Online Library → Henry Moseley → The mechanical principals of engineering and architecture → online text (page 1 of 52)

Font size

THE

MECHANICAL PRINCIPLES

ENGINES KING

ARCHITECTURE.

HENRY MOSELET, I. A. F.R.S.

,

CHAPLAIN IN ORDINARY TO THE QUEEN, CANON OF BRISTOL, VICAR OF OLVESTON ;

CORRESPONDING MEMBER OF TUG INSTITUTE OF FRANCE, AND FORMERLY PROFESSOB

OF NATURAL PHILOSOPHY AND ASTRONOMY IN KING'S COLLEGE, LONDON.

Second American from Second London Edition

WITH ADDITIONS BY

D. H. M A H A N , LL.D.

U. S. MILITARY ACADEMY.

V.MTIJ ri, LUSTRATIONS ON WOOD.

NEW YORK :

JOHN WILEY & SON, 535 BROADWAY

1

ENTERFO according *o Act of Congress, <jt\ the year 1856, Ij 1

WILEY & HA-LSTRD,

fc the Clerk's Office of the District Court of the United States, for the Southu a District

of New York.

EDITOR'S PREFACE.

THE high place that Professor Moseley occupies in the

scientific world, as an original investigator, and the clear-

ness and elegance of the methods he has employed in this

work have made it a standard text book on the subjects it

treats of. In undertaking its revision for the press, at the

request of the publishers of this edition, it has been deemed

advisable, in view of the class of students into whose hands

it may fall, to make some slight addition to the original.

This has been done in the way of Notes thrown into an

Appendix, the matter of which has been gathered from

various authorities ; but chiefly from notes taken by the

editor, whilst a pupil at the French military school at Metz,

of lectures delivered by General Poncelet, at that time, 1829,

professor in that school. It is a source of great pleasure to

the editor to have this opportunity of publicly acknowledg-

ing his obligations to the teachings of this eminent savan,

who is distinguished not more for his high scientific attain-

ment, and the advancement he has given to mechanical

science, than for having brought these to minister to the

wants of the industrial classes, the intelligent success of

whose operations depends so much upon mechanical science,

by presenting it in a form to render it attainable by the most

ordinary capacities.

Hi

iv EDITOR'S PREFACE.

The editor would remark that lie has carefully refrained

from making any alterations in the text revised, except cor-

rections of typographical errors, and in one instance where,

from a repetition of apparently one of these, he apprehended

some difficulty might be offered to the student if allowed

to remain exactly as printed in the original.

UNITED STATES MILITARY ACADEMY,

Went Point March 8, 1866.

PKEFACE TO THE SECOND EDITION.

I HAVE added in this Edition articles : first, " On the

Dynamical Stability of Floating Bodies ;" secondly, " On

the Kolling of a Cylinder ;" thirdly, " On the descent of a

body upon an inclined plane, when subjected to variations of

temperature, which would otherwise rest upon it ;" fourthly,

u On the state bordering upon motion of a body moveable

about a cylindrical axis of finite dimensions, when acted

upon by any number of pressures."

The conditions of the dynamical stability of floating

bodies include those of the rolling and pitching motion of

ships. The discussion of the rolling motion of a cylinder

includes that of the rocking motion to which a locomotive

engine is subject, when its driving wheels are falsely

balanced, and that of the slip of the wheel due to the same

cause. The descent of a body upon an inclined plane

when subjected to variations in temperature, which other-

wise would rest upon it, appears to explain satisfactorily the

descent of glaciers.

The numerous corrections made in the text, I owe chiefly

to my old pupils at King's College, to whom the lectures

of which it contains the substance, were addressed. For

VI PREFACE TO THE SECOND EDITION.

several important ones I am, however, indebted to Mr

Eobinson, Master of the School for Shipwrights' Apprentices,

in Her Majesty's Dockyard, Portsea ; to whom I have also to

express my warm acknowledgments for the care with

which he has corrected the proof sheets whilst going through

the press.

May, 1855

PREFACE.

IN the following work, I have proposed to myself to apply:

the principles of mechanics to the discussion of the most

important and obvious of those questions which present

themselves in the practice of the engineer and the architect ;

and I have sought to include in that discussion all the

circumstances on which the practical solution of such ques-

tions may be assumed to depend. It includes the substance

of a course of lectures delivered, to > the students of King's

College in the department of engineering and architecture,

during the years 1840, 1841, 1842.*

In the first part I have treated of those portions of the

science of STATICS, which have their application in the theory

of machines and the theory of construction.

In the second, of the science of DYNAMICS, and, under this

head, particularly of that union of a continued pressure with

a continued motion which has received from English writers

the various names of "dynamical effect," "efficiency," "work

done," "labouring force," "work," &c. ; and "moment

d'activite"," "quantite d' action," "puissance mecanique,"

" travail," from French writers.

Among the latter this variety of terms has at length given

place to the most intelligible and the simplest of them,,

* The first 170 pages of the work were printed for the use of my pupils in the-

year 1840. Copies of them were about the same time in the possession of

several of my friends in the Universities.

Vlll PREFACE.

