id(max_elem)

min_elem = max_elem - count3 + 1

elementprops_create( "uff_traceline", 67, 25, 20, 28, 2, 34, [40], [4], [ @

"DUMMY"], "Element " // str_from_integer(min_elem) // ":" // str_from_integer(

max_elem) )

$# Start Pre-Test ASET Analysis

msc_delete_old_files( "pre-a", ".bdf", ".op2" )

jobfile.open( "pre-a", "ANALYZE NO JOBFILE" )


@

, "17:37:19"] )



@


jobfile.writec( "JOBNAME", "pre-a" )

















@

"xplore_files\" )


)



@

"s\RibbonIcons" )



)


@

"files" )



@



@



@



@



@

"rocor_files\lib" )


@

"rocor_files\plb" )


@

































































































jobfile.writec( "FMS INPUT 2", "ASSIGN OUTPUT2='preake.op2' ,UNIT=24,DELETE" )

jobfile.writec( "FMS INPUT 3", "ASSIGN OUTPUT2='preadpr.op2',UNIT=42,DELETE" )

jobfile.writec( "FMS INPUT 4", "$" )

jobfile.writec( "FMS INPUT 5", "$ Mode input included in bulk data (.dmig " //

@

"or .pch)" )


jobfile.writec( "FMS INPUT 7", "ASSIGN OUTPUT4='pre_arun.op4'," )







@





jobfile.writec( "CASE INPUT 1", "PARAM,RUNTYPE,ASET" )

jobfile.writec( "CASE INPUT 2", "PARAM,UT1UT2,NO" )

jobfile.writec( "CASE INPUT 3", "$" )





jobfile.writec( "BULK INPUT 1", "INCLUDE 'pre-g.pch'" )

jobfile.writec( "BULK INPUT 2", "$ change aset disp, spcf, and ese patran " //

@

"output" )

jobfile.writec( "BULK INPUT 3", "DTI,PREALAB,1,CR,1,DW,83,84,2" )

jobfile.writec( "BULK INPUT 4", ",PRE-A" )

jobfile.writec( "BULK INPUT 5", "DTI,PREALAB,2,ER,ENDREC" )

jobfile.writec( "BULK INPUT 6", "$ change aset computed mode ke % label fo" //

@

"r patran output" )

jobfile.writec( "BULK INPUT 7", "DTI,PREAKE,1,CR,1,DW,83,89,7" )

jobfile.writec( "BULK INPUT 8", ",PREAKE,1,KINETIC,ENERGY" )

jobfile.writec( "BULK INPUT 9", "DTI,PREAKE,2,ER,ENDREC" )

jobfile.writec( "BULK INPUT 10", "$ change aset computed mode DPR label fo" //

@

"r patran output" )

jobfile.writec( "BULK INPUT 11", "DTI,PREADPR,1,CR,1,DW,83,89,7" )

jobfile.writec( "BULK INPUT 12", ",PREADPR,1,DPR--RA,W" )

jobfile.writec( "BULK INPUT 13", "DTI,PREADPR,2,ER,ENDREC" )

STRING BULK_LINE[14]

STRING ASET_NODE[VIRTUAL]

SYS_ALLOCATE_STRING(ASET_NODE, 22)

FOR(i=1 to count3)

BULK_LINE = "BULK INPUT " // str_from_integer(13+i)

ASET_NODE = "ASET1 123 " // str_from_integer(node_map(i))

jobfile.writec( BULK_LINE, ASET_NODE)

END FOR








jobfile.close( )

$# Create "pre-a" loadcase

loadcase_create2( "pre-a", "Static", "", 1., [""], [0], [0.], "", 0., TRUE )

$# Create "pre-a" subcase

mscnastran_subcase.create( "103", "pre-a", "" )

mscnastran_subcase.create_char_param( "LOAD CASE", "pre-a" )


mscnastran_subcase.create_char_param( "SUBCASE SUBTITLE", "pre-a" )





mscnastran_subcase.create_char_param( "pre-a", "STRUCTURAL" )



"SPCFORCES=0" )









"Lanczos" )




@

, "44" )



@

"Mass" )


$# Start "pre-a" Run

mscnastran_job.associate_subcases( "103", "pre-a", 1, ["pre-a"] )

analysis_submit( "MSC.Nastran", "pre-a", TRUE )

