1990

ESAMI DI MATURITA’ TECNICA INDUSTRIALE

Indirizzo: MECCANICA

Tema di : MECCANICA APPLICATA ALLE MACCHINE

Occorre realizzare una gru a bandiera formata da una colonna verticale di altezza 3 m e daun braccio orizzontale in grado di garantire uno sbraccio di 2.5 m rispetto all’asse della colonna.

Per il sollevamento del carico, avente un valore massimo di 5000 N, è prevista l’installazionedi un paranco, portato da un carrello scorrevole lungo il braccio orizzontale della gru. La massacomplessiva dell’equipaggio mobile, a gancio scarico, è pari a 25 kg ed il carico viene sollevato conuna accelerazione massima di 1 m/sec2.

Il candidato, analizzate le sollecitazioni indotte nella struttura e fissati opportunamente i datioccorrenti, effettui il dimensionamento di massima della gru, eseguendone infine lo schizzo quotato.

Relazione di calcolo

Schizzo quotato

Durata massima della prova: 6 oreDurante lo svolgimento della prova è consentito l’uso di tavole numeriche, manuali tecnici, del regolo calcolatore e dicalcolatrici tascabili.

Relazione di calcolo

PremessaIn questa relazione, pur facendo riferimento per quanto possibile alla normative vigenti (CNR-UNI10011), si introdurranno delle semplificazioni di calcolo tali da consentire a studenti del quinto annodi corso di un Istituto Tecnico un orientamento facilitato alle problematiche connesse con laprogettazione richiesta.La relazione perciò, anche se corretta dal punto di vista sostanziale, non è conforme alle norme di leggee ha una valenza puramente didattica.

Acciai usati nelle costruzioni in acciaio

I valori di Famm e Jamm si riferiscono alle condizioni di carico tipo I, per le condizioni di carico tipo II letensioni ammissibili sono da assumersi pari a:

1125. ⋅ ⋅σ τamm amm e 1.125

Le verifiche sono obbligatorie per ambedue le condizioni di carico

Azioni agenti sulle costruzioni

Le azioni agenti sulle costruzioni si classificano in principali e complementari

Azioni principali< carichi permanenti;< carichi di esercizio;< neve;< spinta delle terre;< effetto dinamico;< coazioni impresse.

Azioni complementari< vento;< variazioni termiche;< ritiro;< fenomeni viscosi;< imperfezione dei vincoli;< difetti di montaggio.

Condizioni di carico

Condizione di carico ICumula sulla struttura le azioni principali nel modo più sfavorevole

Condizione di carico IICumula sulla struttura le azioni principali e complementari nel modo più sfavorevole

1 In genere tuttavia, le gru a bandiera di tipo leggero hanno colonne a sezione pentagonale irregolarecon spigoli smussati.

2 Nell’analisi dei carichi si è trascurato sia l’effetto del vento sia la forza d’inerzia longitudinaleconseguente alla frenatura del paranco. (vedi CNR-UNI 10021)

3 Indicata con g l’accelerazione di gravità, con a l’accelerazione di sollevamento e con PC il caricomassimo sollevabile, la forza di inerzia FI dovuta al sollevamento del carico vale:

FP

ga NI

C= ⋅ ≅ ⋅ =5000

101 500

4Appunto perchè si è trascurato sia l’effetto del vento, sia l’effetto dinamico dovuto alla frenatura si èpensato arbitrariamente di maggiorare il carico del 15%

Vista l’entità del carico massimo sollevabile ci si orienta su una gru a bandiera di tipo leggero concolonna realizzata con un tubo di acciaio cilindrico1 e con braccio in profilato a doppio T e rinforzatocon un tirante. ( tabella )

Analisi dei carichi2

Carico massimo sollevabile PC 5000 NPeso dell’equipaggio mobile PM 250 NForza di inerzia di sollevamento3 FI 500 N

Carico Totale PT 5750 N

Carico di riferimento4

P P NT= ⋅ = ⋅ =ψ 5750 115 6612.

Dimensionamento della colonna

Indicata con h l’altezza della colonna e con l la lunghezza del braccio, il momento flettente alpiede della colonna vale: ( figura )

M P l NmF = ⋅ = ⋅ =6612 2 5 16530.

