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Table of Contents

Dedication

List of Figures

List of Tables

List of Abbreviations and Spbols

Acknowledgments

Abstract

Chapter 1 Introduction

1.1 General

1.2 Literature Review

1.3 Scope of Worfic

Chapter2 Theory

2.1 Introduction

2.2 Rayleigh' s Governing Equation

2.3 Rayleigh's Solution Clamped Boundaries and E q d Tension

2.4 Gorman's Solution for Two Side Fied Two Sides Free

2.5 Discussion

Chapter 3 Holographic Interferometry

3.1 Introduction

3.2 Creation of a Hologram

3.3 Holographie Interferomeûy

3.3.1 Real Time Interferometry

3.3.2 Time Averaged uiterferometry

iii

vi

viii

ix

xii

xiii

Chapter 4 Experimentd Procedure

4.1 ïntroduc tion

4.2 Mateial Properties

4.3 E x m e n t a l Fixhue

4.4 Transient Fxequency Testing

4.5 Determination of Normal Mode Shapes

Chapter 5 Results and Discussion

5.1 Experimentd Results

5.2 Dimensional Analysis

5.3 Analysis of Frequency Data

5.4 Analysis of Mode Shape Data

Chapter 6 Conclusions

6.1 Conclusions

6.2 Future work

Chapter 7 References

Appendix A Test Noies

Appendix B Design Drawings

Appenaix C Dimensional Andysis Work

List of rigures

Uniaxidy Tensioned Membrane

Membrane Element and Loading

Nodal Lime Patterns

Membrane Representation

Experimental Holographie Setup

Spatial Filter Assembly

Stress Strain Curve for the Material

Experirnental Membrane Holâer

Transient Test Setup

Typical Experimental Mode Shape

Variation of Frequency Parameter with Aspect Ratio

N o d z e d Frequencies and Nomialized Area vs Aspect Ratio

Nonnalized Vibration Frequencies

5.5 (a)

5.5 (b)

5.6

A-1

A-2

A-3

A-4

A-5

A 4

B.l

B.2

Mode Shapes for the Aspect Ratio of 1

Mode Shapes for the Aspect Ratio of 1

Displacement Profles

Static Aspect Ratio 1 Image

1 st Mode Aspect Ratio 1 Image

2nd Mode Aspect Ratio 1 Image

2nd Mo& Aspect Ratio 1 Image

3rd Mode Aspect Ratio 1 Image

4th Mode Aspect Ratio 1 Image

Bottom Clamp - Membrane Test Apparatus

Side Plates - Membrane Test Apparatus

vi i

List of tables

4.1 Sizes of the Membranes

5.1 Experimental Natural Frequencies

5.2 Significant Variables and their Dimensions

5.3 Experimental Pi Group 1 Frequencies

List of Abbreviations and Symbols

A characteristic ampiitude of vibration, (m)

ac acceleration of the membrane in the z direction. It is a function

of the , (m/s2)

Length of the membrane in the x direction, (m)

Length of the membrane in the y direction, (m)

Wave speed in the membrane, WS>

3

Ea!, f l e x d rigidity of the membrane 7

E Youngs moduius for the membrane material, (N/m2)

& Modified Youngs modulus for the membrane material (E*t), ( ' lm)

f(t) Input force, (N)

Tx, TY The tension per unit width on the membrane, (N/m)

t the , (sec)

th Thickness of membrane, (m) ix

Out of plane displacement of the membrane, (m)

out of plane displacement as a funcion of position and t h e . (m)

x component of the assumed mode shape, (m)

X position coordinate, (m)

y component of the assumed mode shape, (m)

Y position coordinate, (m)

Circular fiequency of the membrane vibrations, (Rad/s)

mass density of the membrane matenal, (Kg/m3 )

mass density of the air, (Kglm9

Surface density of the membrane material, (Kg/m2 )

Dimeosionless hequeocy parameter b UJ 8 b P, Dimensionless density ratio -

P

E TT3 Dimensionless tension/ stiffness parameter T

b Aspect Ratio

A T T ~ Dimensionless amplitude ratio 5

TT6 Mode number

To Dr. Mitcheli, Dr. Hazeli, Dr. Cochkanoff and Dr. Wilke for th& guidance

and kindness in the production of this thesis.

My Parents, Charles and Louise, and my Brother, John for theh support and

Iove.

Finally, but not least, Mr. James Wylde for bis continuing help and fnendship

in the production of this thesis.

xii

The dynamic response of membranes, with two opposite sides fixed and

under tension and the remaining two sides fke, were examined in this thesis.

Methods using non-contacting methods for measuring the natural frequencies

and mode1 shapes of the membrane were developed and the results reported on. Non-

contacting optical techniques were used for the determination of the n a d frequencies

of the target membranes. Then, the use of holographic interferometry was used for the

d e m a t i o n of the mode shapes. The litenhm shows that both methods of testing have

been used for test subjects other than membranes with good results. However, the use of

holographic techniques for the study of membranes is difficult because of the extreme

stability required by holographic techniques. The literature also shows that the analytical

methods for determining the dynamic response of membranes is Limited. The goal of this

thesis is to present as much infoxmation of the dynamic response of membranes as

possible through the use of non-contacting methods.

For test subjects, membranes of vârious aspect ratios were examined and the

results presented for both fkequency data and the correspondhg mode shapes. Qualitative

comparisons were made to the published literature solutions for the problem with

surprising results. The results show that holographic techniques can be used for the study

of membrane dynamic response. The cornparison between the experimental results and

the analytical solutions verified the presence of modes with vertical nodal lines, or modes

in which the nodal lines run parallel with the axis under tension.

Chapter 1

Introduction

1.1 General Developments in the space indusûy have led to an increased interest in the

design and development of large lightweight structures for use in space. Most of these

structures cm be stored in compact packages for launch and then expanded upon

deployment in orbit. A typical example of a Large Space Structure (LSS) that fits the

small launch package concept is a large membrane with small radar elements bonded to

the surface. These structures are rolled up during launch into space and upon obtaining

orbit are unroîied and deployed. In operation, the surface of the membrane is stretched

uniaxially between two parailel battens which hold the ends of the membrane rigidly

fixed. Figure 1.1 shows a typical stretched membrane of this type.

The signals sent and received by such a radar dish will be affected by any

out of plane motion that occurs in orbit. In order to predict this motion it is necessary to

lmow the fiequency and modal response of the membrane beforehand. No analyticai

method is yet known that cm predict the natural frequencies and mode shapes of these

membranes accurately, and vexy iittle experimental work has appeared in the literature to

date. An additional p b l e m associated with this application is that the vacuum found in

space will remove the damping and mass loading effects normally caused by air. This

means that it is difficult to apply much of the experimental work that does exist to the

space borne membrane.

T = Load onthemednane permiiitrÿia

T T T T T

Figure 1.1 - The Uniaxially Tensioned Membrane

The characteristics of certain other types of vibrating membrane structures

have been examined ushg both analytical and experimental methods over the years. Up

to this point the analytical methods used to study membranes have dealt exclusively

with cases of fixed boundary conditions on all sides and constant membrane tension. The

best known of these is that of a rectangular membrane with uniforni tension in aii

directions, which was first discussed by Lord Rayleigh (John William Stmtt)[l]. To this

date, a general solution for membrane problems with free edges has not been f o d

Experimental techniques have aiso been used to study the modal properties

of membranes with various boundaq conditions. These r e d i s are ofîen reported in

dimensionless fonn to make them more generaily useful for design proposes. However,

this approach is of limited usefulness due to the numerous variations possible in the

boundaq conditions and geometry of the membranes in question. Experimental results

reported in the literature cannot be used for design guides for all possible cases.

1.2 Literature Review.

The first analytical work in the field of vibrations of membranes was, as

Lord Rayleigh states, conducted by S.D. Poisson and involved the study of sound emitted

for d m heads, i.e. circular membranes. However, his study of the subject was

incomplete. In 1862, the complete solution for the vibration of a circular membrane was

produced by Cledsch . This work was followed by Lord Rayleigh with the publication of

his Theory of Sound in 1877 which included solutions for rectangular membmnes with

all four sides bound and under tension, including the unique square membrane solution.

In addition, his work contained the solution for cirçuîar membranes, under uniforni

tension and fixed boundaries. Lord Rayleigh's work has f o d the basis for most of the

curent research in the field of membrane dynamics. Lord Rayleigh aiso states that, the

confqprations for membranes with theoretical solutions also includes the triangle as

produced by G. Lameat around 1852. M. Bourget was the h t to demonstrate the nonmal

mode shapes of circular membranes. Savart was the first to study the effeçt of forced

vibration response which was later confirmed by Elsa's experimental work

The moI.le recent work on membranes started with Thoshenko's 121

research in which he took the earlier solutions for circular and rectangular membranes

and expanded the solutions. These frequency solutions relieci heavily on the work of

Lord Rayleigh and d'Alembert, as quoted by Lord Rayleigh, and were obtained using the

Rayleigh-Ritz method 131. In 1966 Stephens and Bate [4] produced a text that provided

a good explanation for the theory of membrane vibration. Later, in 1968, Morse and

Ingard [SI presented work covering the analysis of strains and stresses, the effects of air

on the vibration of the membrane, as well as the dif%erential equations that govern motion

dwbg membrane vibration for both circular and rectangular membranes with various

boundary conditions.

J. Mazumdar 161, in 1973, produced a paper on the transverse vibration of

membranes of arbitrary shape by a method of constant-deflection contours. The papa

produced a method for approximate computation of the fundamental fiequency of

membranes of arbitrary shapes. In 1979, K. Sato [7] produced a papa on the forced

vibration andysis of a composite rectangular membrane consisting of strips. This paper

was concerneci with the analysis of reçtanguiar membranes consisting of sûips of

different materials using the Laplace transformation method. The paper gave a solution

to the two cases of a membrane subjected to both sinusoidal forcing and the case of step

function forcing.

In 1993 DJ. Gorman and RK. Singhd 181 produced a papa on thei.

work on non-uniformiy tensioned membranes using the Rayleigh-Ritz method for fiee

vibration andysis to establish the initial stress distributions in corner tensioned

membranes. This paper included verification tests that &monstrate the validity cif the

method against known classical problems with unifonn edge loading. The method

obtains the initial stress distribution by the use of tbree Ajl stress functions

superimposed on one another.