$# Import Test Frequencies and Mode Shapes

Integer chan2

text_open("test_shapes_original.pch", "OR", 0, 0, chan1)

text_open("test_shapes.pch", "NORW", 0, 0, chan2)

while(text_read(chan1, "%OF%%A15%%I9%%I8%%I8%%A16%%I16%", integer_holder(1),

real_holder(1), string_holder(1)) != -1)

If(integer_holder(1) == 0) THEN

IF(integer_holder(2) != 0) THEN


text_write(chan2, "%A15%%I9%%I8%%I8%%A16%%I16%", integer_holder(1),

real_holder(1), string_holder(1))

CONTINUE

End If

integer_holder(1) = integer_holder(3)


text_write(chan2, "%A15%%W9%%I16%%I16%", integer_holder(1), real_holder

(1), string_holder(1))

CONTINUE

End If

If(integer_holder(1) != 0) THEN

integer_holder(1) = node_map(integer_holder(1))


text_write(chan2, "%A15%%I9%%I16%%A16%", integer_holder(1), real_holder

(1), string_holder(1))

CONTINUE

End If

End while



$# Run post-test ASET analysis

msc_delete_old_files( "post-a", ".bdf", ".op2" )

jobfile.open( "post-a", "ANALYZE NO JOBFILE" )


@

, "17:39:10"] )



@


jobfile.writec( "JOBNAME", "post-a" )
















@





@

"files" )


)



@

"s\RibbonIcons" )



)



@



@

"rocor_files\plb" )


@

"rocor_files\lib" )


@



@



@



@

"xplore_files\" )































































































jobfile.writec( "FMS INPUT 2", "ASSIGN OUTPUT4='post_arun.op4',UNIT=34,FOR" //

@

"M='FORMATTED' DELETE" )

jobfile.writec( "FMS INPUT 3", "ASSIGN OUTPUT2='posttphi.op2',UNIT=26,DELETE"

)

jobfile.writec( "FMS INPUT 4", "ASSIGN OUTPUT2='postake.op2' ,UNIT=27,DELETE"

)

jobfile.writec( "FMS INPUT 5", "ASSIGN OUTPUT2='posttke.op2' ,UNIT=28,DELETE"

)

jobfile.writec( "FMS INPUT 6", "ASSIGN OUTPUT2='postadpr.op2',UNIT=43,DELETE"

)

jobfile.writec( "FMS INPUT 7", "ASSIGN OUTPUT2='posttdpr.op2',UNIT=44,DELETE"

)





@





jobfile.writec( "CASE INPUT 1", "PARAM,RUNTYPE,TCOR" )


jobfile.writec( "CASE INPUT 3", "PARAM,GRAM,NONE" )

jobfile.writec( "CASE INPUT 4", "PARAM,DMIGPHI,NO" )

jobfile.writec( "CASE INPUT 5", "PARAM,BKEXPTST,YES" )

jobfile.writec( "CASE INPUT 6", "PARAM,GCOMP,NO" )



jobfile.writec( "BULK INPUT 1", "$ change gset disp, spcf, and ese patran " //

@

"output" )

jobfile.writec( "BULK INPUT 2", "DTI,POSTLAB,1,CR,1,DW,83,84,2" )

jobfile.writec( "BULK INPUT 3", ",POST-A" )

jobfile.writec( "BULK INPUT 4", "DTI,POSTLAB,2,ER,ENDREC" )

jobfile.writec( "BULK INPUT 5", "$ change test mode shape label for patran" //

@

" output" )

jobfile.writec( "BULK INPUT 6", "DTI,POSTTPHI,1,CR,1,DW,83,89,7" )

jobfile.writec( "BULK INPUT 7", ",POSTTPHI,1,DISPLACE,MENT" )

jobfile.writec( "BULK INPUT 8", "DTI,POSTTPHI,2,ER,ENDREC" )

jobfile.writec( "BULK INPUT 9", "$ change aset computed mode ke % label fo" //

@

"r patran output" )

jobfile.writec( "BULK INPUT 10", "DTI,POSTAKE,1,CR,1,DW,83,89,7" )