La sollecitazione di compressione, sempre al piede della colonna, vale:

P N= 6612

La lunghezza libera h0 della colonna, considerata come una trave incastrata al piede, risulta pari a:

h h m0 2 6= ⋅ =

Fissata, in prima battuta, una snellezza massima pari a 100, la colonna potrebbe essere realizzata con untubo in acciaio Fe 360 con le seguenti caratteristiche geometriche ( tabella ):

1 Sarebbe meglio usare il termine ‘Momento quadratico di superficie’

2 La formula riportata dalla CNR-UNI 10011 che determina la tensione massima indotta dalla pressoflessione è:

ω

νσ

ψ

⋅+

−⋅

⋅

⋅ ⋅

P

S

M

P

SW

E1

dove

v =

=

15

1

. I condizione di carico

1.5 / 1.125 II condizione di carico tensioni ammissibili

stati limiteν

R ($1) è un coefficiente di parziale adattamento plastico

3 Mancando la consueta proporzionalità fra le deformazioni (ad esempio la freccia f) e il carico P, nonvale il principio di sovrapposizione degli effetti. Per cui la deformazione dovuta a più carichi non è uguale allasomma delle deformazioni dovute ai carichi singoli. Anche i teoremi di Maxwell e di Castigliano, fondati sullaproporzionalità suddetta, non valgono in questo caso (O.Belluzzi Scienza delle Costruzioni Vol.1 Zanichelli)

Diametro esterno De 191 mmSpessore s 4.5 mmSezione S 26.4 cm2

Momento d’inerzia1 I 1147 cm4

Momento resistente W 120 cm3

Raggio giratorio D 6.59 cm

La snellezza 88 massima, con riferimento alla sezione sopra definita, vale:

λρ

= = =h0 600

65991

min .

Verifica a presso flessione (norme LL.PP.1972-74)2

• la colonna deve essere considerata caricata di punta

• non è applicabile la semplice sovrapposizione degli effetti3

σω

σ

= +−

=⋅

+−

⋅⋅

⋅

=P

S

M

PS

W

N mmF

E

115

146 6612

2640

16530000

115 6612

250 26400120000

144 2

.

..

/

dove T è il coefficiente di amplificazione del carico relativo alla snellezza massima dell’asta(tabella)

1 La lunghezza libera del braccio è 2.5 m (asta vincolata agli estremi da due cerniere)

FE è la tensione critica euleriana per la snellezza relativa al piano di flessione (tabella)S sezione resistenteW momento resistente

Se M varia lungo l’asta si assume Mi =1.3 Mm , essendo Mm il valore medio di M con lalimitazione:

0 75. max maxM M Mi≤ ≤

La tensione trovata di 144 N/mm2 è inferiore alla tensione ammissibile relativa ad un acciaioFe360 (160 N/mm2).

Dimensionamento del braccio

Il dimensionamento si conduce considerando il braccio come facente parte di una strutturareticolare a tre cerniere e analizzando le due posizioni del carico:

• carico P applicato all’estremità del braccio• carico P applicato in mezzeria

Carico P applicato all’estremità del braccio

Il braccio si comporta come puntone e la forza normale F che lo sollecita vale:

F Pl

aN= = ⋅ =6612

2 5

0 625550

.

.

Fissando, in prima battuta, la snellezza massima del puntone 120, la sezione in oggettodovrebbe avere un raggio giratorio1 maggiore di 2.08 cm.Dalle tabelle si vede che una sezione HE 100 A è in grado di soddisfare alle esigenzeimposte.Verifica a carico di punta (materiale Fe 360 con Famm = 160 N/mm2 )

la snellezza del puntone vale 8 250/2.51=99

σω

=⋅

=⋅

=F

AN mm

181 25550

212022 2.

/

la tensione è inferiore alla tensione ammissibile

Carico applicato in mezzeria del braccio

V VF

NA B= = = =2

6612

23306

Hl

a

PNB = ⋅ =

213775

Il momento flettente in mezzeria vale:

M Vl

Nmf A= ⋅ = ⋅ =2

33062500

24132 5.

La verifica a presso flessione è condotta con la seguente relazione:

σω

σ

= +−

⋅

⋅

=⋅

+⋅

−⋅⋅

⋅≅

H

A

M

H

AW

N mmA if

A

E

115

181 13775

2120

0 75 4132500

115 13775557 2120

7300055 2

.

. ..

/

anche in questo caso la tensione è inferiore a quella ammissibile.