" travail." The English word " work " is the obvious trans-

lation of " travail," and the use of it appears to be recom-

mended by the same considerations. The work of overcoming

a pressure of one pound through a space of one foot has, in

this country, been taken as the unit, in terms of which any

other amount of work is estimated ; and in France, the work

of overcoming a pressure of one kilogramme through a space

of one metre. M. Dupiii has proposed the application of the

term dyname to this unit.

I have gladly sheltered myself from the charge of having

contributed to increase the vocabulary of scientific words,

by assuming the obvious term " unit of work " to represent

concisely and conveniently enough the idea which is attached

to it.

The work of any pressure operating through any space is

evidently measured in terms of such units, oy multiplying

the number of pounds in the pressure by the number of feet

in the space, if the direction of the pressure be continually

that in which the space is described. If not, it follows, by

a simple geometrical deduction, that it is measured by the

product of the number of pounds in the pressure, by the

number of feet in the projection of the space described,*

upon the direction of the pressure ; that is, by the product

of the pressure by its virtual velocity. Thus, then, we

conclude at once, by the principle of virtual velocities, that

if a machine work under a constant equilibrium of the

pressures applied to. it, or if it work uniformly, then is the

aggregate work of those pressures which tend to accelerate

its motion equal to the aggregate work of those which tend

to retard it ; and, by the principle of vis viva, that if the

machine do not work under an equilibrium of the forces

impressed upon it, then is the aggregate work of those which

tend to accelerate the motion of the machine greater or less

* If the direction of the pressure renfain always parallel to itself, the space

described may be any finite space ; if it do not, the space is understood to be

so small, that the direction of the pressure may be supposed to remain parallel

to itself whilst that space is described.

PREFACE. IX

than the aggregate work of those which tend to retard its

motion by one half the aggregate of the vires vivce acquired

or lost by the moving parts of the system, whilst the work is

being done upon it. In no respect have the labours of the

illustrious president of the Academy of Sciences more con-

tributed to the development of the theory of machines than

in the application which he has so successfully made to it of

this principle of vis viva.* In the elementary discussion of

this principle, which is given by M. Poncelet, in the intro-

duction to his Mecanique Industrielle, he has revived the

term vis inertia (vis inertias, vis insita, Newton), and,

associating with it the definitive idea of a force of resistance

opposed to the acceleration or the retardation of a body's

motion, he has shown (Arts. 66. and 122.) the work expended

in overcoming this resistance through any space, to be

measured by one half the vis viva accumulated through the

space ; so that throwing into the consideration of the forces

under which a machine works, the vires inerticB of its moving

elements, and observing that one half of their aggregate vis

viva is equal to the aggregate work of their vires inertice, it

follows, by the principle of virtual velocities, that the differ-

ence between the aggregate work of those forces impressed

upon a machine, which tend to accelerate its motion, and

the aggregate work of those which tend to retard the motion,

is equal to the aggregate work of the vires inerticB of the

moving parts of the machine : under which form the prin-

ciple of vis viva resolves itself into the principle of virtual

velocities. So many difficulties, however, oppose themselves

to the introduction of the term vis inertice, associated with

the definitive idea of a force, into the discussion of questions

of mechanics, and especially of practical and elementary

mechanics, that I have thought it desirable to avoid it. It

is with this view that I have given a new interpretation to

that function of the velocity of a moving body which is

known as its vis viva. One half that function I have inter-

preted to represent the number of units of work accumulated

* See Poncelet, Mecanique Industrielle, troisieme partie.

PREFACE.

in the body so long as its motion is continued. This number

of units of work it is capable of reproducing upon any resist-

ance opposed to its motion. A very simple investigation

(Art. 66.) establishes the truth of this interpretation, and

gives to the principle of vis viva the following more simple

enunciation : " The difference between the aggregate work

done upon the machine, during any time, by those forces

which tend to accelerate the motion, and the aggregate

work, during the same time, of those which tend to retard

the motion, is equal to the aggregate number of units of

work accumulated in the moving parts of the machine

during that time if the former aggregate exceed the latter,

and lost from them during that time if the former aggregate

fall short of the latter." Tims, then, if the aggregate work

of the forces which tend to accelerate the motion of a

machine exceeds that of the forces which tend to retard it,

then is the surplus work (that done upon the driving points,

above that expended upon the prejudicial resistances and

upon the working points) continually accumulated in the

moving elements of the machine, and their motion is thereby

continually accelerated. And if the former aggregate be

less than the latter, then is the deficiency supplied from the

work already accumulated in the moving elements, so that

their motion is in this case continually retarded.

The moving power divides itself whilst it operates in a

machine, first, into that which overcomes the prejudicial

resistances of the machine, or those which are opposed by

friction and other causes, uselessly absorbing the work in its

transmission. Secondly, into that which accelerates the

motion of the various moving parts of the machine, and which

accumulates in them so long as the work done by the moving

power upon it exceeds that expended upon the various

resistances opposed to the motion of the machine. Thirdly,

into that which overcomes the useful resistances, or those

which are opposed to the motion of the machine at the

working point, or points, by the useful work which is done

by it.