jobfile.writec( "BULK INPUT 11", ",POSTAKE,1,KINETIC,ENERGY" )

jobfile.writec( "BULK INPUT 12", "DTI,POSTAKE,2,ER,ENDREC" )

jobfile.writec( "BULK INPUT 13", "$ change test mode ke % label for patran" //

@

" output" )

jobfile.writec( "BULK INPUT 14", "DTI,POSTTKE,1,CR,1,DW,83,89,7" )

jobfile.writec( "BULK INPUT 15", ",POSTTKE,1,KINETIC,ENERGY" )

jobfile.writec( "BULK INPUT 16", "DTI,POSTTKE,2,ER,ENDREC" )

jobfile.writec( "BULK INPUT 17", "$ change aset computed mode DPR label fo" //

@

"r patran output" )

jobfile.writec( "BULK INPUT 18", "DTI,POSTADPR,1,CR,1,DW,83,89,7" )

jobfile.writec( "BULK INPUT 19", ",POSTADPR,1,DPRA-RA,W" )

jobfile.writec( "BULK INPUT 20", "DTI,POSTADPR,2,ER,ENDREC" )


jobfile.writec( "BULK INPUT 21", "$ change test computed mode DPR label fo" //

@

"r patran output" )

jobfile.writec( "BULK INPUT 22", "DTI,POSTTDPR,1,CR,1,DW,83,89,7" )

jobfile.writec( "BULK INPUT 23", ",POSTTDPR,1,DPRT-RA,W" )

jobfile.writec( "BULK INPUT 24", "DTI,POSTTDPR,2,ER,ENDREC" )

jobfile.writec( "BULK INPUT 25", "INCLUDE 'test_freqs.dmi'" )

jobfile.writec( "BULK INPUT 26", "INCLUDE 'test_shapes.pch'" )

FOR(i=1 to count3)




END FOR







jobfile.close( )

$# Create "post-a" loadcase

loadcase_create2( "post-a", "Static", "", 1., [""], [0], [0.], "", 0., TRUE )

$# Create "post-a" subcase

mscnastran_subcase.create( "103", "post-a", "" )

mscnastran_subcase.create_char_param( "LOAD CASE", "post-a" )


mscnastran_subcase.create_char_param( "SUBCASE SUBTITLE", "post-a" )





mscnastran_subcase.create_char_param( "post-a", "STRUCTURAL" )



"SPCFORCES=0" )









"Lanczos" )





@

, "44" )



@

"Mass" )


$# Start "post-a" Run

mscnastran_job.associate_subcases( "103", "post-a", 1, ["post-a"] )

analysis_submit( "MSC.Nastran", "post-a" , TRUE)

$# Extract and average MAC values

text_open("post_arun.op4", "OR", 0, 0, chan1)

Real MAC_line(VIRTUAL), MAC1

Integer i, i2

sys_allocate_array(MAC_line, 1, 15, 1, 15)

count = 0

while( text_read(chan1, "%OF%%3I%%A%", integer_holder(3:5), real_holder(1),


IF(string_holder(1) == "1MACTA") THEN

i = integer_holder(4)

i2 = integer_holder(3)

For(count = 1 to i2)

text_read(chan1, "%3I%", integer_holder(1), real_holder(1),

string_holder(1))

text_read(chan1, "%" // str_from_integer(i) // "F%", integer_holder(1)

, MAC_line(count, 1:15), string_holder(1))

End For

End IF

End While

INTEGER count4

Real MAC_Avg(15)

For(count4 = 1 to i)

MAC1 = 0


IF(MAC_line(count, count4) > MAC1) THEN MAC1 = MAC_line(count, count4)

End For

MAC_Avg(count4) = MAC1

End For

$# sys_free_array(MAC_line)


REAL theta, MAC_vec(VIRTUAL)

sys_allocate_array(MAC_vec, 1, 361, 1, 15)