The 1993 paper by V.H. Cortinez and P.A. Laura 191 introduced an exact

solution for the vibrations of non-homogeneous rectangular membranes where the two

subdomains of the non-homogeneous rectangular membrane are of rectangular sbape.

The solution obtained by the use of the method proposed within the paper is in

agreement with the solutions produced by both the classical Kantorovich method and the

optirnized Kantorovich method. The solution is produced by the separation of the two

parts of the membrane and the use of two separate differential equations to describe

them.

Later in 1993, DJ. Gorman, R.K. Singhal, W.B. Graham, and J.M.

Crawford [IO] published a paper investigating the stress distributions in corner tensioned

rectangular membranes using two methods. The first was the use of superimposed Airy

stress functions and the second method was a finite element andysis conducted with a

commercial program. In 1994 D J. Gorman, R.K. Singhal, W.B. Graham, and J.M.

Crawford [ l l] conducted a theoreticai and experimentai study of the free vibration of a

rectangular membrane under a uniaxial loading. The loading was achieved by two rigid

transverse bars attached at the extremities of the membrane on the short edges. This

paper, which is of great relevance to the current investigation, included the only known

experimental vibration test results recorded in both air and vacuum. Those results were

compared with results fiom both a finite element model and a theoretical analysis.

In 1993 R.K. Singhal and D.J. Goman 1123 studied the effects of linearly

varying tension and light flexural rigidity on the free vibration of a rectangular membrane

with two fke edges. The paper pmduced a mathematical solution for the problem by

using a beam analogy dong with the Rayliegh-Ritz method. The biggest concern within

this approach is the beam eigenfunctions that arie chosen to model the deflected sample of

the membrane. These functions will change depending on the modes (i.e., symmetric or

antisymetric) under study.

Expimental results are notably in short supply in the fiterature, possibly

due to the difficulty in ob-g data for light weight structures. In 1965 W.E. Nickola

[13] investigated the suitability of using the Moiré method for detennining the dynarnic

response of thin membranes for both transient and periodic behavior. The methods of

specimen preparation, test apparatus and test results are included for both rectangular and

circular membranes. The results were compared with theoretical methods for the cases of

periodic response in which good results were obîairted. This was the first recorded case

of non-contacting methods being used in the study of membranes.

The experimental technique of holographic interferometry would seem

ideally suited for such studies. There is a large body of literature dealing with

holography, beginning with the classic 1965 papa on photography by laser by Emmett

N. Leith and Juris Upatnicks 1141. This paper explained the method of producing a three

dimensional image using a highly coherent light source such as that produced by a laser.

The basis of the procedure is that the interference pattern of two coherent light waves are

recorded on a fine grained photopphic film in place of an actuai image.

In 1970 the h t holographic images of a membrane's mode shapes were

obtained by S. Liem [15]. These images consisted of the lower modes of a circular

membrane. This was the first use of holographic techniques in the recordiiag of the

modes of vibration of a membrane and is in part the basis of the method used here to

obtain the data for this thesis. In 1973, CR. Hazeil and S.D. Liem [16] produced a paper

on the vibration of plates by real-time stroboscopie holography. This paper

demonsîrated application of real-time holography to the analysis of vibrating surfaces

and gave details of the advantages of this method. The methods and apparatus of the

experiniental method are discussed including the introduction of an initial group of

interference f i g e s . The advantages of this technique include the abiliîy to manipulate

the driving frequency and the chiving forces, the need of onïy one hologram to shidy

several vibration modes of the experimental object, full field coverage of the object and

the abiiïty to obtain the sign of the deflection with respect to an extemal reference. The

rnethod does however have several disadvantages and these are also discussed. These

include that the results cm be affected by the addition of unwanted fringe patterns caused

by factors such as ngid body motion, thermal effects or emulsion shrinkage.

13 Scope of Work.

This thesis will present the results of an experimental study of a

rectangular membrane fixed on two opposing sides, with the remaining two sides free.

The main focus wiii be to study mode shapes and natual fkquencies for stretched

membranes of varying aspect ratios and tensions. The mode shapes will be determined

ushg holographic interkromeüy to avoid adding mass loading effects on the membrane

that conventional sensors, such as accelerometers, would cause. The frequency data will

be collected using a non-contacting optical vibration transducer. The frequency analysis

of this data will be conducted using an HP 3582A spectrum anaiyzer. Tests wiii be

canied out using both impact excitation and sinusoidal Erequency sweeps generated with

a driving speaker. The same vibration sensor will be used in the mode shape imaging

process to confirm the operathg fkequency before exposure of the holographic plate.

The variation of natural frequencies of the membranes will be studied for

various aspect ratios and tensions. Due to the limited number of holographic plates

available and the uncertainty of obainifig more supplies of plates, only a single series of

mode shape &ta will be obtained, al1 at a constant tension. The second stage of the mode

shape experiments relating to the changing of the tension on the membrane were not

attempted. This will remain a possible area for future work.

In addition to the membrane data there wiil be accelerometers attached to

the h e to ensure that the driving frequencies used to excite the membrane do not

excite the experimental h e as well. If a frequency is found to excite the f m e , then

mass will be added to the h e in order to shift the response of the frame's natural

frequenckk out of the area of interest for the membrane.

Chapter 2

Theory

2.1 Introduction

The characteristic which differentiates a membrane from a thin plate is

that in a membrane bending forces are considered negligible compared to the tensile

loading. In the thin plate analysis, bending, torsional and shearing loads are considered

in addition to the compressive and tensile loading. This difference is the primary reason

that the existing analytical and numerical methods for plates are inapplicable for

membrane problems. The flat plate solutions have been well developed. However, the

appiication for membranes is at best partial.

The first serious work on membrane theory appeared in the Theory of

Sound, Volume 1 by Lord J. W. S. Rayleigh 113 fïrst published 1877. In this text Lord

Rayleigh explains theoretical membranes as

"a perfectly flexible and innnitely thin lrimina of solid matter, of uniform

material and thickness, which is stretched in al1 directions by a tension so great as to

remain sensibly unaltered during the vibrations and displacements contemplated."

Lord Rayleigh limited his analysis to "the investigation of the transverse

vibrations of membranes of different shapes, whose boundaries are fixed".

The requirement that al1 sides of the membrane must be fixed means that

Rayleigh's solutions are not applicable to the problem at hand. However his formulation

of the governhg differential equation is the basis for much of the analytical work that has

appeared in the literature. This chapter, will examine the development of Rayleigh's

govemhg equation, his well known solution for a rectangular membrane with equal

tension in both directions and then using an approximate numerical technique proposed

by Gomm et ai for determining the naturd fhquencies of an axially tensioned

membrane with two sides free. Perceived faults in Gorman's technique will be discussed

and fkaily a new anaiyticd mode1 wiil be proposed.

2.2 Rayleigh's Governing Equation

In developing bis goverrllng equation Rayleigh made use of a force

balance on a differential element of a membrane. He also made several assurnptions, as

follows:

a. The material is a homogeneous, perfectly flexible

membrane;

b. It is bounded in the XY plane for purposes of notation;

c. There are very large uniform tensions per unit length, T,

and T,acting on the edges of the membrane elernent;

d. The membrane moves in pure translation in the Z direction,

perpendicular to the XY plane;

e. The equilibrium position is in the XY plane. As a

consequence, any point on the membrane can be defieci

by its equilibrium position in the coordinate system P(x,y)

and its lateral displacement, w = w(x,y,t), and

w is very small in cornparison to the dimensions of

the membrane. Consequently the tensions Tx and Ty

can be assumed to remain constant during the motion.

To find the equaiîon of motion for the membrane consider a smaU area of

a membrane under unifom tension, as shown in Figure 2.1

Figure 2.1 - Membrane Element and Loading

Taking the area of the element as dxdy, it cm then be said that the forces

acting on it are given as Txdx and Tdy, where T is a force per unit length. Now consider

the Z components of the forces acting dong either direction. First for the x direction

tension forces:

Therefore

similarly in the y direction:

Therefore

The total of the Z direction forces acting on the element can be expressed as:

The total of the Z direction forces must be equal to the mass of the element, multiplîed by

its Z direction acceleration, which gives the following governing equation:

Where Pt is the "surface density" or mass per unit area for the membrane material.

2.3 Rayleigh's Solution for Clamped boundaries and Equal Tendon.

Rayleigh found an exact solution for equation (2.1) ody for the case of

qua1 tensions in the X and Y directions and with al i four sides fixed In this case T, =

Ty = T and equation (2.1) may be riewritten as:

w here,

Solving equation (2.2) for circular fiequency of vibration in the normal modes will give

equation (2.4), as seen in reference [l]:

where m and n are integers given by

( m,n= 1,2,3, ..... ,m )

The corresponding eigenfunction for this equation is given by:

m m x u (*y) = X Y,, = sin ( Y ) sin (F) These eigenfunctions al1 have sbiaight nodal lines as shom in Figure 2.2.

This solution is of interest because of the appearance of the ~ T / P , tam in the solution

and because it predicts straight nodal lines parailel to the membrane boundaries for the

various modes. These appear to be characteristic features of most membrane solutions,

and might plausibly be expecîed to be important in the problem of interest here.

Figure 2.2 - Nodal Line Patterns for Rayleigh's Membrane with ail Sides Fixed

2.4 Goman's Solution for Two Sides Fixed, Two Sides Free.

D.J. Gorrnan [19] proposed an approximate f?equency solution for the

case of an axially tensioned membrane with two sides fiee that is very similar to the one of

interest here. His approach took a varying tension caused by the mass of the membrane

and gravitational forces into account. The mode1 is that of a rectangular membrane of

dimensions a and b as shown in Figure 2.3 below.

Fret Edge

x

Frrt Edge

Fina Edge

(d

Figure 2.3 - Membrane Representation

a. - for Symmetric Modes

b. - for Antisymmetric Modes

Here the edges x = o and x = a are fiee and the edges y = O and y = b are

k e d i.e. simply supporteci. The membrane is held upright in the vertical plane and due to

the initial tensioning and the gravitational forces present there will be linearly varying

tension acting in the y-direction which is expressed by the equation

Where

Ty is the bearly varying tension with gravitational forces included,

TO is the initial tension on the membrane,

a is the Linear variation in tension, and

y is the position of the co-ordinate in the y-direction on the membrane.