PREFACE. XI

Between these three elements there obtains in every

machine a mathematical relation, which I have called its

MODULUS. The general form of this modulus I have discussed

in a memoir on the " Theory of Machines " published in the

Philosophical Transactions for the year 1841. The deter-

mination of the particular moduli of those elements of

machinery which are most commonly in use, is the subject

of the third part of the following work. From a combination

of the moduli of any such elements there results at once the

modulus of the machine compounded of them."

"When a machine has acquired a state of uniform motion,

work ceases to accumulate in its moving elements, and its

modulus assumes the form of a direct relation between the

work done by the motive power upon its driving point and

that yielded at its working points. I have determined by a

general method' 35 ' the modulus in this case, from that statical

relation between the driving and working pressures upon

the machine which obtains in the sfate bordering upon its

motion, and which may be deduced from the known condi-

tions of equilibrium and the established laws of friction. In

making this deduction I have, in every case, availed myself

of the following principle, first published in my paper on the

theory of the arch, read before the Cambridge Philosophical

Society in Dec. 1833, and printed in their Transactions of

the following year: "In the state bordering upon motion

of one body upon the surface of another, the resultant

pressure upon their common surface of contact is inclined

to the normal, at an angle whose tangent is equal to the

coefficient of friction."

This angle I have called the limiting angle of resistance.

Its values calculated, in respect to a great variety of surfaces

of contact, are given in a table at the conclusion of the

second part, from the admirable experiments of M. Morin,f

into the mechanical details of which precautions have been

introduced hitherto unknown to experiments of this class,

* Art. 152. See Phil. Trans., 1841, p. 290.

f Nouvelles Experiences sur le Frottement, Paris, 1833.

Xll PEEFACE.

and which have given to our knowledge of the laws of

friction a precision and a certainty hitherto unhoped for.

Of the various elements of machinery those which rotate

about cylindrical axes are of the most frequent occurrence

and the most useful application; I have, therefore, in the

first place sought to establish the general relation of the

state bordering upon motion between the driving and the

working pressures upon such a machine, reference being

had to the weight of the machine.* This relation points out

the existence 'of a particular direction in which the driving

pressure should be applied to any such machine, that the

amount of work expended upon the friction of the axis may

be the least possible. This direction of the driving pressure

always presents itself on the same side of the axis with that

of the working pressure, and when the latter is vertical it

becomes parallel to it ; a principle of the economy of power

in machinery which has received its application in the

parallel motion of the marine engines known as the Gorgon

Engines.

I have devoted a considerable space in this portion of my

work to the determination of the modulus of a system of

toothed wheels ; this determination I have, moreover,

extended to bevil wheels, and have included in it, with the

influence of the friction of the teeth of the wheels, that of

their axes and their weights. An approximate form of this

modulus applies to any shape of the teeth under which they

may be made to work correctly ; and when in this approxi-

mate form of the modulus the terms which represent the

influence of the friction of the axis and the weight of the

wheel are neglected, it resolves itself into a well known

theorem of M. Poncelet, reproduced by M. ISTavier and the

Rev. Dr. Whewell.f In respect to wheels having epicy-

* In my memoir on the " Theory of Machines " (Phil. Trans. 1841), I have

extended this relation to the case in which the number of the pressures and

their directions are any whatever. The theorem which expresses it is given in

the Appendix of this work.

f In the discussion of the friction of the teeth of wheels, the direction of the

mutual pressures of the teeth is determined by a method first applied by me to

PREFACE. xiij

cloidal and involute teeth, the modulus assumes a character

of mathematical exactitude and precision, and at once

establishes the conclusion (so often disputed) that the loss of

power is greater before the teeth pass the line of centres

than at corresponding points afterwards ; that the contact

should, nevertheless, in all cases take place partly before

and partly after the line of centres has been passed. In the

case of involute teeth, the proportion in which the arc of

contact should thus be divided by the line of centres is

determined by a simple formula ; as also are the best

dimensions of the base of the involute, with a view to the

most perfect economy of power in the working of the

wheels.

The greater portion of the discussions in the third part of

my work I believe to be new to science. In the fourth part

I have treated of " the theory of the stability of structures,"

referring its conditions, so far as they are dependent upon

the rotation of the parts of a structure upon one another, to

the properties of a certain line which may be conceived to

traverse every structure, passing through those points in it

where its surfaces of contact are intersected by the resultant

pressures upon them. To this line, whose properties I first

discussed in a memoir upon " the Stability of a System of

Bodies in Contact," printed in the sixth volume of the Carrib.

Phil. Trans., I have given the name of the line of resist-

ance ; it differs essentially in its properties from a line

referred to by preceding writers under the name of the

curve of equilibrium or the line of pressure.

The distance of the line of resistance from the extrados of

a structure, at the point where it most nearly approaches it,

I have taken as a measure of the stability of a structure,* and

that purpose in a popular treatise, entitled Mechanics applied to the Arts,

published in 1834.