MAC_vec(1,1:15) = MAC_Avg(1:15)

file_delete("test_shapes.pch")

file_delete("pre-a.f06")

file_delete("post-a.f06")




file_delete("pre-a.bdf")

file_delete("post-a.bdf")

file_delete("post-a.op2")

file_delete("postadpr.op2")

file_delete("postake.op2")

file_delete("posttdpr.op2")

file_delete("posttke.op2")

file_delete("posttphi.op2")

file_delete("pre-a.op2")

file_delete("preadpr.op2")

file_delete("preake.op2")

file_delete("post-a.log")

file_delete("pre-a.log")

analysis_main_tasks.delete_job( "MSC.Nastran", "pre-a" )

analysis_main_tasks.delete_job( "MSC.Nastran", "post-a" )

ga_group_members_delete("")

text_open("Mode_vector.txt", "NORW", 0, 0, chan2)

text_write(chan2,"%11G%", integer_holder, MAC_vec(1,1:(count4-1)),

string_holder)


theta = 0

$# Looping for optimization

theta = 30

FOR(theta=121 to 360 by 1)

$# Variables for nodematch

String match_out[VIRTUAL], string3[VIRTUAL]

String deleted_nodes[VIRTUAL], not_found_nodes[VIRTUAL]

Real distance_out(VIRTUAL), uff_coords2(VIRTUAL)

sys_allocate_array(uff_coords2, 1, nodecount, 1, 3)

count = 0


count += 1

uff_coords2(count,1) = uff_coords(count,1)*mth_cosd(theta) - uff_coords(

count,2)*mth_sind(theta)

uff_coords2(count,2) = uff_coords(count,1)*mth_sind(theta) + uff_coords(

count,2)*mth_cosd(theta)

uff_coords2(count,3) = uff_coords(count,3)

End While

count = 0


count += 1

string_write("[%3E%]", int1 , uff_coords2(count,1:3), string1, node_coord

)

fem_create_nodes_1( "Coord 0", "Coord 0", 2, "#", node_coord, string3 )

db_get_max_node_id(max_node)


str_max = "Node "// str_from_integer(max_node)

fem_geom_match_closest(str_max, groupnodes, @

entity_count, match_out, distance_out, length_out)

node_map(count) = str_to_integer(str_substr(match_out, 5, 6))



String match_out[VIRTUAL]

Real distance_out(VIRTUAL)

fem_delete_free_node(str_max, deleted_nodes, not_found_nodes )

End while

$# Clear variables from nodematch


sys_free_string(deleted_nodes)

sys_free_string(not_found_nodes)


sys_free_array(uff_coords2)

$# Populate ASET elements

Integer uff_trace2(VIRTUAL)

sys_allocate_array(uff_trace2, 1, 2, 1, count2*8)

count3 = 0

for(i=1 to count2*8)

IF(uff_trace(i) == 0) THEN CONTINUE

IF(uff_trace(i+1) == 0) THEN CONTINUE

count3 += 1

uff_trace2(1, count3) = node_map(uff_trace(i))

uff_trace2(2, count3) = node_map(uff_trace(i+1))

End for

String string_holder2[VIRTUAL](2), traceformat[12]

sys_allocate_string(string_holder2, 2000)

traceformat = "Node %" // str_from_integer(count3) // "I%"


string_holder2(1))


string_holder2(2))

fem_create_elems( "Bar ", "Bar2", "#", "Standard", TRUE, string_holder2(1),

string_holder2(2), "", "", "", "", "", "", "" )

elementprops_delete( ["uff_traceline"] )

sandia_utils.delete_n_metadata( 1, ["uff_traceline"] )

elementprops_create( "uff_traceline", 67, 25, 20, 28, 2, 34, [40], [4], [ @

"DUMMY"], "Element " // str_from_integer(min_elem) // ":" //

str_from_integer(max_elem) )

sys_free_array(uff_trace2)

sys_free_string(string3)

sys_free_string(string_holder2)

$# Start Pre-Test ASET Analysis

msc_delete_old_files( "pre-a", ".bdf", ".op2" )

jobfile.open( "pre-a", "ANALYZE NO JOBFILE" )


jobfile.write_spl( "/* Jobfile for PATNAS created %A% at %A% */", ["25-Apr

-17" @

, "17:37:19"] )


jobfile.writec( "DATABASE", "C:\Users\crabbitt\Desktop\PCL Practice\Part2\"

// @


jobfile.writec( "JOBNAME", "pre-a" )















jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\msce"

// @

"xplore_files\" )

jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\bin\

exe" )

jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\bin"