This very spezifïcally models a particular Large Space Structure (LSS)

that Gorrnan was worlcing on at the time. The introduction of gravitational forces was

important as the surface of the membrane in question was covered with a large number of

electronic cells attached by means of thin plates to the surface. The solution handles the

symmetric modes and the anti-symmeûic modes separately, where the definition of

symmetric or anti-symmetric modes is taken about the long central axis of the

membrane. Only symmetric modes are considered in the discussion that follows.

No aualytical solution to this problem was possible because of both the

linearly varying tension in the membrane and the presence of two fkee edges. Therefore,

Gorman used the Rayleigh-Ritz energy method to obtain an approximate fhquency

solution. He chose to mode1 the deflected membrane shape as a double series using

orthogonal fk+fiee beam eigenfunctions in the x-direction muïtiplied by sin functions

in the y-direction. For the symmetric modes only the symmetric beam modes were

used. The general mode shape was given by equation (2.7):

Where

L is an unknown series of coefficients for amplitude, and

@ In (x) are the eigenfunctions.

Gonnan assumed that a single beam flexurd mode shape, ( x ) , could

serve very weîi to approximate the corresponding membrane mode shape in the vertical

direction. The first two of these functions are:

= c0ItStmt (say I ) (2.8)

and

(2.9)

Where B is the beam and equals 4.730 for the fundamental mode. It c m also be

shown that:

The sine functions in equation (2.7) satis@ the edge support conditions of

a simply supported thin plate or membrane . The fliee-free beam eigenfunctions satisfy

the free edge conditions.

Gorman then took the rather unusual step of including the plate bending

energy in the expression for the potential energy of the vibrating membrane. His total

potential energy was expressed as:

v=v,+v,+v, (2.11)

The bending energy is represented by vE and is aven by the equation:

(2.12)

Where D is the bending stiffhess of the plate and v is Poison's ratio for the membrane

material. The Elastic energy from tension is represented by VT and is given by the

equation:

The Inertial force energy is represented by VI and is given by the equation:

The integration of the energy t m s will produce an expression for the

total potential energy in the system in terms of the 2k unknown series coefficients E,.

Values for these coefficients were then found by mhhhhg the rate of change for the

total potential energy with respect to each of the series coefficients E,. This gave rise to

2k homogeneous algebraic equations relating the coefficients kom which the eigenvdue

matrix was obtained. The eigenvalues were then obtained by findirig the values for h

which cause the deterininant of the ma& to vanish. Then, setting one of the imknown

coefficients equaX to unity the resdting set of non-homogeneous equations could be

solved to establish the coefficients and the mode shape associateci with any particular

eigenvalue.

2.5 Discussion

Gorinans method would appear, at first glance, to predict a series of mode

shapes similar to those given by Rayleigh's solution for four sides fixed with both

vertical and longitudinal nodal lines. Recail that Gorman used a single beam mode shape

to mode1 the vertical deflection of his membrane. Therefore the number of vertical nodal

lines that would be observed would simply be a function of the beam mode used, and

could be easiiy changed by simply substituting different a s s d mode shapes . However, a closer examination of this technique reveals that the vertical

mode shape assumed can have little or no effect on the frequency coefficient obtained.

To illustrate this consider the fiaction that results when using Rayleigh's method to

detemine the vibration Çequency of the fundamental mode and neglect the plate bending

term in Goman's solution, which is known to be srnail. Also, for the sake of simpficity

and because it is not important to the present investigation, let us assume tbat the variable

tension coefficient is allowed to approach zero. Rayleigh equated the system's kinetic

and potential energy as follows:

The kinetic energy was given in equation (2.14) as:

The potential energy

equation (2.13) is:

due to constant tension obtained by ailowing a to go to zero in

The natural frequency obtained in this method is therefore proportional to the ratio of:

Now assume that w is a product of two functions with variables X(x) and Yb) sepamble

such that:

w=x(x) Y ( Y )

Differentiating this expression, squaring and substituting into equation (2.15) gives:

Because the variables are separable, the integrals in equation (2.16)can be canied out

independentiy. Equation (2.16) therefore can be sirnplified to:

The x integrals in equation (2.17) are identical and therefore cancel out, fkom which it

can be concluded that the naturai frequencies would be a function of the Y components of

the chosen mode shapes and indepen&nt of the X function.

For an assumed sinusoida1 mode shape in the y-direction, similar to that

used in equation (2.7), equation (2.17) becomes:

Substituthg the acîual sine expressions and differentiating the tenns gives:

When evaluated, this expression collapses into:

This is the well known equation for the vibrating string as given in 1173.

Any approach based on the above mentioned energy terms will not predict

distinct vertical modes and a l l of the vertical modes are expected to occur at the same

vibration frequency. Goman's inclusion of plate bending energy in his formulation of the

problem appears to be an attempt to circumvent this problem and generate a solution with

separate and distinct vertical mode frequencies. As expected, the plate bending energy

terms wiil be vanishingly small in most membrane vibration problems therefore this

rem does not apply. Some other mechanism will be required to predict such modes, if

they indeed exist. In any case, it is assumed that there are no bending stresses in a

membrane and it is W c u l t to justify the use of plate bending energy for this analysis.

Chapter 3

Holographie Interferometry

Holography is a rnethod for recording and reproducuig complex

wavefonns traveling through space. The waveform is recorded exactly and contains al1

of the original information of amplitude and phase. The extent of this reproduction is

such that there is no known optical test or procedure to distinguish between the original

object and the reconsbvcted image of the object. The most common use of the

holographic meîhod is the capture, in a photographie medium, of opticai waveforms that

are reflected fkom a target object and the reconstruction of these waveforms to produce an

image of the target objecî. When this captured image is viewed, the observer will see the

original object as when k t viewed. Therefore the observer will see a three dimensional

reproduction of the original object

Holography was proposed in 1948 when Demis Gabor first defined the

procedure. At the t h e Gabor, a Hungarian, was working in London. His first

experiments were limiteci by the requitement of a highly coherent light source and the

most powerful source at the tirne were highly fïitered mercwy vapor lamps with very

short coherence lengths. Therefore, holographic interferometry had to wait until the

discovery of the continuous wave laser in 1962 before becoming a practical technique

thanks, rnainly, to the work of Leith and Upatnieks. These men clearly reproduced three

dimensional images using the relatively new light source of the laser. From this point in

time onwards holographic interferometry has becorne an accepted tool for science and

industry.

The principle of a hologram is based on the interfmnce effect between

two beams of cohexent light (light of only one frequency, phase and amplitude). One

beam illiiminates the object and one beam is used as a reference. The object beam is

distorted by the object, or target and then reflected towards the photogtaphic medium.

This reflected beam wiîl contain a l l of the information about the object, both static and

dynamic in nature by distorting the original beam of light in both amplitude and phase.

The reflected object beam will then interfere with the reference beam, which has retained

the original phase and amplitude, at the photographic plate. This interference patteni

between the two beams of light is then captured in the photographic emulsion on the plate

in the form of a complex f i g e pattem, which contains all the information necessary to

reproduce the holopphic image. When the reference beam reilluminaîes the developed

holograpaic plate, the holographie image wiU then become visible to the naked eye. This

is accomplished by the light king diffracteci by the h g e patteni in the photographic

emulsion to reconstruct an image of the original object in three dimensions at the original

location in space.

3.2 Creation of a hologram

To mate a hologram the laser is tuned to emit a coherent collimated light

wave (a single sine wave) at a single wavelength. This beam of light emerges h m the

laser head and is then split into two beam. Using a variable beam splitter, or a partially

silvered mirror which wiii reflect part of the beam and ailow the other part of the beam to

pass through the mirror, the transmission, or reflection ratio is adjustable. Une of the

beams, hereafter referred to as the reference beam, will then be refiected towards the

holographic plate holder and expanded by a lens locateâ within a spatial filter. The

spatial filter is a device that contains a lem assembly and a mask with a pinhole in series.

This pinhole is placed directly in the location of the focal point of the lens and will then

block, or filter, the side bands of light created by passing through the lens (See Figure

3.2). The expanded light wave then illuminates the holographic plate and wiii be used in

both the creation and remation of the hologram. The other beam of light, hereafter

referred to as the object beam, will then be refiected towards the object of the hologram,

passed through another spatial filter, and then refiected by the object towards the

holographic plate holder. This reflection from the object wili cause the light that is

reflected to change in intensity and phase with respect to the reference beam. The two

beams then combine at the holographic plate holder to expose the photogmphic emulsion

of the photographic plate. This exposure is the result of both constructive and destructive

interference patterns created by these two beams of light.

The reconstruction of the hologram c m then take place after the

photographic plate has been developed. This is accomplished by reiîluminating the plate

with the same light source as the onginal reference beam This beam will then pass

through the holographic plate and be diffiacted by the recorded image on the plate to

reproduce the original object. It should be noted that the length of the paths that the

object and reference beams travel should be the same distance, or as close as possible

within the coherence length of the light source. (See Figure 3.1)

Figure 3.1 - Experimental Holographie Setup for the Vibration Isolation Table

Figure 3.2 - Spatial Filter Assembiy

3.3 Holographic Interferometry

Holographic interferometry is a technique that allows the study of dynamic

motion and response. This is a precise method of measuring displacement through non-

contacthg methods. This method is extremely sensitive to motion and can detect motion

as smaU as one halfthe wave-length of visible light. The many techniques of hotographic

interferometry are ail conceptuaiiy similar and for the purpose of the work presented here

only two of the methods di be discussed in detail.

It is the superposition of two coherent light waves, one reflected from the

object and the other being the reference beam interfiering to produce a holographic image.

These two overlapping images will produce visible fhge patterns on the object.

Fringes are areas within the image that are da& and light and represent motion of the

object. This is sirnilar to the idea of a "double exposure" in photography. These fringe

patterns can then be used to determine the amplitude and shape of dynarnic motion of the

object.