* This idea was suggested to me by a rule for the stability of revetement

walls attributed to Vauban, to the effect, that the resultant pressure should

intersect the base of such a wall at a point whose distance from its extrados is

iths the distance between the extrados at the base and the vertical through

the centre of gravity.

X1T PREFACE.

have called it the modulus of stability; conceiving thia

measure of the stability to be of more obvious and easier

application than the coefficient of stability used by the

French writers.

That structure in respect to every independent element

of which the modulus of stability is the same, is evidently

the structure of the greatest stability having a given quantity

of material employed in its construction ; or of the greatest

economy of material having a given stability.

The application of these principles of construction to the

theory of piers, walls supported by counterforts and shores,

buttresses, walls supporting the thrust of roofs, and the

weights of the floors of dwellings, and Gothic structures,

has suggested to me a class of problems never, I believe,

before treated mathematically.

I have applied the well known principle of Coulomb to

the determination of the pressure of earth upon revetement

walls, and a modification of that principle, suggested by M.

Poncelet, to the determination of the resistance opposed to

the overthrow of a wall backed by earth. This determina-

tion has an obvious application to the theory of foundations.

In the application of the principle of Coulomb I have

availed myself, with great advantage, of the properties of

the limiting angle of resistance. All my results have thus

received a new and a simplified form.

The theory of the arch I have discussed upon principles

first laid down in my memoir on " the Theory of the Stability

of a System of Bodies in Contact," before referred to, and

subsequently in a memoir printed in the "Treatise on

Bridges" by Professor Hosking and Mr. Hann.* They

differ essentially from those on which the theory of Coulomb

is founded ;f when, nevertheless, applied to the case treated

* I have made extensive use of the memoir above referred to in the following

work, by the obliging permission of the publisher, Mr. Weale.

f The theory of Coulomb was unknown to me at the time of the publication

of my memoirs printed in the Camb. Phil. Trans. For a comparison of the

two methods see Mr. Hann's treatise.

PKEFACE. XT

by the French mathematicians, they lead, to identical results,

I have inserted at the conclusion of my work the tables of

the thrust of circular arches, calculated by M. Garidel from

formulae founded on the theory of Coulomb.

The fifth part of the work treats of the "strength of

materials," and applies a new method to the determination

of the deflexion of a beam under given pressures.

In the case of a beam loaded uniformly over its whole

length, and supported at four different points, I have deter^

mined the several pressures upon the points of support by a

method applied by M. Navier to a similar determination in

respect to a beam loaded at given points.*

In treating of rupture by elongation I have been led to a

discussion of the theory of the suspension bridge. This

question, so complicated when reference is had to the weight

of the roadway and the weights of the suspending rods, and :

when the suspending chains are assumed to tte of uniform

thickness, becomes comparatively easy when the section of

the chain is assumed so to vary its dimensions as to be every

where of the same strength. A suspension bridge thus

constructed is obviously that which, being of a given

strength, can be constructed with the least quantity of

materials ; or, which is of the greatest strength having a

given quantity of materials used in its construction.!

The theory of rupture by transverse strain has suggested

a new class of problems, having reference to the forms of

girders having wide flanges connected by slender ribs or by

open frame work : the consideration of their strongest forms

leads to results of practical importance.

In discussing the conditions of the strength of breast-

summers, my attention has been directed to the best positions

of the columns destined to support them, and to a comparison

* As in fig. p. 487. of the following work.

f That particular case of this problem, in which the weights of the suspending

rods are neglected, has been treated by Mr. Hodgkinson in the fourth vol. of

Manchester Transactions, with his usual ability. He has not, however, suc-

ceeded in effecting its complete solution.

XVI PREFACE.

of the strength of a beam carrying a uniform load and sup-

ported freely at its extremities, with that of a beam similarly

loaded but having its extremities firmly imbedded in

masonry.

In treating of the strength of columns I have gladly

replaced the mathematical speculations upon this subject,

which are so obviously founded upon false data, by the

invaluable experimental results of Mr. E. Hodgkinson,

detailed in his well known paper in the Philosophical

Transactions for 1840.

The sixth and last part of my work treats on " impact ;"

and the Appendix includes, together with tables of the

mechanical properties of the materials of construction, the

angles of rupture and the thrusts of arches, and complete

elliptic functions, a demonstration of the admirable theorem

of M. Poncelet for determining an approximate value of the

square root of the sum or difference of two squares.

In respect to the following articles of my work I have tc

acknowledge my obligations to the work of M. Poncelet,

entitled Mecanique Industrielle. The mode of demonstration

is in some, perhaps, so far varied as that their origin might

with difficulty be traced ; the principle, however, of each

demonstration all that constitutes its novelty or its value

belongs to that distinguished author.

30,* 38, 40, 45, 46, 47, 52, 58, 62, 75, 108,f 123, 202,

267,t 268, 269, 270, 349, 354, 365.

* The enunciation only of this theorem is given in the Mec. Ind., 2me partie,

Art. 38.

f Some important elements of the demonstration of this theorem are taken

from the Mec. Ind., Art. 79. 2me partie. The principle of the demonstration

is not, however, the same as in that work.