)

jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\icon"

// @

"s\RibbonIcons" )

jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\icons

" )

jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\

alters" )

jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\help"

// @

"files" )


jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\shar"

// @



// @




// @


jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\mscp"

// @



// @

"rocor_files\lib" )


// @

"rocor_files\plb" )


// @

































































































jobfile.writec( "FMS INPUT 2", "ASSIGN OUTPUT2='preake.op2' ,UNIT=24,DELETE

" )

jobfile.writec( "FMS INPUT 3", "ASSIGN OUTPUT2='preadpr.op2',UNIT=42,DELETE

" )


jobfile.writec( "FMS INPUT 5", "$ Mode input included in bulk data (.dmig "

// @

"or .pch)" )


jobfile.writec( "FMS INPUT 7", "ASSIGN OUTPUT4='pre_arun.op4'," )





jobfile.writec( "EXEC INPUT 1", "INCLUDE 'C:\MSC.Software\Patran_x64\20160"

// @





jobfile.writec( "CASE INPUT 1", "PARAM,RUNTYPE,ASET" )

jobfile.writec( "CASE INPUT 2", "PARAM,UT1UT2,NO" )

jobfile.writec( "CASE INPUT 3", "$" )





jobfile.writec( "BULK INPUT 1", "INCLUDE 'pre-g.pch'" )

jobfile.writec( "BULK INPUT 2", "$ change aset disp, spcf, and ese patran "

// @

"output" )

jobfile.writec( "BULK INPUT 3", "DTI,PREALAB,1,CR,1,DW,83,84,2" )

jobfile.writec( "BULK INPUT 4", ",PRE-A" )


jobfile.writec( "BULK INPUT 5", "DTI,PREALAB,2,ER,ENDREC" )

jobfile.writec( "BULK INPUT 6", "$ change aset computed mode ke % label fo"

// @

"r patran output" )

jobfile.writec( "BULK INPUT 7", "DTI,PREAKE,1,CR,1,DW,83,89,7" )

jobfile.writec( "BULK INPUT 8", ",PREAKE,1,KINETIC,ENERGY" )

jobfile.writec( "BULK INPUT 9", "DTI,PREAKE,2,ER,ENDREC" )

jobfile.writec( "BULK INPUT 10", "$ change aset computed mode DPR label fo"

// @

"r patran output" )

jobfile.writec( "BULK INPUT 11", "DTI,PREADPR,1,CR,1,DW,83,89,7" )

jobfile.writec( "BULK INPUT 12", ",PREADPR,1,DPR--RA,W" )

jobfile.writec( "BULK INPUT 13", "DTI,PREADPR,2,ER,ENDREC" )

STRING BULK_LINE[14]

FOR(i=1 to count3)




END FOR







jobfile.close( )

$# Start "pre-a" Run

mscnastran_job.associate_subcases( "103", "pre-a", 1, ["pre-a"] )

analysis_submit( "MSC.Nastran", "pre-a", TRUE )

$# Import Test Frequencies and Mode Shapes

text_open("test_shapes_original.pch", "OR", 0, 0, chan1)

text_open("test_shapes2.pch", "NORW", 0, 0, chan2)

REAL temp1234(3)

while(text_read(chan1, "%OF%%A15%%I9%%I8%%I8%%A16%%I16%", integer_holder(1)

, real_holder(1), string_holder(1)) != -1)





CONTINUE

End If



text_write(chan2, "%A15%%W9%%I16%%I16%", integer_holder(1),


CONTINUE

End If




temp1234(1) = str_to_real(string_holder(2))/1100



CONTINUE

End If


temp1234(2) = str_to_real(string_holder(2))/1100



CONTINUE

End If



real_holder(1) = temp1234(1)*mth_cosd(theta) - temp1234(2)*mth_sind

(theta)



text_write(chan2, "%A15%%I9%%I16%%E16.5%", integer_holder,

real_holder, string_holder)

real_holder(1) = temp1234(1)*mth_sind(theta) + temp1234(2)*mth_cosd

(theta)





real_holder(1) = str_to_real(string_holder(2))/1100





CONTINUE

End If

CONTINUE

End If

End while



text_open("test_shapes2.pch", "OR", 0, 0, chan1)

text_open("test_shapes.pch", "NORW", 0, 0, chan2)

while(text_read(chan1, "%OF%%A15%%I9%%I8%%I8%%A16%%I16%", integer_holder(1)