3.3.1 Real-time Holographic Interferomeûy . The process of real-tirne holographic interferomeûy is that of using an

original static hologram of the object in the original location of exposure and the object in

its original location and under a dynamic load. This process requires that a static

holographic image of the object be first obtained. This static hologram is easily cfeated

by the exposure and subsequent development of an image of the object un&r a no load

situation.

The "real time intderence pattem" is then produced by the re-

illumination of the object, while under a dynamic load, with the original coherent light

source. The reflected light waves will then interfere with those created by reconstruction

of the static hologram to produce the desired f i g e patterns. These h g e patterns

represent real-the motion on the holographic object and the fiinges wiU sbift with any

change in the excitation fiequency of the object. This method is an extremely powerful

analytical tool for the determination of the frequencies and mode shapes of the object

under study. However, the usefiilness of this method is limited. The dynamic nature of

the object makes the capture of specific modes difficult. Also, amplitude is diffïcult to

measure, versus the t h e averaged method, due to twice the number of fikges for a given

amplitude.

3.3.2 The-averaged Holographic Interferomeûy

The method of time-averaged holographic interferometry was first

reporteci in reference [Su. In this method the recording of a hologram is made as the

object undergoes dynamic motion. The recording is made in exactly the same way as

was used in the making of a static image, however, the object of study is vibrating in a

steady-state nomial mode during the exposure. The information necessary to reconstruct

the image of the object under study is then contained in the photographic emulsion and

the photographic plate is developed as before. The reconstruction of the image will

show a permanent recording of the fnnges for the dynamic state the object was under at

the tirne of exposure.

As the "image" is pennanently contained in the photographic emulsion on

the plate, this method does not suffer from the problems of real-time holography, namely

the image will be permanent and the number of fringes contâined within is more

manageable. The only problem with this technique is that the frequencies must be

previously hown by experirnentation or analytical 1 numencal solutions in order to

obtain the pmper mode shapes.

Chapter 4

Experimental Procedure.

4.1 Introduction

The object for the experimental program was to determine mode shapes

and natural fiequencies of a number of membranes of varying aspect ratios. The exact

shape of the deflected membrane as it vibrated was of great interest to test the validity of

the assumptions made in the analytical studies reviewed earlier. The question of whether

or not separate and distinct vertical modes at Werent natural frequencies actually exist

was also to be exlimined. To this end seven different membranes of various aspect ratios

were studïed. The membranes were al l made of the same mylarm material and were

relatively smaü to accommodate the holographie apparatus. Table 4.1 below shows the

sizes of the membranes and their aspect ratios.

Table 4.1 - Sizes of Test Membranes

4cm b, cm Aspect Ratio

10.16 15.24 0.667

10.16 10.16 1

15.24 10.16 1.5

10.16 5.08 2

12.7 5.08 2.5

15.24 5.08 3

17.78 5.08 3.5

Each of these test subjects was loa&d with a constant tension per unit width of 214.5

N/m during the trials.

The experimental work consisted of three main parts. In part one the

material properties of the membrane were &termined. In the second part, îhe natuial

frequencies of vibration were determined. In the final part the mode shapes were

obtained using holographic interferometry. Details of each of these processes wfl be

proviâed in this chapter.

4.2 Material Properties.

The test membranes were made of duPont Mylarm sheet with a nominal

thickness of 76.O*lO a m and a nominal density of 1404.0 kg/m3 fiom manufacturem

~ ~ c a t i o n s . The actual thickness of the sheet was found by direct measurement using

mirrometers to be 68.75*10 " m. The thickness measurement was made by placing four

thickness of material in the micrometer at one tirne. The micrometer had a measurement

uncertainty of +/- 0.01 mm. This meant that the uncertainty in the thickness

measurernent was 10*10 m spread over four thickness of material, or 2.5*10 " m.

The measured thickness is significantly at odds with the manufactures quoted value.

The density was also directly measured by weighing a relatively large

section of the test material (O.Sm by 0.5m) on a high percussion Metlerm electronic

balance. The balance had an accuracy of +/- 10 -' kg. The experimental density was

found by dividing the weight obtained by the calcdated volume. The experimental

density of 1403 kg/m3 was almost exactly the quoted value of 1404 Kglm3.

These two experimental values combinai to give an apparent 'surface

density' of O.O96S*lO " kg/m2 . This surface density is assigned the symbol Pt through

out the thesis.

Young's modulus for the matenal was determined by testing thin sûips of

the material in an Instron mode1 1225 tensile testhg machine. Ten trials were conducted

in dl using sarnples approximate1y 250 mm long by 25 mm wide. The average Youngs

modulus from these tests was found to be 3.95 Gpa, which compares very favorably to

3.79 Gpa that the manufacturer claims. A typical stress strain curve obtained is shown in

Figure 4.1.

Stress Strain Curve for Membrane

Figure 4.1- Stress-Strain Curve for the Membfane Material.

43 Experimental Fixture,

The membrane was held during the experiments in an adjustable apparatus

built around an optical double post stand, wMch has y axis control that can be used to

raise and lower the attachent plate. A photograph of the membrane holder is shown in

Figure (4.2).

Figure 4.2 - The Experimental Membrane Holder - a. Side plates, b. Top wire

rollers, c. Center plate, d. Bottom clamp, e. Membrane rouer-battens, f. Weight

containers, g. Membrane batten attachments, h. Membrane batten side plates, i.

Membrane

The parts for the membrane holder were built fiom steel in order to

maintain a high degree of rigidity. This was done in order to obtain a stand that was heavy

and rigid enough to dampen out any unwmted noise from reaching the test membrane, as

wel as to ensure that the stand did not respond to the fkequencies and amplitudes used to

excite the membranes under sîudy.

An 8" speaker was used to drive the vibrations during the tests. The

speaker was bolted direcdy to the vibration isolation table for convenience, but not to the

membrane holder so as to avoid the problem of exciting the membrane holder through

direct contact with the shaker device. The signal source used for the tests was a Hewlett-

Pakard 33120A, ISMHz digital function generator, and the signal was amplified by a

Bruel and Kjaer Type 2706 power amplifier before being fed to the speaker.

4.4 Transient Frequency Testing.

The method used to d e t e d e the normal mode fkequencies of the

membranes was transient response testing similar to that often used for large structures.

In tbis type of test, an impact is applied to the structure and the response of the structure

is record4 using a sensor and spectnim analyzer. For most applications a piezoelectric

accelerometer is used as the sensor. The accelerometer is a piezoelectric crystal encased

in a steel casing and must be attached to the structure under testing. For large structures

this does not present a great problem as the inmase in mass is negligible and will not

significantly shift the frequencies under study. However, the mass of a membrane is very

srnaii by cornparison with an accelerometer, and the increase in macs caused by attaching

the measuring device could cause a signifiant shift in the hquencies under study.

Therefore testing of membranes requires either a contacting sensor that is very light, or a

sensor that does not contact the surface of the membrane and therefore does not add to

the mass of the membrane.

This study used a non-contacting optical sensor manufactured by the

Fotonicm Corporation to obtah vibration data. This sensor was rielatively simple and

easy to use. In îhis device, a flexible fiber optic cable carries light to a sensor tip that

contains upwards of 800 optical fibers. Of these 800 fibers 400 transmit light to the test

subject. The light is then reflected off the test subject and the remaining 400 fibers

capture the light and then carry the reflected light back to a photodiode. A variable

voltage signal is generated that is proportional to amount of reflected light captured,

which is roughly proportional to the distance between the sensor tip and the target.

In these tests the structure was exited to vibrate and the fotonicu" seasor

was used to monitor the motion. The voltage signal from the sensor was fed to a HP

3582A spectnim analyzer which performs a Fast Fousier Transform (FlT) and displays

the muencies and amplitudes, of the normal mode responses.

The test procedure itself was relatively simple. The equipment used is

listed below :

a. Spectrurn analyzer , a Hewlett Packard mode1 3582A;

b. fotonic sensor;

c. The test membrane holder;

d. Weight for the membrane holder in the form of lead shot;

e. The membrane to be tested; and

f. An empty squirt bottle.

The procedure was as follows:

s tep1 The experimental fixture was set up with the fotonic sensor placed in h n t of the

membrane, pointing directiy at it with a distance of separation no greater than 1 mm (see

Figure 4.3). The sensor output is then connected to channef A of the spectnim anaiyzer.

The speaker was comected to the excitation generator and power amplifier;

step 2

The fotonic sensor was t m e d on and the spectnim analyzer was set for the fiequency

range and amplitude desireci, The frequency range desired for this test should be

approximately twice the highest or suspected highest natural frequency desired and for

the case at hand 1 KHz was selected. The amplitude was set at the lowest range possible

since the vibrations of the membrane are very small in amplitude.;

step 3

The spectnun andyzer ûigger levels were set and the data averaging function was

turned on;

SteP 4 The squirt bottïe was used to apply a non-contacting impulse signal to the membrane. ,

This was done by a short hard squeeze on the bottle, which produces an air wave that

strikes the membrane and causes the membrane to vibrate at the natural frequencies.

This was repeated four times for the averaging function on the spectnun analyzer to

capture four distinct signals ;

s-5 The initial Çequencies were studied to determine the range of the b t six natural

frequencies and the spectnun analyzer ' s frequency range and amplitude settings were

adjusted as requireà;

Sm6 Step 5 was repeated 8 to ten times and the data recorded; and

step 7

The final step was to repeat the testing for the natural frequencies by running a dynamic

sine sweep through the suspected frequencies found in step (f.) using the speaker

assembly and a digital function generator. The input signal was stepped through a range

of hquencies in one hertz intervals while the membrane response was monitored on the

spectnim analyzer for peaks in the spectnun. &ce a peak was obsemed the step on the

input signal was decreased to l/lOth of a hertz.

It was important to e1iminat.e the structure response fiom the membrane

response. To accomplish this an accelerometer was comected to the membrane holder

and the structure was then subjected to an impact test. The frequencies obtained fiom

the accelerometer were compareci to the membrane fiequencies obtained using the

fotonic sensor and any matching fiequencies noted. These fiequencies were ignored for

the membrane test results.

The following Figure 4.3 shows the set-up for the transient response

testing .