\ In this and the three following articles I have developed the theory of the

MECHANICAL PRINCIPLES

ENGINES KING

ARCHITECTURE.

HENRY MOSELET, I. A. F.R.S.

,

CHAPLAIN IN ORDINARY TO THE QUEEN, CANON OF BRISTOL, VICAR OF OLVESTON ;

CORRESPONDING MEMBER OF TUG INSTITUTE OF FRANCE, AND FORMERLY PROFESSOB

OF NATURAL PHILOSOPHY AND ASTRONOMY IN KING'S COLLEGE, LONDON.

Second American from Second London Edition

WITH ADDITIONS BY

D. H. M A H A N , LL.D.

U. S. MILITARY ACADEMY.

V.MTIJ ri, LUSTRATIONS ON WOOD.

NEW YORK :

JOHN WILEY & SON, 535 BROADWAY

1

ENTERFO according *o Act of Congress, <jt\ the year 1856, Ij 1

WILEY & HA-LSTRD,

fc the Clerk's Office of the District Court of the United States, for the Southu a District

of New York.

EDITOR'S PREFACE.

THE high place that Professor Moseley occupies in the

scientific world, as an original investigator, and the clear-

ness and elegance of the methods he has employed in this

work have made it a standard text book on the subjects it

treats of. In undertaking its revision for the press, at the

request of the publishers of this edition, it has been deemed

advisable, in view of the class of students into whose hands

it may fall, to make some slight addition to the original.

This has been done in the way of Notes thrown into an

Appendix, the matter of which has been gathered from

various authorities ; but chiefly from notes taken by the

editor, whilst a pupil at the French military school at Metz,

of lectures delivered by General Poncelet, at that time, 1829,

professor in that school. It is a source of great pleasure to

the editor to have this opportunity of publicly acknowledg-

ing his obligations to the teachings of this eminent savan,

who is distinguished not more for his high scientific attain-

ment, and the advancement he has given to mechanical

science, than for having brought these to minister to the

wants of the industrial classes, the intelligent success of

whose operations depends so much upon mechanical science,

by presenting it in a form to render it attainable by the most

ordinary capacities.

Hi

iv EDITOR'S PREFACE.

The editor would remark that lie has carefully refrained

from making any alterations in the text revised, except cor-

rections of typographical errors, and in one instance where,

from a repetition of apparently one of these, he apprehended

some difficulty might be offered to the student if allowed

to remain exactly as printed in the original.

UNITED STATES MILITARY ACADEMY,

Went Point March 8, 1866.

PKEFACE TO THE SECOND EDITION.

I HAVE added in this Edition articles : first, " On the

Dynamical Stability of Floating Bodies ;" secondly, " On

the Kolling of a Cylinder ;" thirdly, " On the descent of a

body upon an inclined plane, when subjected to variations of

temperature, which would otherwise rest upon it ;" fourthly,

u On the state bordering upon motion of a body moveable

about a cylindrical axis of finite dimensions, when acted

upon by any number of pressures."

The conditions of the dynamical stability of floating

bodies include those of the rolling and pitching motion of

ships. The discussion of the rolling motion of a cylinder

includes that of the rocking motion to which a locomotive

engine is subject, when its driving wheels are falsely

balanced, and that of the slip of the wheel due to the same

cause. The descent of a body upon an inclined plane

when subjected to variations in temperature, which other-

wise would rest upon it, appears to explain satisfactorily the

descent of glaciers.

The numerous corrections made in the text, I owe chiefly

to my old pupils at King's College, to whom the lectures

of which it contains the substance, were addressed. For

VI PREFACE TO THE SECOND EDITION.

several important ones I am, however, indebted to Mr

Eobinson, Master of the School for Shipwrights' Apprentices,

in Her Majesty's Dockyard, Portsea ; to whom I have also to

express my warm acknowledgments for the care with

which he has corrected the proof sheets whilst going through

the press.

May, 1855

PREFACE.

IN the following work, I have proposed to myself to apply:

the principles of mechanics to the discussion of the most

important and obvious of those questions which present

themselves in the practice of the engineer and the architect ;

and I have sought to include in that discussion all the

circumstances on which the practical solution of such ques-

tions may be assumed to depend. It includes the substance

of a course of lectures delivered, to > the students of King's

College in the department of engineering and architecture,

during the years 1840, 1841, 1842.*

In the first part I have treated of those portions of the

science of STATICS, which have their application in the theory

of machines and the theory of construction.

In the second, of the science of DYNAMICS, and, under this

head, particularly of that union of a continued pressure with

a continued motion which has received from English writers

the various names of "dynamical effect," "efficiency," "work

done," "labouring force," "work," &c. ; and "moment

d'activite"," "quantite d' action," "puissance mecanique,"

" travail," from French writers.

Among the latter this variety of terms has at length given

place to the most intelligible and the simplest of them,,

* The first 170 pages of the work were printed for the use of my pupils in the-

year 1840. Copies of them were about the same time in the possession of

several of my friends in the Universities.

Vlll PREFACE.

