Experimental Modelling in Engineering

194 Pages·1982·3.594 MB·English

by  F. W. David and H. Nolle (Auth.)

The cover picture shows a 1 :100 scale model of the hotel rig Alexander L
Kielland, constructed from perspex. The model is designed to accurately
represent the ballasting and buoyancy configurations of the actual vessel for
the purposes of devising a method for uprighting the capsized vessel in
Norway.
The method was implemented by the Structural Dynamics Group of
Southampton at the end of 1980 in a salvage attempt that was later halted.
Tank testing was used extensively in devising a ballasting sequence and in
the course of the actual operation in Norway. The model was also gimbal
mounted and used as a training device for diver instruction.
Experimental Modelling
in Engineering
F. W. David and H. Nolle
Butterworths
London Boston Sydney Wellington Durban Toronto
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price given by the Publishers in their current price list.
First published 1982
© F. W. David and H. Nolle, 1982
British Library Cataloguing in Publication Data
David, F. W.
Experimental modelling in engineering.
1. Engineering design - Models
I. Title II. Nolle, H.
620'.0042'0724 TA177
ISBN 0-408-01139-4
Printed in England by Manse/I (Bookbinders) Ltd., Witham, Essex
PREFACE
Experiments using models have been carried out for many years in a quest to discover
and understand the behaviour and the properties of physical systems. Until the turn of
this century, experimental technique followed largely an ad hoc procedure. This produced
results of varying success, but usually the experiments were performed with narrow
specific objectives in mind, and consequently the results would turn out to have very
limited generality.
The origins of the more systematic approach to modelling and model testing may be
traced back to scientific inquiry and the study of natural phenomena, particularly in the
field of fluid flow. The practical advantages of modelling under these so called
"conditions of similarity" were recognized soon after and put to use in a number of fields
of engineering. From these origins the art and science of experimental modelling have
continued to evolve and today form an integral part of practically all branches of Physics
and Engineering.
Fourier, in the early part of the 19th century, and Rayleigh some 50 years later were
the first to formally establish the principles for similarity of physical systems,
conditions which are now derived by the well-known process called "dimensional analysis".
On these foundations further theoretical refinements have been attempted, from time to
time; however, these contributions, mainly of an abstract nature, have not added
substantially to the existing knowledge. On the other hand, the basic theories of
dimensional analysis and similarity have found ever increasing application to cover the
needs of the fast developing engineering activities in two ways: firstly, they have been
applied to experimental techniques, and secondly, they have been used to assist in
analytical investigations of physical phenomena generally.
The history of scientific records reveals that the seed for the fundamental theorem
for similarity the now well-known ir-theorem was sown by a Frenchman A. Vaschy in
about 1896. This Theorem was subsequently re-defined by a number of other writers, but
it was not till Buckingham's formulation was published in 1914 that the Theorem and its
practical significance became more widely known and accepted by the engineering community.
Since then a great deal of literature has been accumulated, mostly concerned with
specialist's analyses and procedures, and with some theoretical formulations. Among the
latter, two concepts have been advanced that have, however, very limited, if any,
practical application. They are the directional or vector properties of variables and
the dual role of mass. Both concepts are briefly discussed in this text, and some worked
examples of the first are used to illustrate the severe limitations of its usefulness for
experimental modelling.
In this volume the principles of experimental modelling are presented methodically
and in such a generalized way that they may lend themselves to application in practically
all fields of technology.
Though only the basic concepts and procedures are developed, these are copiously
illustrated by examples taken from a wide field of practical applications. By careful
selection, fully worked examples from disciplines in which modelling is well established
a * tradition', so to speak are balanced by others taken from areas of technology
in which the modelling technique is relatively new, yet potentially powerful, e.g. in the
performance prediction of all-terrain vehicle-soil systems.
Chapters 1 to 4 present a compact yet comprehensive review of ideas pertaining to
physical units and dimensionality and the theory of non-dimensional formulations.