Figure 4.3 - Transient Test Set-up

4.5 Detemination of Normal Mode Shapes

The normal mode shapes were determined us- time-averaged holographie

inteflerometry as descnbed in the previous section. The method however was slightly

modifieci to take into account several unique problems associateci with

membranes. The f h t situation that presented itself was that the membrane response was

greatly affected by the tension applied to the membrane. Any change in the initial

tension applied to the membrane would cause slow creep u t i l equilibnum was

established. In order to limit the effect that this would have on the experiments, a

routine was established so that no testing was done for at least one hour after an

adjustment was made to the membrane, membrane holder, or the weights attached to the

membrane. This ensured that the system had the to corne to a steady state before

running a dynaniic test. The first problem also related directly to the second problem.

Once the system came to a rest it was observed that the natural muencies may have

k e n shifted as much as 10 to 15%. To create a hologram at the original test frequency

with such a shift in the natural frequency would quite naturally result in not obtaining the

desired mode shape. In order to combat this problem the transient frequency response

test and the sine sweep test were nui once again before the hologram was produced.

The procedure for the creation of holographic images of the mode shapes

at the fiequencies under study requjreû the same equipment as in Section 4.4. Befoxe

each holographic recording was made the driving speaker was tuned to match the

previously identifiai natural fiequency. The beam ratios were then set and the

holographic plate was exposed while the vibration continued. After processing, the plate

was reatrned to the holder on the table and the image was reconstmcted using only the

reference beam of the holographic setup. A digital camera was then used to record

photographie images of the time averaged interfemgrams.

Chapter 5

Results and Discussion.

5.1 Ekperimentai Results

The natural fiequemies for the various membranes saidied are tabulated

below in table 5.1. Mode shapes were obtained for a number of these fkequencies, which

are identified in the table with an asterisk. Approximately 40 individual mode shape

trials were made in all, severai trials for each mode identified. A single representative

mode shape is shown in Figure 5.1, and the remainder of the mode shape idonnation

can be found in appendix A.

TABLE 5.1 - Experimentai Natwai Fxquencies vs Aspect Ratios

Mode

Number

Aspect Ratio

0.667

1 .O

1.5

2

2.5

3

3.5

(au in Hz)

1 2 3 4 5 6

* Are frequencies for which images have b e n recorded.

** Frequency not recorded.

Load I -,

Load

Free edge A-A, B-B Fixed edge C-C, D-D Node line 1-1

Figure 5.1 - Typical Experimental Mode Shape,

The Second Mode of Aspect Ratio 1

5.2 Dimensional Anaiysls.

The probIem was assessed by developing a set of dimensionless

parameters that described the problem. The use of such dimensionless parameters will

always reduce the number of variables in a problem by a factor equal to the number of

basic dimensions involved. In this problem it is assumed that their are nine significant

variables, as shown below.

Table 5.2 - SignLncant Variables and theu Dimensions

Name of variable

Surface density of the membrane

Length of membrane

Width of membrane

Stiffness per unit width

Tension per unit width

Density of the medium

Frequency of vibration

Mode of vibration

Characteristic amplitude of vibration

Symbol

P,

b

a

E,

Units

M C

L

L

M/t2

M/tZ

M/L3

llt

Dimensionless

L

In table 5.2, their are nine variables and t h e dimensions (M, L, t),

meaning that their will be six dimensionless groups in ail. One of the variables (mode

number) is already dimensionless A certain amount of discretion can be used in setting

up particular groups of dimensionless parameters to describe the problem. Based on

previous work appearhg in the literature and personal intuition the following groups

were chosen.

Frequency Ratio

Pa b ?=F Density Ratio

Et T T 3 = ~

Initial Strain Ratio

Aspect Ratio

A =5=b Amplitude Ratio

rT6 = Mode Number Mode Number

The frequency tests that were descfibed earlier were conducted with m,

held constant whik al1 other groups varied. The densiîy ratio was constant for the

majority of tests but vaned slightly for the narrow aspect ratio test where the b variable

was allowed to change. No information was recorded that wouid allow m, to be

evaluated f?om trial to trial.

5 3 Analys& of Frequency Data.

In Chapter 2 it was shown that the fundamental naturai frequency of an

axially loaded membrane could reasonably be expected to approach that of a vibrating

string. In fact, equation (2.18) can be easily re-arranged to predict a lower bound for the

TI dimensional group of exactly (3.1416). This of course would ody apply for

an ideal membrane vibrating in a vacuum. For our tests it is expected that the ml

group wili be lower than this theoretical value due to the entrained rnass of the

surrounding air.

For the experimental tests the material properties were:

a. Young's modulus (E) was found to be 3.95 Gpa. The

modifïed Young's modulus used in developing the Pi

groups is found by multiplying E by the membrane

thickness, so that E was 272250.00 Nh.

Surface densiîy P, for the unpainted matenal was found

to be 9.65*104 K g M . With the addition of paint to make

the surface reflective for the holographie experiments this

increased to 15.121 1*10-8 Kglm2.

c. The thickness of the niaterial, or t h was measured to be

68.75*106 m for the unpainted surface.

d. Finally, the load per unit length or T = 214.25 N/m

Using these matenal parameters it is possible to convert the data in table

5.1 into dimensional form. The dimensional data is shown in table 5.3.

TABLE 53 - Experimentaï Pi Group 1 Frequencies vs Aspect Ratios

Aspect ratio

Mode Number

* Estimated value.

42

The dimensional frequency data in Table 5.3 is shown in graphical f o m in

Figure 5.2. Examination of thXs &ta shows that the fundamental Erequency parameter

approaches but never exceeds the value of 3.14 that the simple theory predicts. The

fluctuations at f k t glance appear to be random. It is possible that certain modes of

vibration were missed in the testing. It is known that signifiçant creep o c c m d in the

sarnples once tension was appiied, causing natural frequencies to decline with t h e .

Therefore their may be some unçertainty in the actual tension that was applied to the

membranes at the time of the measurements.

Experimental Results 10 ..-.-+

/. 9 1 .-

Figure 5.2 - Variation of the Frequency Parameter with Aspect Ratio

t

0

Upon close examination, the fluctuation of the fundamental mode

frequencies seems to be related to a function of membrane area. To compare the

relationship between the fundamental mode fiquencies and the area of the membrane a

plot of both the normalized fundamental hquencies and the normalized membrane a m

is shown in Figure 5.3. The normalized fundamental frequency is found by taking the

l I I I 1 I

0.67 1 1.5 2 2.5 3 3.5

Aspect Ratio

fundamental frequency for each aspect ratio and dividing by the fundamental frequency

for the aspect ratio of 0.667. Similarly, the nomialized membrane area is found by taking

the area for each aspect ratio and dividing by the m for aspect ratio 0.667.

l~ormalized Frequency and Area vs AR

1' "u,, Nom Area 1 . .

Aspect Ratio

Figure 5.3 - Normalized Fundamental Frequencies and

Normüizeà Membrane Area vs Aspect Ratio.

The shape of the nomialized curves in Figure 5.3 are very similar. Under

these c~umstances it seems reasonable to assume that the acniai tension in the

experimental membranes varied significantly due to creep in a m e r proportional to the

area of each membrane . In order to make meaningful frequency comparisons, it will be

necessary to eliminate this effect. Figure 5.4 shows noRnalized frequency data for each

membrane obtained by dividing the higher mode experhental frequencies for each by

the observed fundamentai frequency.

1 Normalized Frequency vs Modes

Mode 1 ~ o d e 2 Mode 3

A Mode 4

c 0.67 1 1.5 2 2.5 3 3.5

Aspect M o

Figure 5.4 - No11I181iZed Vibration Frequencies

The normalized frequencies shown in Figure 5.4 for the most part follow

predictable patterns. In particulat, the ~ u e n c i e s for modes 3,4, and 5 appear to be

integer multiples of the fundamental fkequency for membranes with aspect ratios of 2 or

higher. This is exactly as would be expected fkom the string equation mode1 for modes

with horizontal nodal lines. The mode at approWmately 1.6 the funâamental fiequency is

unexpected, and thought to be a "vertical" mode with a vertical nodal line running

parallel to the tende loading. Modes 2,3,4, and 5 for membranes with aspect ratios of

less than 2 would also appear to be vertical modes since their frequencies never reach a

multiple of 2 or greater.

5.4 Anaiysis of the Mode Shape Data

The time averaged interferograms used to record the vibration mode

shapes may be interpreted as contour maps, where each bright h g e represents a line of

constant displacement. Each fkinge represents an out of plane displacement of one half a

wave length of light used in the recording, approximately 262.2 nanometers for the argon

laser used in this case.

AU of the mode shapes recorded in the study seem to suffer h m either

some flaw in the experimental boundary conditions or the participation of unwanted

modes in the observed response. All of the modes, regardless of the aspect ratio, showed

motion along one fiee edge and no motion dong the other. A m a t deal of care was

taken in the experirnent to equalize the tension applied along each fixed edge, so the non-

symmetric appearance of the experimental mode shapes was sornewhat perplexing. This

must represent some sort of systematic problem with either the experimental fixture or

the driver system since a variety of different membranes was used in the trials.

However, a good deal of useful information can still be obtained from the

triais. In examining families of mode shapes recorded it was obvious that distinct vertical

modes of vibration do exist at separate frequencies for membranes with aspect ratios of

less than 2. A farnily of such modes is shown in Figure 5.5 for the aspect ratio of 1. It is

quite apparent that the number of vertical nodal lines grows by one for each higher mode

of vibration. It is unfortunate but no modes with horizontal nodal lines were recorded in

this study.

The spacing between the vertical nodal lines in Figures 5.5 (a) through (d)

seems to vary as you move across each image. Although it is difficult to prove

analytically, Uiis would intuitively seem to suggest a variation in the axial tension at

different points across the membrane. The position of the nodal lines was extremely

reproducible and did not seem to Vary with vibration amplitude.