" travail." The English word " work " is the obvious trans-

lation of " travail," and the use of it appears to be recom-

mended by the same considerations. The work of overcoming

a pressure of one pound through a space of one foot has, in

this country, been taken as the unit, in terms of which any

other amount of work is estimated ; and in France, the work

of overcoming a pressure of one kilogramme through a space

of one metre. M. Dupiii has proposed the application of the

term dyname to this unit.

I have gladly sheltered myself from the charge of having

contributed to increase the vocabulary of scientific words,

by assuming the obvious term " unit of work " to represent

concisely and conveniently enough the idea which is attached

to it.

The work of any pressure operating through any space is

evidently measured in terms of such units, oy multiplying

the number of pounds in the pressure by the number of feet

in the space, if the direction of the pressure be continually

that in which the space is described. If not, it follows, by

a simple geometrical deduction, that it is measured by the

product of the number of pounds in the pressure, by the

number of feet in the projection of the space described,*

upon the direction of the pressure ; that is, by the product

of the pressure by its virtual velocity. Thus, then, we

conclude at once, by the principle of virtual velocities, that

if a machine work under a constant equilibrium of the

pressures applied to. it, or if it work uniformly, then is the

aggregate work of those pressures which tend to accelerate

its motion equal to the aggregate work of those which tend

to retard it ; and, by the principle of vis viva, that if the

machine do not work under an equilibrium of the forces

impressed upon it, then is the aggregate work of those which

tend to accelerate the motion of the machine greater or less

* If the direction of the pressure renfain always parallel to itself, the space

described may be any finite space ; if it do not, the space is understood to be

so small, that the direction of the pressure may be supposed to remain parallel

to itself whilst that space is described.

PREFACE. IX

than the aggregate work of those which tend to retard its

motion by one half the aggregate of the vires vivce acquired

or lost by the moving parts of the system, whilst the work is

being done upon it. In no respect have the labours of the

illustrious president of the Academy of Sciences more con-

tributed to the development of the theory of machines than

in the application which he has so successfully made to it of

this principle of vis viva.* In the elementary discussion of

this principle, which is given by M. Poncelet, in the intro-

duction to his Mecanique Industrielle, he has revived the

term vis inertia (vis inertias, vis insita, Newton), and,

associating with it the definitive idea of a force of resistance

opposed to the acceleration or the retardation of a body's

motion, he has shown (Arts. 66. and 122.) the work expended

in overcoming this resistance through any space, to be

measured by one half the vis viva accumulated through the

space ; so that throwing into the consideration of the forces

under which a machine works, the vires inerticB of its moving

elements, and observing that one half of their aggregate vis

viva is equal to the aggregate work of their vires inertice, it

follows, by the principle of virtual velocities, that the differ-

ence between the aggregate work of those forces impressed

upon a machine, which tend to accelerate its motion, and

the aggregate work of those which tend to retard the motion,

is equal to the aggregate work of the vires inerticB of the

moving parts of the machine : under which form the prin-

ciple of vis viva resolves itself into the principle of virtual

velocities. So many difficulties, however, oppose themselves

to the introduction of the term vis inertice, associated with

the definitive idea of a force, into the discussion of questions

of mechanics, and especially of practical and elementary

mechanics, that I have thought it desirable to avoid it. It

is with this view that I have given a new interpretation to

that function of the velocity of a moving body which is

known as its vis viva. One half that function I have inter-

preted to represent the number of units of work accumulated

* See Poncelet, Mecanique Industrielle, troisieme partie.

PREFACE.

in the body so long as its motion is continued. This number

of units of work it is capable of reproducing upon any resist-

ance opposed to its motion. A very simple investigation

(Art. 66.) establishes the truth of this interpretation, and

gives to the principle of vis viva the following more simple

enunciation : " The difference between the aggregate work

done upon the machine, during any time, by those forces

which tend to accelerate the motion, and the aggregate

work, during the same time, of those which tend to retard

the motion, is equal to the aggregate number of units of

work accumulated in the moving parts of the machine

during that time if the former aggregate exceed the latter,

and lost from them during that time if the former aggregate

fall short of the latter." Tims, then, if the aggregate work

of the forces which tend to accelerate the motion of a

machine exceeds that of the forces which tend to retard it,

then is the surplus work (that done upon the driving points,

above that expended upon the prejudicial resistances and

upon the working points) continually accumulated in the

moving elements of the machine, and their motion is thereby

continually accelerated. And if the former aggregate be

less than the latter, then is the deficiency supplied from the

work already accumulated in the moving elements, so that

their motion is in this case continually retarded.

The moving power divides itself whilst it operates in a

machine, first, into that which overcomes the prejudicial

resistances of the machine, or those which are opposed by

friction and other causes, uselessly absorbing the work in its

transmission. Secondly, into that which accelerates the

motion of the various moving parts of the machine, and which

accumulates in them so long as the work done by the moving

power upon it exceeds that expended upon the various

resistances opposed to the motion of the machine. Thirdly,

into that which overcomes the useful resistances, or those

which are opposed to the motion of the machine at the

working point, or points, by the useful work which is done

by it.