Commencing in Chapters 5 and 6 a range of commonly encountered questions and problems are
examined, arising in the course of dimensional analysis and during the early stages of
model design. In Chapter 7 the fundamental principle of similarity of physical systems
is defined, and a number of essential concepts are discussed: these are the 'scaling
factor', 'homology' and the 'characteristic variable'. Further, in Chapter 7, a set of
guidelines to procedure in modelling are developed. These include also the modelling of
possible input functions; compatibility conditions for them are examined which will
ensure that the linear relations in homology are preserved. Further, the subject of
"distortion" is considered and particular care is taken in defining the meaning and
significance of model distortion, a term often rather loosely used and much abused. The
concept of distortion is illustrated by detailed examples. The Chapter concludes with a
treatise of the commonly encountered similarity conditions which provide a foundation
knowledge to most of the modelling problems in engineering practice.
Scale effects the errors and uncertainties arising from scale-dependence of
physical interactions, or from incomplete knowledge or understanding of the phenomenon of
the system under investigation is the final topic in the theory of modelling, and is
presented in Chapter 8.
In Chapter 9, theory and learning from past experiences have been applied in a
series of fully worked cases involving model testing. These are taken from recently
published technical literature, and all have been chosen for their general interest in
their respective fields. In most instances the reader is shown experimental data which
are used to assess the model behaviour and relevant scaling laws. Where appropriate, the
results are subjected to a critical appraisal, and limitations of their applicability are
pointed out. In a sense, the examples constitute precedents in modelling and
experimental procedure, and as such should be of particular interest to those engaged in
engineering and the applied sciences. But the practical bounds of the modelling
techniques are determined to a large extent by the degree of competence, experience and
imagination of the experimenter, and it is hoped that the selected range of applications
in this Chapter will serve also as a guide in respect of procedural matters and as a
source of inspiration and ideas for new and as yet unexplored problems.
Chapter 10 provides further material for exercises of varying difficulty, covering a
wide field of technological and physical situations.
The modelling conditions for input functions discussed in Chapter 7 are dealt with
in greater detail in Chapter 12; as well, further information is given on some
fundamental relationships between physical quantities used in electromagnetic systems,
and some definitions of such physical quantities.
In Chapter 13, solutions to the exercise problems have been given, with accompanying
comment, where appropriate, to enhance the teaching value of the exercise.
The practice of experimental modelling continues to gain increasing acceptance at all
levels of research, development and industrial design activity; and new applications for
the technique are developed to this day. It would be outside the objectives of this
fundamental treatise to provide a general listing of original works dealing with modelling
and model testing. Such references number many hundreds and have already been subject to
comprehensive literature searches and listings of titles in many specialist areas of
technology. At the conclusion of this book the reader is referred to the various sources
where such listings appear. The short list of journal titles represents a selection
which usually carries articles on the subject of modelling, and their latest issues should
provide a useful starting point in a search of the most up to date publications.
In the numerous footnotes references are given to papers from which the various
examples in the text were evolved. These papers usually include a bibliography that
would be of interest to readers who wish to study in greater detail some of the special
problems under discussion.
The origin of this book stems from a lecture series presented at Monash University as
part of a final year undergraduate unit called Engineering Morphology. In these lectures,
the principles of similarity were established, to serve as a foundation for a broader
study of 'shape1 and 'form' in Nature and in man-made systems, and the forces that have
influenced their evolution towards optimality and, not infrequently, beyond optimality
and onwards to the final stage in the evolutionary cycle: extinction. In the course of
these developments, the authors were inspired by the classic work of D'Arcy Thompson, "On
Growth and Form"*, which is as fresh today as it was in 1917 when it was first published.
* Cambridge University Press, 1966.
From there, the contents of some of the lecture notes themselves evolved into deeper, more
specialized study of systems encountered specifically in Physics and Engineering.