A I Load i3 l

Load I

Free edge A-A, B-B Fixed edge C-C, D-D Node line 1-1

p Load

1 A Load

2 0d Mode

Figure 5 J(a) and (b) - Mode Shapes for the Aspect Ratio of 1.0

3 Mode 4& Mode

Free edge A-A, B-B Fixed edge C-C, D-D Node h e 1-1, II-II, Iü-ïïï

Figure 5J(c) and (d) - Mode Shapes for the Aspect Ratio of 1.0

The interferograms can be easily anaiyzed to provide profiles of

displacement running in bot . the x and y direction. This infornation can be used to

check the validity of the assumptions of the various assumed mode shapes in the

analytical literature. Two such profiles are shown in Figure 5.6 taken almg the x and y

directions for the second mode of vibration of the square membrane.

Figure 5.6 (c) indicates that the assumption of a sinusoidal profile in the y

direction is justifid The distorted mode shape ninning in the x dùezîion was not

expected at the outset of the experhents, and is more difficult to model with simple sine

and cosine functions. It is unclear what effect this distortion might have on the vibration

frequencies recordeci, since in Gorman's model the predicted vibration frequenci*es were

not a function of the x direction.

Load 1 !f

Load

Free edge A-A, B-B Fixed edge C-C, D-D Nodeiine 1-1

Figue 5.6 - (a) Experimentai Mode Shape - 2nd Mode of Aspect Ratio 1

X Direction Cross Plot

-1 O 2337485771 767881 8410

O

Membrane position (% of 10.16 cm)

Figure 5.6 - (b) Displacement &ofBe in the x - Direction for the

2" Mode of Aspect Ratio 1

Y Direction Cross Plot

Membrane position(% of 10.16 cm)

Figure 5.6 - (c) Displacement ProfBe in the y - Direction for the

Pd Mode of Aspect Ratio 1

Chapter 6

Concltusions and Future Work

6.1 Conclusions

The major findings are listed below:

a. That Holographie techniques can be used to record

membrane mode shapes. This is, however, very M c u l t due to the light

weight of the membrane. Great care in the production of the hologram

makes this possible. This has been previously done only once [22] and

that was for membranes bounded arouud the total perimeter of the

membrane. This thesis is the first to produce a hologram of a membrane

with fiee edges.

b. Vertical modes, defined as modes ninning parallel to the

loading, exist and cm be produceci in addition to the horizontal modes

expected.

c. The horizontal modes as predicted by Gormans paper

are similar to string theory.

d. It seerns likely that in order to produce the vertical modes

some tensile force loading must be generated in the perpendicular

direction to îhat of the applied tension. Without this force there WU be no

vertical modes. This tensile force is most likely generated tbrough

Poisson's effect although there was no noticeable wrinkling or narrowing

of the membrane perpendicular to the loading. This is generated by an

unknown mechanism which stores energy related to the vertical mode

shapes.

e. The system seerns to be very sensitive to imperfections in

the boundary conditions and methods of excitation due to the smal l size of

the overall test subject. Despite best efforts the observed mode shapes

were non-symmetrical in appearance. Specifically nodal lines did not run

vertically but were at a slight angle to the tension applied and offset fkom

?he expected positions.

f. The effects of entrained air and air mass damping seemed

to have a significant effect on the observed frequencies. These

frequencies were approximately 27.0% below TT as predicted by

Goman's theory. These effects wiLl bear upon the cornpanson of the

analytical and experimental results due to the fact that the analytical

predictions are for a membrane in a vacuum. The experiments were

perfonned in air and suffer the full effect.

& The fundamental frequency seems to vary with the

relaxation in the tension on the membrane.

f. Finally, the analyticd method produced by Gorman does

predict the order of modes with horizontal no& lines but fails to predict

the modes with vertical node lines.

6.2 Fu- Work

The future work wil l list both the items of experimentation and andytical

work that merit fiirther study. The future work for this thesis can be m k e n into two

parts. First is the additional experimental work. Second is the improvements to the

analytid solution. The additional experimental work is listed below:

a. The variation of other parameters aside from the Aspect

Ratio. This represents a huge body of work and includes the tension on

the membrane, the thickness of the membrane, the density of the medium

surrounding the membrane, and the mataial properties of the membrane.

b, The development of a method to isolate the membrane

un&r study from imperfections in the boundaq conditions as well as the

source of excitation.

c. The correction of the relaxation of tension to the

membrane. This cm be affectecl by the use of puileys or bearings to

mate a true fixed tension.

The future work on analytical methods concentrates on the production of

an improved anaiytical solution. This includes:

a. Development of an improved analytical solution which

accounts for the iinknown energy storage method which generates the

tensile loading necessary to produce the vertical modes.

Chapter 7

References

Rayleigh, Lord J. W. S. T'hem of Sound, volume 1, [pages 306 - 35 11,

Published by Dover books, 1945, f is t publisbed 1877

Tiinioshenko, S. Vibration Problems in Enpineerin~ 3d Edition, D.

VanNostand Company, Inc., 1955

Volterra, E. and Zachmanogiou, E.C. Dvnamics of Vibration, 1965 C. E.

M e r d Books Inc.

Stevens and Bate, Acoustics and Vibrational Physics 2" Ed., Edward

Amold Publishers Ltd., London, 1966

Morse and hgard, Theoretical Acoustics, McGraw - Hill Book Company,

New York, 1968

Mazumdar, J. Transverse Vibration of Membranes of Arbitrary Shape by

the Method of Constant-Deflection Contours, Journal of Sound and

Vibration (1 973) 27(1), 47-57

Sato, K. Forced Vibration Analysis of a Composite Rectangular

Membrane Consisting of Strips, Journal of Sound and Vibration (1979)

63(3), 41 1 4 1 7

Singhal, R.K. and Gonnan, D J. A Superposition - Rayleigh-Ritz method

for Free Vibration Analysis of Non-uniforrnly Tensioned Membranes,

Journal of Sound and Vibration (1993) 162(3), 489-501

Cortinez, V.H. and Laura, P A Vibrations of Non-homogeneous

Rectangular Membranes, Journal of Sound and Vibration (1992) l56(2),

21 7-225

Gorman, D.J. Singhal, R.K. Graham, WB. Crawford, J.M. Investigation

of the Stress Distributions in Corner Tensioned Rectangular Membranes,

AIAA Journal, Vol. 31. No. 12, December 1993

Goman, D.J. Singhal, R.K. Graham, W.B. Crawford, J.M. Investigation

of the Free Vibration of a Rectangular Membrane,, AIAA Joumal, Vol.

32. No. 12, December 1994

Singhal, R.K. and Gorman, D J. Effects of Linearly varying Tension and

Light Hexural Rigidity on Free Vibration of Rectangular Membrane with

Two Free Edges, 14" Canadian Congress of Applled Mechanics, Queen's

University, 30 May4 June 1993

NickoIa, W.E. The Dynamic Response of Thin Membranes by the Moire

Method, IBM Corp., 1966 SESA Spring Meeting, Detroit Michigan, May

4-6 1966

Leith, Emrnett N. and Upatnicks, Juris Photography by Laser,, Scientif'c

Amencan, June 1965 Volume 212 Number 6

Liem, S. Vibration Analvsis bv Hol~gra~hic Interferomeûv, MaSc

Thesis, Mechanical Engineering Department, UBC, B.C., 1970

u61 H u l l , C.R. and Liem, S.D. Vibration of Plates by Real-the

Stroboscopie Holography, Experirnental Mechanics, Vol. 13, No. 8, pp.

339-344, August 1973

~171 Thomson, W.T. and Dahleh, M.D. Theory of Vibration, Prentice Hall,

Upper Saddle River, NJ, 1998

Test notes

The following tests are of the 4*4 (fmed-fied, free-free) membrane test notes

Static tests for the naîural fiequencies of a 4*4 test subject using a fotonic sensor:

Test #

Mode

No*

1 2 3 4 5 6

Test # 1

(al1 in Hz)

1 2 3 4 5 6 avg.

Mode

No*

(aI1 in Hz)

2 3 4 5 6 avg.

Load on the three load wires was 740 grms.

Test numberl plate number 1 - Static image of 4*4 Test

Frequency None - Static Image

Amplitude of input signal None - Static Image

Beam ratio 3/1

Exposure time and laser power setting 12 sec / 270 mw

DeveIoping/stop/Fixing bath 2 min / 30 sec / 2 min

Notes Static image of 1 : 1 aspect ratio.

Free edge A-& B-B

Fixed edges C-C, D-D

Figure A-1 - Static Image for AR 1

Test numbed plate number . 2 - (1st mode 4^4 test)

Frequency 148.1 Hz

Amplitude of input signal 100 microv

Beam ratio 2.811

Exposure time and laser power setting 14 sec 1 280 mw

Developing/stop/Fixing baths 2 min/ 30 sec 1 2 min

Notes The first noticeable thing is that the

fiequency has decreased fiom the previous tests. This is still the first mode (i.e. the

fundamental eequency) as the image corresponds with that of both the first 3 fiames and

the predicted shape. The decrease in the fiequency can be accounted for by the

manipulation of the load bearing Ares. This means that if you touch the wires you will

distort the fictional losses across the load fiame. Therefore, you will increase or

decrease the natural fiequencies as you increase or decrease the stiflkess of the

membrane.

Image not reproduced

Test numbed plate number 3 - (1st mode 4*4 test)

Frequency 146.2 Hz

ArnpIitude of input signal 1 50 microv

Beam ratio 3/1

Exposure t h e and laser power setting 14 sec/ 280 mw

Developing/stop/Fixing baths 2 min 130 s e d 2 min

Notes This second atternpt at the 1' mode

of the 4 by 4 (AR 1) membrane is a beîter representation of the proper amplitude to use

for the membrane size. The image still has a shift to the right, which seems to becoming

a repeatable pattern.

Free edge A-A, B-B

Fixed edges C-C, D-D

Figure A-2 - Image of 1" Mode for AR 1

Test numberl plate number . . 4 - (2nd mode 4*4 test)

Frequency 172 Hz

Amplitude of input signal 2 mv

Beam ratio 3/1

Exposure time and laser power setting 12 sec / 270 mw

Developing/stop/Fixing bath 2 min /30 sec / 2 min

Notes This was the first attempt of a

dynamic hologram that produced viable results. This plate, however, d l had some

problems. The amplitude of the exciting force used on the membrane was set too high.

This, in tuni, caused the number of fiinges within the image to be too large (4W). The

solution for this was to decrease the amplitude of the exciting force used on the

membrane by a factor of 10 times. This, it is hoped, will decrease the number of fi-inge

lines enough that each may then be viewed as an individual and distinct line.