PREFACE. XI

Between these three elements there obtains in every

machine a mathematical relation, which I have called its

MODULUS. The general form of this modulus I have discussed

in a memoir on the " Theory of Machines " published in the

Philosophical Transactions for the year 1841. The deter-

mination of the particular moduli of those elements of

machinery which are most commonly in use, is the subject

of the third part of the following work. From a combination

of the moduli of any such elements there results at once the

modulus of the machine compounded of them."

"When a machine has acquired a state of uniform motion,

work ceases to accumulate in its moving elements, and its

modulus assumes the form of a direct relation between the

work done by the motive power upon its driving point and

that yielded at its working points. I have determined by a

general method' 35 ' the modulus in this case, from that statical

relation between the driving and working pressures upon

the machine which obtains in the sfate bordering upon its

motion, and which may be deduced from the known condi-

tions of equilibrium and the established laws of friction. In

making this deduction I have, in every case, availed myself

of the following principle, first published in my paper on the

theory of the arch, read before the Cambridge Philosophical

Society in Dec. 1833, and printed in their Transactions of

the following year: "In the state bordering upon motion

of one body upon the surface of another, the resultant

pressure upon their common surface of contact is inclined

to the normal, at an angle whose tangent is equal to the

coefficient of friction."

This angle I have called the limiting angle of resistance.

Its values calculated, in respect to a great variety of surfaces

of contact, are given in a table at the conclusion of the

second part, from the admirable experiments of M. Morin,f

into the mechanical details of which precautions have been

introduced hitherto unknown to experiments of this class,

* Art. 152. See Phil. Trans., 1841, p. 290.

f Nouvelles Experiences sur le Frottement, Paris, 1833.

Xll PEEFACE.

and which have given to our knowledge of the laws of

friction a precision and a certainty hitherto unhoped for.

Of the various elements of machinery those which rotate

about cylindrical axes are of the most frequent occurrence

and the most useful application; I have, therefore, in the

first place sought to establish the general relation of the

state bordering upon motion between the driving and the

working pressures upon such a machine, reference being

had to the weight of the machine.* This relation points out

the existence 'of a particular direction in which the driving

pressure should be applied to any such machine, that the

amount of work expended upon the friction of the axis may

be the least possible. This direction of the driving pressure

always presents itself on the same side of the axis with that

of the working pressure, and when the latter is vertical it

becomes parallel to it ; a principle of the economy of power

in machinery which has received its application in the

parallel motion of the marine engines known as the Gorgon

Engines.

I have devoted a considerable space in this portion of my

work to the determination of the modulus of a system of

toothed wheels ; this determination I have, moreover,

extended to bevil wheels, and have included in it, with the

influence of the friction of the teeth of the wheels, that of

their axes and their weights. An approximate form of this

modulus applies to any shape of the teeth under which they

may be made to work correctly ; and when in this approxi-

mate form of the modulus the terms which represent the

influence of the friction of the axis and the weight of the

wheel are neglected, it resolves itself into a well known

theorem of M. Poncelet, reproduced by M. ISTavier and the

Rev. Dr. Whewell.f In respect to wheels having epicy-

* In my memoir on the " Theory of Machines " (Phil. Trans. 1841), I have

extended this relation to the case in which the number of the pressures and

their directions are any whatever. The theorem which expresses it is given in

the Appendix of this work.

f In the discussion of the friction of the teeth of wheels, the direction of the

mutual pressures of the teeth is determined by a method first applied by me to

PREFACE. xiij

cloidal and involute teeth, the modulus assumes a character

of mathematical exactitude and precision, and at once

establishes the conclusion (so often disputed) that the loss of

power is greater before the teeth pass the line of centres

than at corresponding points afterwards ; that the contact

should, nevertheless, in all cases take place partly before

and partly after the line of centres has been passed. In the

case of involute teeth, the proportion in which the arc of

contact should thus be divided by the line of centres is

determined by a simple formula ; as also are the best

dimensions of the base of the involute, with a view to the

most perfect economy of power in the working of the

wheels.

The greater portion of the discussions in the third part of

my work I believe to be new to science. In the fourth part

I have treated of " the theory of the stability of structures,"

referring its conditions, so far as they are dependent upon

the rotation of the parts of a structure upon one another, to

the properties of a certain line which may be conceived to

traverse every structure, passing through those points in it

where its surfaces of contact are intersected by the resultant

pressures upon them. To this line, whose properties I first

discussed in a memoir upon " the Stability of a System of

Bodies in Contact," printed in the sixth volume of the Carrib.

Phil. Trans., I have given the name of the line of resist-

ance ; it differs essentially in its properties from a line

referred to by preceding writers under the name of the

curve of equilibrium or the line of pressure.

The distance of the line of resistance from the extrados of

a structure, at the point where it most nearly approaches it,

I have taken as a measure of the stability of a structure,* and

that purpose in a popular treatise, entitled Mechanics applied to the Arts,

published in 1834.

* This idea was suggested to me by a rule for the stability of revetement

walls attributed to Vauban, to the effect, that the resultant pressure should

intersect the base of such a wall at a point whose distance from its extrados is

iths the distance between the extrados at the base and the vertical through

the centre of gravity.