, real_holder(1), string_holder(1)) != -1)






CONTINUE

End If



text_write(chan2, "%A15%%W9%%I16%%I16%", integer_holder(1),


CONTINUE

End If



str_assign(string_holder(2), 14, 2, "D-")

text_write(chan2, "%A15%%I9%%I16%%A16%", integer_holder(1),


CONTINUE

End If

End while



$# Run post-test ASET analysis

msc_delete_old_files( "post-a", ".bdf", ".op2" )

jobfile.open( "post-a", "ANALYZE NO JOBFILE" )

jobfile.write_spl( "/* Jobfile for PATNAS created %A% at %A% */", ["25-Apr

-17" @

, "17:39:10"] )


jobfile.writec( "DATABASE", "C:\Users\crabbitt\Desktop\PCL Practice\Part2\"

// @


jobfile.writec( "JOBNAME", "post-a" )
















// @




jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\help"

// @

"files" )

jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\

alters" )

jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\icons

" )

jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\icon"

// @

"s\RibbonIcons" )

jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\bin"

)

jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\bin\

exe" )



// @



// @

"rocor_files\plb" )


// @

"rocor_files\lib" )


// @



// @



// @


jobfile.writec( "File Search Path", "C:\MSC.Software\Patran_x64\20160\msce"

// @

"xplore_files\" )































































































jobfile.writec( "FMS INPUT 2", "ASSIGN OUTPUT4='post_arun.op4',UNIT=34,FOR"

// @

"M='FORMATTED' DELETE" )

jobfile.writec( "FMS INPUT 3", "ASSIGN OUTPUT2='posttphi.op2',UNIT=26,

DELETE" )

jobfile.writec( "FMS INPUT 4", "ASSIGN OUTPUT2='postake.op2' ,UNIT=27,

DELETE" )

jobfile.writec( "FMS INPUT 5", "ASSIGN OUTPUT2='posttke.op2' ,UNIT=28,

DELETE" )

jobfile.writec( "FMS INPUT 6", "ASSIGN OUTPUT2='postadpr.op2',UNIT=43,

DELETE" )

jobfile.writec( "FMS INPUT 7", "ASSIGN OUTPUT2='posttdpr.op2',UNIT=44,

DELETE" )




jobfile.writec( "EXEC INPUT 1", "INCLUDE 'C:\MSC.Software\Patran_x64\20160"

// @





jobfile.writec( "CASE INPUT 1", "PARAM,RUNTYPE,TCOR" )


jobfile.writec( "CASE INPUT 3", "PARAM,GRAM,NONE" )

jobfile.writec( "CASE INPUT 4", "PARAM,DMIGPHI,NO" )

jobfile.writec( "CASE INPUT 5", "PARAM,BKEXPTST,YES" )

jobfile.writec( "CASE INPUT 6", "PARAM,GCOMP,NO" )



jobfile.writec( "BULK INPUT 1", "$ change gset disp, spcf, and ese patran "

// @

"output" )

jobfile.writec( "BULK INPUT 2", "DTI,POSTLAB,1,CR,1,DW,83,84,2" )

jobfile.writec( "BULK INPUT 3", ",POST-A" )

jobfile.writec( "BULK INPUT 4", "DTI,POSTLAB,2,ER,ENDREC" )

jobfile.writec( "BULK INPUT 5", "$ change test mode shape label for patran"

// @

" output" )

jobfile.writec( "BULK INPUT 6", "DTI,POSTTPHI,1,CR,1,DW,83,89,7" )

jobfile.writec( "BULK INPUT 7", ",POSTTPHI,1,DISPLACE,MENT" )

jobfile.writec( "BULK INPUT 8", "DTI,POSTTPHI,2,ER,ENDREC" )

jobfile.writec( "BULK INPUT 9", "$ change aset computed mode ke % label fo"