Image not reproduced

Test numberl plate number . . 5 - (2nd mode 4*4 test)

Frequency 170 Hz

Amplitude of input signal 2OOmicrov

Beam ratio 3.211

Exposure time and laser power setting 14 sec1280 mw

Developing/stop/Fixing bath 2 min 130 sec1 2 min

Notes This holograrn is the second attempt

at the &st mode and still has the problem of the amplitude of the exciting force being too

high for the fnnge patterns to properly reproduce. This time, however, the Enge lines

are clear enough for the shape of the mode to be visible. The decreasing of the amplitude

resulted in an improved image. This also brought to light the shifting of the enge

pattern shape to the right. The possible reasons for this shift are delaminating of the

membrane, the unequal distribution of the stress field, and the fixed-fixed boundaries not

being parallel. The solution for the image quality was to again decrease the amplitude of

the exciting force by half. A possible test for this is to purposely distort the boundary

conditions by placing one of the edges at a known angle. A possible test for the unequal

stress field is to pull or load one side of the membrane more than the other side. With the

delarninating of the membrane the only solution is to replace the membrane in total.

Image not reproduced

Test numbed plate number 6 - (2nd mode 4*4 test)

Frequency 170 Hz

Amplitude of input signal 1 O0 rnicrov

Beam ratio 3/1

Exposure tirne and laser power setting 14 sec / 280 mw

Developing/stop/Fixing baths Zmin/30sec /Zmin

Notes Better resolution with the decrease in

the amplitude of the exciting force. The image is still shifted to the right. The same

reasoning for the shift as in plate no. 2 with the same methods for testing.

Free edge A-A, B-B

Fixed edgcs C-C, D-D

Node Line 1-1

Figure A-3 - Image of 2.* Mode for AR 1

Test number/ plate number . 7 - (2nd mode 4*4 test)

Frequency 146.2 Hz

Amplitude of input signal 150 microv

Beam ratio 3/1

Exposure time and laser power setting 14 sec 1280 mw

Developing/stop/Fixing baths 2 min/ 30 sec 1 2 min

Notes This test is of the second mode of the

1: 1 aspect ratio for the membrane. This test improves on the number and visibility of the

fiinge patterns, however the shift to the right is stili present and the number of Snges is

still excessive so the most likely solution is to reduce the amplitude of the exciting force.

Image not reproduced

Test numberf plate number . . 8 - (2nd mode 4*4 test)

Frequency 206 Hz

Amplitude of input signal 1 50 microv

Beam ratio 2.9/1

Exposure time and laser power setting 14 sec 1280 mw

Developing/stop/FWng b a h 2 min 1 30 sec / 2 min

Notes This test produced the best resuIts for

the 1: 1 aspect ratio of the membrane. The nurnber of discernible ninges is 14 and the

spacing is quite easy to see, however the shift to the right is still present at approximately

the 115 point.

Free edge A-A, B-B

Fixed edges C-C, D-D

Node Line 1-1

Figure A-4 - Image of 2& Mode for AR

Test number/ plate number . 9 - (3rd mode 4*4 test)

Frequency 223.8 Hz

Amplitude of input signal 150 microv

Bearn ratio 3 .25/1

Exposure time and laser power setting 14 sec / 280 mw

Developing/stop/Fixing baths 2 min / 30 sec 12 min

Notes This image is the worst of al1

produced thus far. It is under exposed and somewhat hazy. However, the image does

show the three node lines of the fourth mode. These lines are, at best, faint and can only

be discerned by looking at the upper batten. The first node line is very dose to the right

hand edge of the membrane and, as before, is shifted to the right. The second line is

M e r to the left of the first but is still shifted to the nght. The third line is to the left of

the second but, again, it is still shifted to the right. Increasing exposure time c m solve

the problem of image quality.

Image not reproduced

Test number/ plate number 10 - (3rd mode 4*4 test)

Frequency 218 Hz

Amplitude of input signal 100 microv

Beam ratio 3/1

Exposure t h e and laser power setting 14 sec / 280 mw

Developing/stop/Fixing bath 2 min / 30 sec / 2 min

Notes No Image was produced.

Test numberl plate number 11 - (3rd mode 4*4 test)

Frequency 214 Hz

Amplitude of input signal 70 microv

Beam ratio 2.86/1

Exposure time and laser power setting 12 sec / 280 mw

Developing/stop/Fixing b a h 2rninl3Osecl2min

Notes This plate was the third mode of the

1: 1 aspect ratio and although the image in the photo is out of focus this was a good

reproduction. The hologram shows the two node lines of the third mode quite well and,

as before, shows that the node lines are shifted to the right of the membrane. The

problem with the image is the focus of the carnera and not the plate itself

Image not reproduced

Test numberl plate number . . 12 - (3rd mode 4*4 test)

Frequency 215 Hz

Amplitude of input signal 80 microv

Beam ratio 3. I V 1

Exposure time and laser power setting 7 sec / 280 rnw

Developing/stop/Fixing baths 2 min / 30 sec / 2 min

Notes This is a better example of the third

mode of the 1 : 1 aspect ratio of the membrane as the fi-inge lines are more readily

discemible and the image is of sharp quality. The shifk to the nght is again present in this

image.

Load

Free edge A-A, B-B

Fixed edges C-C, D-D

Node Lines 1-4 II-II Figure A-5 - Image of 3d Mode for AR 1

Test numberl plate number . 13 - (4th mode 4*4 test)

Frequency 235 Hz

Amplitude of input signal 1 O0 microv

Beam ratio 3.19/1

Exposure time and laser power setting 7 sec / 280 rnw

Developing/stop/Fixing baths 2 min / 30 sec / 2 min

Notes . This is the second plate of the fourth

mode of the 1: 1 aspect ratio of the membrane. The image in this plate is sharper and

clearer than the previous fourth mode image. The node lines are, again, shifted to the

right of the membrane.

Load

Load

Free edge A-& B-B

Fixed edges C-C, D-D

Node Lines 1-4 II-II, ïïï-lïï Figure A 4 - Image of 4& Mode for AR 1

Test numberl plate number 14 - (2nd mode 4*4 skewed test)

Frequency 142.8 Hz

Amplitude of input signal 100 microv

Beam ratio 311

Exposure time and laser power setting 5 sec / 250 mw

Developing/stop/Fixing baths 2min/30sec/Smin

Notes This plate represents the second

mode of the 1: 1 aspect ratio with the following differences. First the lower batten has

been skewed with the right hand side being higher than the left. This was done to see if a

shift could be forced in the position of the node line fiom that seen in test 8. As seen in

the images below there was no noticeable shifi fkom the original position.

Image not reproduced

Test nurnberl plate number 15 - (2nd mode 4*4 skewed test)

Frequency 173.2 Hz

Amplitude of input signal 100 microv

Beam ratio 3/1

Exposure time and laser power setting 5 sec 1250 mw

Developing/stop/Fixing baths 2 min / 30 sec 1 2 min

Notes This plate represents the second

mode of the 1 : 1 aspect ratio with the following diferences. First the lower batten has

been skewed with the left-hand side being higher than the right. This was done to see if a

shift codd be forced in the position of the node line fiom that seen in test 8. As seen in

the images beIow there was no noticeable shift fkom the original position.

Image not reproduced

The following tests are of the 4*6 ( fd- fmed, free-free) membrane.

Static tests for the natural fiequencies of a 4*6 test subject using a fotonic sensor:

Test #

Mode

No.

1 2 3 4 5 6

Test #

Mode

No.

1 2 3 4 5 6

(aii in Hz)

3 4 5 6 avg.

(al1 in Hz)

3 4 5 6 avg.

Load on the t h e load wires was 740grams.

Test number/ plate number

Frequency

Amplitude of input signal

Beam ratio

Exposure time and laser power setting

Developing/stop/Fixing baths

Notes

Test number/ plate nurnber

Frequenc y

Amplitude of input signal

Beam ratio

Exposure time and laser power setting

Developinglstop/Fixing baths

Notes

16 - (Static 4*6 test)

O &

O microv

4.22/1

5 secI23Omw

2 min / 30 sec/ 2 min

No Image developed.

17 - (Static 4*6 test)

OHz

O microv

31 1

5 sec / 230 rnw

2minl30 sec12min

The image is the first of the 4*6

membrane tests, which give a 1: 1.5 aspect ratio. The image is that of a static membrane

and it is slightly underexposed. The image does, however, show that the static image has

very little deflection due to the ambient noise present in the room. The shape of the one

fnnge is that of a half moon with the right edge showing motion and the left edge

remaining stationary. This shape hints that the mode shapes for this aspect ratio will,

once again, shift to the right.

Image nst reproduced

Test numbed plate number 18 - (Static 4*6 test)

Frequency O&

Amplitude of input signal O microv

Beam ratio 3.11/1

Exposure time and laser power setting 5 sec1230 mw

Developing/stop/Fixing bath 2 min / 30 sec / 2 min

Notes This image is the second static image

of the 4*6 membrane tests, which give a 1: 1.5 aspect ratio. The image is still slightly

under exposai. The image does, however, show the same static image of the 1 : 1.5 aspect

ratio with very little deflection due to ambient noise. The same half moon finge pattern

is visible.

Image not reproduced

Test numberl plate number . 19 - (2nd mode 4*6 test)

Frequency 125.2 Hz

Amplitude of input signal 100 microv

Beam ratio 3.29/1

Exposure time and laser power setting 5 sec/23Omw

Developing/stop/Fixing baths 2 min 1 30 sec / 2 min

Notes No Image deveioped.

Test number/ plate number . 20 - (2nd mode 4*6 test)

Frequency 126.7 Hz

Amplitude of input signal 100 microv

Beam ratio 3.08/1

Exposure time and laser power setting 5 secI230mw

Developing/stop/Fixing baths 2 min130 sec12 min

Notes This is the second mode of the 1 : 1.5

aspect ratio of the membrane. The image shows that the node line is shiRed to the right

of the membrane the same as in the 1: 1 aspect ratio membrane. However, the amplitude

of the membrane in this plate exposwe is slightly fiigh and should be reduced in order to

obtain a better representation of the actual deflection of the membrane surface.