The authors wish to take this opportunity to acknowledge the contribution of their
colleagues, Professors John D.C. Crisp and Kenneth H. Hunt and Dr. George Sved, for their
encouragement and valuable advice and suggestions during the final stages in the writing
of the manuscript. The preparation of the work was greatly facilitated by the support of
the Department of Mechanical Engineering, Monash University, for which the authors express
their sincere appreciation. They are indebted also to Mrs. L. Ryan, for her conscientious
interest taken in the typing of the manuscript.
This text is written for undergraduate and graduate students in Engineering or in the
Sciences, and for practising professionals who wish to familiarize themselves in some
depth with the theory and application of experimental modelling techniques, or who wish to
refresh their knowledge in these areas. The reader is expected to possess some knowledge
of applied mathematics and a working knowledge of Physics or Applied Mechanics usually
offered in the first year courses of tertiary institutions.
1
1. INTRODUCTION: MODELLING BASED ON CONDITIONS OF SIMILARITY
The technique of modelling may be applied to any physical system that requires
experimental investigations for the solution of specific problems.
A physical system is understood to be an assembly of engineering components and/or
parts of the natural physical world which respond individually or in their entirety to
physical inputs by converting them to physical outputs. An example of a 'system' would
be an all-terrain vehicle, driven over snow covered ground. Its performance would be
gauged by the available draw-bar pull (the output) for a given drive axle torque (the
input). The output in this case is a measure of the capability of the vehicle to move a
trailer or other payload across the terrain, and hence a measure also of the effectiveness
of the vehicle design. The behaviour of such systems may be investigated by experimental
techniques, i.e. by subjecting the system to planned test conditions and observing or
measuring the system response. For systems that are large and of a complex nature such
investigations are preferably carried out on a replica of the system, called the model,
made to a smaller scale for reasons of economy, convenience and savings in time. Such is
the case with the design and development of aircraft, tall structures, oceanic vessels,
large dams, harbours and bridges, and many other technologically advanced systems where
performance and behaviour have to be predicted with confidence to a high degree of
accuracy. There are, however, also instances where the original system, called the
prototype, is very small and where a scaled-up model system is then used to advantage.
This variability of physical size is one of the most important features of modelling, and
its practical value will be fully explored in later chapters.
In analogy with physical size, the question may be asked of time does the test on
the model have to take as long as do the processes involved in the prototype? How, for
instance, does one study the gradual silting up of harbours or the erosion of river banks,
processes that may take a number of years? Does the experimenter have to take measure-
ments for years? The answer is no, because the modelling technique allows the
'telescoping' or 'expansion' of time scales for the modelling process, similar to the
scaling of size. Thus years may be reduced to hours or days, and the desired information
may be obtained in a relatively short and convenient time span.
A mandatory condition on which experimental modelling is based is that the model
system and the prototype system obey the same physical laws. Furthermore, the model
system must be constructed so as to embody all the relevant features and parts of the
prototype system. (If cognizance is taken of the fact that some physical features of the
prototype may not be relevant to the phenomena under investigation, the design and
construction of the model can usually be simplified, without impairing the model
performance.) If these conditions are satisfied, one may expect that a unique i.e.
numerically single-valued relationship exists between the behaviour of the prototype
system on the one hand, and the model system on the other. Consequently, the results of
any experiments carried out on the model system could directly be related to the prototype
system, thus providing a sought-after solution of the problem for the prototype.
The unique relationship between prototype and model is broadly referred to as
similarity. The condition of similarity between systems may be established by a
procedure called Dimensional Analysis hereinafter abbreviated to DA. As will be shown
later, this procedure is based on the fact that all general relationships in Physics and
Engineering are expressed by homogeneous equations. When methods or time for finding a
generally valid solution of a problem are not immediately available, DA may be applied
with advantage, i.e. use may be made of this property of dimensional homogeneity which
allows a problem to be attacked solely from the 'outside'. (By this is meant that
information is gleaned through use of a technique that does not reach into the core of a
physical phenomenon and therefore does not necessarily disclose or explain the underlying
physical processes.) If, however, a problem is to be fully explored and if answers are
to be found to the question "why?", this can only be done by analytical treatment and/or
experimental investigations.