X1T PREFACE.

have called it the modulus of stability; conceiving thia

measure of the stability to be of more obvious and easier

application than the coefficient of stability used by the

French writers.

That structure in respect to every independent element

of which the modulus of stability is the same, is evidently

the structure of the greatest stability having a given quantity

of material employed in its construction ; or of the greatest

economy of material having a given stability.

The application of these principles of construction to the

theory of piers, walls supported by counterforts and shores,

buttresses, walls supporting the thrust of roofs, and the

weights of the floors of dwellings, and Gothic structures,

has suggested to me a class of problems never, I believe,

before treated mathematically.

I have applied the well known principle of Coulomb to

the determination of the pressure of earth upon revetement

walls, and a modification of that principle, suggested by M.

Poncelet, to the determination of the resistance opposed to

the overthrow of a wall backed by earth. This determina-

tion has an obvious application to the theory of foundations.

In the application of the principle of Coulomb I have

availed myself, with great advantage, of the properties of

the limiting angle of resistance. All my results have thus

received a new and a simplified form.

The theory of the arch I have discussed upon principles

first laid down in my memoir on " the Theory of the Stability

of a System of Bodies in Contact," before referred to, and

subsequently in a memoir printed in the "Treatise on

Bridges" by Professor Hosking and Mr. Hann.* They

differ essentially from those on which the theory of Coulomb

is founded ;f when, nevertheless, applied to the case treated

* I have made extensive use of the memoir above referred to in the following

work, by the obliging permission of the publisher, Mr. Weale.

f The theory of Coulomb was unknown to me at the time of the publication

of my memoirs printed in the Camb. Phil. Trans. For a comparison of the

two methods see Mr. Hann's treatise.

PKEFACE. XT

by the French mathematicians, they lead, to identical results,

I have inserted at the conclusion of my work the tables of

the thrust of circular arches, calculated by M. Garidel from

formulae founded on the theory of Coulomb.

The fifth part of the work treats of the "strength of

materials," and applies a new method to the determination

of the deflexion of a beam under given pressures.

In the case of a beam loaded uniformly over its whole

length, and supported at four different points, I have deter^

mined the several pressures upon the points of support by a

method applied by M. Navier to a similar determination in

respect to a beam loaded at given points.*

In treating of rupture by elongation I have been led to a

discussion of the theory of the suspension bridge. This

question, so complicated when reference is had to the weight

of the roadway and the weights of the suspending rods, and :

when the suspending chains are assumed to tte of uniform

thickness, becomes comparatively easy when the section of

the chain is assumed so to vary its dimensions as to be every

where of the same strength. A suspension bridge thus

constructed is obviously that which, being of a given

strength, can be constructed with the least quantity of

materials ; or, which is of the greatest strength having a

given quantity of materials used in its construction.!

The theory of rupture by transverse strain has suggested

a new class of problems, having reference to the forms of

girders having wide flanges connected by slender ribs or by

open frame work : the consideration of their strongest forms

leads to results of practical importance.

In discussing the conditions of the strength of breast-

summers, my attention has been directed to the best positions

of the columns destined to support them, and to a comparison

* As in fig. p. 487. of the following work.

f That particular case of this problem, in which the weights of the suspending

rods are neglected, has been treated by Mr. Hodgkinson in the fourth vol. of

Manchester Transactions, with his usual ability. He has not, however, suc-

ceeded in effecting its complete solution.

XVI PREFACE.

of the strength of a beam carrying a uniform load and sup-

ported freely at its extremities, with that of a beam similarly

loaded but having its extremities firmly imbedded in

masonry.

In treating of the strength of columns I have gladly

replaced the mathematical speculations upon this subject,

which are so obviously founded upon false data, by the

invaluable experimental results of Mr. E. Hodgkinson,

detailed in his well known paper in the Philosophical

Transactions for 1840.

The sixth and last part of my work treats on " impact ;"

and the Appendix includes, together with tables of the

mechanical properties of the materials of construction, the

angles of rupture and the thrusts of arches, and complete

elliptic functions, a demonstration of the admirable theorem

of M. Poncelet for determining an approximate value of the

square root of the sum or difference of two squares.

In respect to the following articles of my work I have tc

acknowledge my obligations to the work of M. Poncelet,

entitled Mecanique Industrielle. The mode of demonstration

is in some, perhaps, so far varied as that their origin might

with difficulty be traced ; the principle, however, of each

demonstration all that constitutes its novelty or its value

belongs to that distinguished author.

30,* 38, 40, 45, 46, 47, 52, 58, 62, 75, 108,f 123, 202,

267,t 268, 269, 270, 349, 354, 365.

* The enunciation only of this theorem is given in the Mec. Ind., 2me partie,

Art. 38.

f Some important elements of the demonstration of this theorem are taken

from the Mec. Ind., Art. 79. 2me partie. The principle of the demonstration

is not, however, the same as in that work.

\ In this and the three following articles I have developed the theory of the

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

Online Library → Henry Moseley → The mechanical principals of engineering and architecture → online text (page 1 of 52)