Image not reproduced

Test numberl plate nurnber 21 - (2nd mode 4*6 test)

Frequency 126 Hz

Amplitude of input signal 50 microv

Beam ratio 3.75/1

Exposure time and laser power setting 5 sec / 250 mw

Developing/stop/Fixing baths 2min /30 sec12 min

Notes This is the second exposure of the

second mode of the 1 : 1.5 aspect ratio of the membrane. The image shows that the node

line is shifted to the right of the membrane the same as the 1 : 1 aspect ratio. However the

amplitude of the membrane is high and should be reduced to obtain a better image.

Image not reproduced

Test number/ plate number

Frequency

Amplitude of input signal

Bearn ratio

Exposure time and laser power setting

Developing/stop/Fixing baths

Notes

Test number/ plate number

Frequency

Amplitude of input signal

Beam ratio

Exposure time and laser power setting

Developing/stop/lFixing baths

Notes

22 - (3rd mode 4*6 test)

137.7 Hz

260 microv

3.2/1

6 sec / 260 mw

2min/30sec/2min

No Image developed.

23 - (3rd mode 4*6 test)

130.8 Hz

500 microv

3/ 1

7 sec/25Omw

2min/30sec/2min

The image in this plate is that of the

third mode of the 1 : 1.5 aspect ratio of the membrane and is slightly under exposed as the

lower batten does not reproduce in the photograph of the plate. The image does show up

when viewed directly through the plate. The image shows the two node lines forrning on

the membrane skewed to the right as before.

Image not reproduced

Test numberl plate number . 24 - (4th mode 4*6 test)

Frequency 146.5 Hz

Amplitude of input signal 400 microv

Beam ratio 3/1

Exposure time and laser power setting 7 sec / 250 mw

Developing/stop/Fixing baths 2 min / 30 sec / 2 min

Notes The image is that of the fourth mode

of the 1: 1.5 aspect ratio of the membrane and is slightly under exposed, as the lower

batten does not show up in the photograph of the image. The image does show up when

viewed directly through the plate. The image shows the three node lines forming on the

membrane skewed to the right as before.

Image not reproduced

The following tests are of the 6*4 (frred-fied, free-free) membrane.

Static tests for the natural âequencies of a 6*4 test subject using a fotonic sensor:

Test # 1

Mode

No.

(al1 in Hz)

2 3 4 5 6 avg .

Load on the five load wires was 666grams.

Test number/ plate number . 25 - (static 6*4 test)

Frequency OHZ

Amplitude of input signal O microv

Beam ratio 3.1411

Exposure time and laser power setting 8 sec 1250 mw

Developing/stopB;ixing baths 2 min 1 30 sec / 2 min

Notes This produced a very faint static

image of the 1:0.667 aspect ratio and does show that no visible fkinge patterns are on the

image.

Image not reproduced

Test numbed plate number

Frequency

Amplitude of input signal

Beam ratio

rn 26 - (6*4 2." mode test)

193.9 Hz

550 microv

3/1

Exposure time and laser power setting 8 sec/250mw

Developing/stop/Fixing bath 2 min / 30 sec 12 min

Notes This is the first exposure of the

second mode of the 1 :0.667 aspect ratio. The image shows that the node line is again

shifted. However the amplitude of the membrane is too high to see the Enge patterns.

Image not reproduced

Test numberf plate number 27 - (6*4 2d mode test)

Frequency 196.0 Hz

Amplitude of input signal 500 microv

Beam ratio 2.7711

Exposure time and laser power s d n g 8 sec / 250 mw

Developing/stop/Fixing bath 2 min / 30 sec 1 2 min

Notes This is the second exposure of the

second mode of the 1:0.667 aspect ratio. The image shows that the node line is again

shifted. However the amplitude of the membrane is too high to see the fi-inge patterns.

Image not reproduced

Test numberl plate number . s 28 - (6*4 2d mode test)

Frequency 196.0 Eh

Amplitude of input signal 500 microv

Bearn ratio 2.7711

Exposure time and laser power setting 8 sec / 250 mw

Developing/stop/Fixing bath 2 min / 30 sec / 2 min

Notes This is the third exposure of the

second mode of the 1:0.667 aspect ratio. The image shows that the node line is again

shified. However the amplitude of the membrane is too high to see the Enge patterns.

Image not reproduced

Test num ber/ plate num ber Frequency

Amplitude of input signal

Beam ratio

Exposure time and laser power setting

DevelopinghtopIFixing baths

Notes

29 - (6*4 3" mode test) 224 Hz

400 microv

3.1/1

8 sec / 250 mw

2 min / 30 sec / 2 min

This is the first exposure of the third

mode of the 1:0.667 aspect ratio. The image shows that the node line is again shifted.

However the amplitude of the membrane is too high to see the fiinge patterns.

Image not reproduced

Appendix B

Design Drawings

&

Test Equipment Design Notes

The membrane test apparatus was constmcted fiom steel to elinhate

transmission of vibration noise generated by a flexible structure. The steel was wld roiîed

stock readily available fiom the Dal-Tech machine shop. AU manufachiring was done on

site by the resident machinist, Mr. A. Macpherson. The final test structure was assembled

as shown in Figure 4.2 in chapter 4, again presented here for convenience.

Figure 4.2 - The Experimental Membrane Holder.

The bottom clamp of the test apparatus was constructed fiom a block of

steel measuring 10.5" by 2.5" by I " . The clamp was produced to the specifications in

Figure B. 1 below .

Figure B.1- Bottom Clamp - Membrane Test Apparatus

The bottom clamp was attached to the test table using course thread '/d inch

bolts. The wires attached to the membrane battens were pinned in the groves cut into the

clamp.

B.2 The Experimentd Membrane Holder - Side plates.

The side plates were out from H inch plate steel. The side plates were

produced to the specifications in Figure B.2 below.

Figure B.2 - Side Plates - Membrane Test Apparatus

The side plates were attached to the center plate using course thread %

inch bolts. The membrane rolier-battens were attached to the side plates by placing the

threaded portions of the roiler batkm through the dots out in the side plates and fastened

by means of lock nuts.

B.3 The Experimental Membrane Holder - Top wire roliers.

The top wire roliers were constructed using % inch round stock steel. The

length of each of the rollers was 14" with the frrst inch of each end turned down to ?4 inch

and then threaded. Five 1/8 inch groves were cut into the rollers at 1.5 inch intervals.

These groves were centered on the rouer.

B.4 The Experimental Membrane Holder - Center plate.

The two center plates of the membrane test apparatus were constructed

using two blockç of steel one measuring 12" by 2.5" by 1' and the other measuring 10" by

2.5" by 1'. Three ?h inch holes were drilled into the 2.5 inch side of the bar at 1.5"

intervals with the center hole at the center of the bar. Two ?A inch diameter holes were

then d d e d 2" into each end of the bar separated by 1 ". These four holes were then

tapped and the side plates were attached to 12" plate and the membrane batten side plates

were attached to 10" plate. 330th of the plates were attached to the optical stand.

B.5 The Experimental Membrane Holder - Membrane der-battens.

The membrane roller-battens were constructed using % inch round stock

steel. The length of the rollers was 14" with the fkst inch of each end tunieci d o m to ?4

inch and then threaded. The roller-battens were then attached to the membrane batten

side plates by placing the threaded portions into the groves cut in the batten side plates

then lock nuts were used to hold them.

B.6 The Experimentai Membrane Holder - Membrane batten attachments.

The membrane batten attachments were made fi-om 118 inch 1" wide

aluminum stock. They were glued on both ends and both sides of the membrane ends and

then holes for the wires were driiled at 1.5'' intervais.

B.7 The Experirnental Membrane Holder - Membrane batten side plates.

The membrane batten side plates were cut fiom '/4 inch plate steel. They

were of an eccentric "T" shape with a ?4 inch hole drilled in the short end of the "Tfl and a

4" long % inch wide dot cut into the other end. These cuts gave the membrane a length

varying between 4" to 8". The center bar of the "T" was 8" long. These side battens were

attached as descr-ibed above.

B.8 The Experimental Membrane Holder - Weight containers.

The weight containers used for the test apparatus were simple spray cm

tops attached to the wires providing the tension to the membrane. The mass was provided

by the use of lead shot placed in the can tops.

To interpret the information presented within this thesis the developrnent

of a set of dimensional parameters that describe the problem under study was required.

The use of such dimensional parameters WU always reduce the nuniber of variables in a

problem equal to the number of base dimensions involved. In this problem it is assumed

that there are nine significant variables, as shown fïrst in table 5.2 riepeated below.

TabIe 5.2 - Simiif;cant Variables and their Dimensions

Name of variable

Surface density of the membrane

Length of membrane

Width of membrane

S m e s s per unit width

Tension per unit width

Densiîy of the medium

Frequency of vibration

Mode of vibration

Characteristic amplitude of vibration

Symbol

Pt

b

a

E T

Pa

Cr)

id

A

The method of dimensionai parameters is to set three repeating variables

and then form groups with them for each of the remaining variables and to fonn an

equation in which ail of the dimensions cancel out. By doing this the resultant equation

becomes dîmensionless.

The tbree variables that are chosen to remain in any of the groups are:

Name of variable Symbol Uniîs

Surface density of the membrane p, MIL2

Length of membrane b L

Tension per unit width T M/t2

For exarnple, the fïrst dimensionless parameter fkquency can be

established using as the required variable and with a choice of the repeating

variables n, can be developed. The equation cm be wxitten:

Three auxiliary equations with three unknowns can be written. Solving

simultaneously x, y, and z are found to be 1, -1/2, and +1/2 respectively and this gives.

Frequency Ratio

By foliowing this technique, or by inspection, the remaining paramekm

can be determined as follows.

Second Parameter: The Density

Density Ratio

Third Parameter: The Stfiess per unit width

E ml=$ Initial Strain Ratio

Fourth Parameter: The Aspect Ratio

Fifth Parameter: The Amplitude Ratio

Aspect Ratio

Amplitude Ratio

Sixth Parametex The mode nurnber

TT6= Mode N d e r Mode Number

Al1 of the above parameters meet the Pi theorem in that each has at least

one of the three repeating variables dong wiîh a variable which is used but once.