In short, DA is completely useless as a tool for the discovery of fundamental
physical (numerical) laws. However, analytical treatment is in many instances
facilitated by a preceding DA study, particularly in cases involving non-linear
2
differential equations with given boundary and initial conditions, and especially if this
study is followed by experimental work on the prototype or on suitably scaled models.
The technique that deals with the relationships of prototype and model is called
modelling. Modelling, in the narrow sense, is an experimental technique and its
application is based essentially on dimensional considerations. The technique thus puts
to use certain dimensional properties of the variables appearing in the problem which are
arranged in non-dimensional groupings. The advantages of this process are that it will
reveal where simplifications are possible in the design of the experiment; it also
facilitates the systematic and concise presentation of experimental data, and a certain
generalization of the results of a limited test programme.
It should be re-emphasized, moreover, that results of model tests are empirical
results only which do not necessarily disclose the physical laws in force. As will be
seen from some examples in modelling in Chapter 9, it is normal practice to generalize the
results of experimental investigation by first graphing the experimental data and then
drawing through the points a curve of best fit. The curve is meant to represent an
analytical equation relating the variables of the problem, and the values of the constants
of the equation are found by the usual techniques that minimize some selected function of
the errors, or simply by using a measure of good judgement when drawing the line that fits
the points. The choice of the equation is entirely arbitrary, and the equation does not,
in most cases, express a general physical law.
The effectiveness of the technique may be enhanced through the use of information
about the system under investigation that has analytical or empirical origin. When such
information can be found, both model design and subsequent experimental work can be
simplified to a degree, depending on the amount of information available. This becomes
important in all areas of engineering where one faces limitations brought about, for
example, by complex and little understood properties of materials, shortcomings of
laboratory instrumentation, insufficiencies in financial and technical resources and by
the inescapable lack of time. A carefully devised model and experimental programme
usually can overcome most, if not all, of these difficulties, and it is the purpose of
this text to give the reader a thorough introduction to all aspects of experimental
modelling that may help to achieve economically and technically acceptable solutions.
3
2. UNITS AND DIMENSIONS
2.1 Primary and Secondary Quantities
The measurement of physical quantities is an essential part of the practice of
Physics and Engineering. These quantities obey the laws of Physics, and are therefore
related to each other in definite ways. For example, Newton's second law of motion
relates the quantities of force, mass and acceleration, and hence, also those of length
and time for all dynamical measurements in Newtonian Mechanics. Taking into account the
basic definition of acceleration, it follows, that of the remaining four quantities
(force, mass, length and time) only three are mutually independent, and for measuring
purposes any units for these three may be freely chosen. The concept of a physical
"unit" gives rise to the concept of a "scale" of units for the purpose of making
comparisons. In fact, all physical measurements are, by definition, comparisons between
two like quantities; the measuring tape, the mercury thermometer, the clock and any
digital counter being examples of scaling devices for measuring the magnitudes of physical
quantities.
In the foregoing example mass, length and time may be selected as the three basic
independent measures to establish the magnitude of any particular force. However, before
measurements can be made, a measuring system must be adopted defining the units for mass,
length and time (or acceleration, in lieu of the latter two). The choice of the
respective units can be entirely arbitrary, and is governed usually by the practicalities
of establishing and maintaining stable references against which all physical measurements
may be compared. Once the units are defined, substitution of any set of measured values
of length and time (or acceleration) and mass into Newton's equation of motion gives the
numerical value of the magnitude of the force acting on the mass. The units of force
would then be defined in terms of the chosen measuring system.
This illustration of the method for determining the magnitude of a quantity in
Mechanics reveals the elements of physical measurement generally. It is now essential to
formally define the properties required of such measuring systems, to permit the measure-
ment of any physical quantity. Historically, it is interesting to note that Descartes
(1596-1650) considered a similar aspect, but from a much broader and philosophical point
of view. He defined the objectives of the exact sciences to be the description of all
happenings in terms of three fundamental concepts, namely space and time as the fovm, and
matter as the substance of the real world, as it was known at his time*.