// @

"r patran output" )

jobfile.writec( "BULK INPUT 10", "DTI,POSTAKE,1,CR,1,DW,83,89,7" )

jobfile.writec( "BULK INPUT 11", ",POSTAKE,1,KINETIC,ENERGY" )

jobfile.writec( "BULK INPUT 12", "DTI,POSTAKE,2,ER,ENDREC" )

jobfile.writec( "BULK INPUT 13", "$ change test mode ke % label for patran"

// @

" output" )

jobfile.writec( "BULK INPUT 14", "DTI,POSTTKE,1,CR,1,DW,83,89,7" )

jobfile.writec( "BULK INPUT 15", ",POSTTKE,1,KINETIC,ENERGY" )

jobfile.writec( "BULK INPUT 16", "DTI,POSTTKE,2,ER,ENDREC" )

jobfile.writec( "BULK INPUT 17", "$ change aset computed mode DPR label fo"

// @

"r patran output" )

jobfile.writec( "BULK INPUT 18", "DTI,POSTADPR,1,CR,1,DW,83,89,7" )

jobfile.writec( "BULK INPUT 19", ",POSTADPR,1,DPRA-RA,W" )

jobfile.writec( "BULK INPUT 20", "DTI,POSTADPR,2,ER,ENDREC" )


jobfile.writec( "BULK INPUT 21", "$ change test computed mode DPR label fo"

// @

"r patran output" )

jobfile.writec( "BULK INPUT 22", "DTI,POSTTDPR,1,CR,1,DW,83,89,7" )

jobfile.writec( "BULK INPUT 23", ",POSTTDPR,1,DPRT-RA,W" )

jobfile.writec( "BULK INPUT 24", "DTI,POSTTDPR,2,ER,ENDREC" )

jobfile.writec( "BULK INPUT 25", "INCLUDE 'test_freqs.dmi'" )

jobfile.writec( "BULK INPUT 26", "INCLUDE 'test_shapes.pch'" )

FOR(i=1 to count3)




END FOR







jobfile.close( )

$# Start "post-a" Run

mscnastran_job.associate_subcases( "103", "post-a", 1, ["post-a"] )

analysis_submit( "MSC.Nastran", "post-a" , TRUE)

$# Extract and average MAC values

text_open("post_arun.op4", "OR", 0, 0, chan1)

Real MAC_line(VIRTUAL), MAC1

Integer i, i2

sys_allocate_array(MAC_line, 1, 20, 1, 20)

count = 0

while( text_read(chan1, "%OF%%3I%%A%", integer_holder(3:5), real_holder(1),


IF(string_holder(1) == "1MACTA") THEN

i = integer_holder(4)

i2 = integer_holder(3)


text_read(chan1, "%3I%", integer_holder(1), real_holder(1),

string_holder(1))

text_read(chan1, "%" // str_from_integer(i) // "F%", integer_holder

(1), MAC_line(count, 1:15), string_holder(1))

End For

End IF

End While

INTEGER count4

Real MAC_Avg(15)

For(count4 = 1 to i)

MAC1 = 0



IF(MAC_line(count, count4) > MAC1) THEN MAC1 = MAC_line(count, count4)

End For

MAC_Avg(count4) = MAC1

End For

$# sys_free_array(MAC_line)


MAC_vec(theta+1,1:15) = MAC_Avg(1:15)

file_delete("test_shapes.pch")





file_delete("pre-a.bdf")

file_delete("post-a.bdf")

file_delete("post-a.op2")

file_delete("postadpr.op2")

file_delete("postake.op2")

file_delete("posttdpr.op2")

file_delete("posttke.op2")

file_delete("posttphi.op2")

file_delete("pre-a.op2")

file_delete("preadpr.op2")

file_delete("preake.op2")

file_delete("post-a.log")

file_delete("pre-a.log")

analysis_main_tasks.delete_job( "MSC.Nastran", "pre-a" )

analysis_main_tasks.delete_job( "MSC.Nastran", "post-a" )

ga_group_members_delete("")

End FOR

sys_free_array(uff_coords)

sys_free_array(node_map)

sys_free_string(ASET_NODE)

sys_free_array(MAC_line)

sys_free_array(uff_trace)

text_open("Mode_vector.txt", "NORW", 0, 0, chan2)

For(i=1 to 361)

text_write(chan2,"%11G%", integer_holder, MAC_vec(i,1:(count4-1)),

string_holder)

End FOR