When subsequently Newton (1642-1727) laid the foundation of Mechanics as a fundamental
science, he arrived at a similar conclusion, but for the narrower field of Mechanics. He
concluded that there are only three types of measuring units required to define both the
nature or dimension and the magnitude of any quantity in Mechanics, i.e. any quantity in
Mechanics can be fully defined by three basic quantities. Newton's special result
(referring to Mechanics) may be generalized to embrace all physical events, viz. the
relationship between the nature or dimension of a physical quantity and the basic physical
quantities is given either by the laws of Physics, or by specific definitions.
As may be observed, these laws and definitions, throughout Physics and Engineering,
have one characteristic in common, namely, they express the dimension of a physical
quantity as the product of the powers of the basic quantities. Take, for example, the
nature of the physical quantities "velocity", "acceleration" and "force" which are defined
as follows:
velocity v = length x time-1 ,
acceleration a = velocity x time"1 = length x time-2 ,
and the dimension of force is given by Newton's second law:
force F = mass x acceleration = mass x length x time-2 .
These expressions give no indication of the magnitude of the respective physical quantities,
but give solely their relationships to the basic quantities, the latter also called the
* Weyl, H. - Space-Time-Matter, Dover Publications, Inc., 1922.
4
primary quantities (P.Q.'s). For these, the symbols M (mass), L (length) and T
(time) will be used. Re-writing the above expressions, gives then
nature of velocity = L x T_1 ,
nature of acceleration = L x T~2 ,
nature of force = M x L x T-2 .
In order to signify that relationships like the above ones express only the nature or
dimension of a physical quantity, regardless of its magnitude, square bracket symbols will
be used, so that the above relationships may be expressed as
[v] = [L x T"1] ,
[a] = [L x T~2] ,
[F] = [M x L x T"2] .
These are read as the dimension of velocity is length divided by time etc. The so
derived quantities are called secondary quantities.
At the 10th General Conference on Weights and Measures, in 1955, the P.Q.'s M , L
and T were adopted as base dimensions for Mechanics*. The concepts of M , L and T
are mutually independent, since as far as the concept of M is concerned, any possible
association with a length dimension is irrelevant, and likewise no time dependency is
involved for M in Newtonian Mechanics. Similarly, the concept of L is time
independent and free of any relationship with mass.
The relationship between the dimensions of a secondary quantity and the P.Q.'s,
mentioned above, can be derived as follows.
For this purpose use is made of the fact that all physical and engineering equations
are dimensionally homogeneous, since they represent comparisons between quantities that
have the same physical nature. Assume now a physical quantity in Mechanics to be x .
Then the relationship with the P.Q.'s may be represented by an unknown function f( )
such that
x = f(V V V '
where x , x and x are the respective amounts of the P.Q.'s M , L and T ,
M LT
contained in x , in a manner to be established. An approximation theorem in mathematics
by Weierstrasst states that if f( ) is a function, continuous within a certain range,
then f( ) can be approximated within that range to any degree of accuracy by a series
expression
x = cfi (x, x . x ) \ = C^ .x a.i x.x t>-i. x ci y + C^.x a?z .x boz .x C^o + ...
M LT 1 M LT 2 M LT
where C.(i =1, 2, ...) are non-dimensional coefficients. Since the equation is
dimensionally homogeneous, it is necessary that
ai=a2=...=a; b1=b2=...=b; Cl=e2- ... =c .
Hence, the equation may simply be written as
x = f(x. XT , x ) = C.xa.x .x° .
M LT M LT
and x are the respective amounts of the P.Q.'s contained in
it follows that
* Kaye, G.W.C. and Laby, T.H. - Tables of physical and chemical constants, 13th Ed.,
London, Longmans, 1966.
t Frank, P. and Mises, R. - Die Differential und Integralgleichungen der Mechanik und
Physik, v.l, 2nd Ed., New York, Dover Publications, Inc., 1961, pp 219-221.