Scelta del tirante

Il tirante è realizzato con un profilato a C avente larghezza congruente con le caratteristiche del profilatoscelto per il braccio ( HE 100 A )Si può scegliere come primo tentativo un profilato avente le dimensioni 80x45x5 con area resistente Apari a 1070 mm2

Verifica a trazione

σ =+

=+

≅V H

AN mm

A A2 2 2 2

23306 13775

107013 /

La tensione è inferiore a quella ammissibile

Progetto della fondazione

Si ipotizzi un plinto delle seguenti dimensioni in cm:

Base quadrata di lato a 80 cmaltezza h 60 cm

Peso del Plinto PP 9216 NForza totale di riferimento P 6612 NPeso della struttura PS 1000 N (stimata)Momento flettente alla base M 4132.5 Nm

L’eccentricità e vale:

eM

P P Pmm

P S

=+ +

= 24 5.

Poiché e > a/6 la sezione risulta parzializzata.La massima tensione di compressione sul terreno vale (vedi Appendice A):

( )σ ct

t SP P P

ae a

N cm=⋅ + +

⋅ −

⋅

=⋅

⋅ −

⋅

=2

32

2 16828

3802

24 5 809 2

./

Ritenendo che il terreno d’appoggio sia in argilla compatta, la tensione trovata è pienamente ammissibile.

Pressione ammissibile nei terreni in N/cm2 (L.Santarella Il Cemento Armato vol.II Hoepli)

terreni di riporto o rimossi 1÷5sabbia umida compatta 10÷15sabbia e argilla compatta 15÷20sabbia asciutta compatta, argilla asciutta,ghiaia a strati con tracce sabbiose 25÷30ghiaia grossa compatta 40rocce sane 50÷100÷200

Verifica al ribaltamento

Il momento ribaltante rispetto allo spigolo (A) vale:

M P la

NmR = ⋅ −

= ⋅ −

=

26612 2 5

0 8

213885.

.

Il momento stabilizzante, rispetto allo stesso spigolo:

M Pa

NmS P= ⋅ = ⋅ =2

9126 0 4 3686.

La gru si ribalta !!!Occorre aumentare il peso del plinto

lato plinto a 120 cmaltezza h 100 cmPeso plinto PP 34560 N

Momento ribaltante

M NmR = ⋅ −

=6612 2 5

12

212563.

.

Nuovo Momento stabilizzante

M NmS = ⋅ =34560 0 6 20736.

Occorre, a questo punto riverificare la pressione massima sul terreno. Con riferimento ad un plintoquadrato di lato 120 cm e altezza 100 cm, l’eccentricità e vale:

e cm= 9 7. La sezione non è parzializzata.

Tensione dovuta a flessione

σ f

M

aN cm=

⋅=

⋅=

6 6 1653000

1205 73 3

2. /

Tensione dovuta a compressione

( )σ c

S PP P P

aN cm=

+ +=

+ +=2 2

26612 1000 34560

1203 /

La tensione totale 8.7 N/cm2 è inferiore alla tensione ammissibile per il terreno considerato.

Calcolo dei bulloni

Ipotizzo di utilizzare, per il fissaggio della colonna al plinto, 4 bulloni M16 classe 10.9.Ciascun bullone dovrà essere serrato in modo da provocare sul gambo della vite una forza di trazioneNS nel gambo della vite

N f A NS kN s= ⋅ ⋅ = ⋅ ⋅ =08 08 700 157 87920. .Re

fkN resistenza caratteristica di un bullone a trazione in N/mm2 (tabella)ARes area della sezione resistente della vite mm2 (tabella)

Il momento massimo trasmissibile dalla flangia vale:

M Nc

M Nm

F S

F

= ⋅ ⋅

= ⋅ ⋅ =

22

2 87920 0 4 70336.

Nettamente superiore al momento massimo insistente alla base dellacolonna (M=6612 Nm)

Verifiche della rigidezza della struttura

Verifica della freccia fV

La deformabilità della struttura viene valutata in base all’entità del rapporto

( )f

h lV

+≅ ÷

1

125

1

800tra la freccia verticale fV e la somma dell’altezza h della colonna e della lunghezza del braccio l dellosbraccio.

Essendo il braccio della gru irrigidito, si può supporre con buona approssimazione che la freccia verticalesia dovuta esclusivamente alla rotazione n dell’estremo B della colonna, considerata come una mensoladi lunghezza h incastrata in A e sollecitata in B da un momento Fl

fM h

EIcm0

2 2

2

1653000 300

2 20600000 11473=

⋅=

⋅⋅ ⋅

=

ϕ =⋅M h

EI

da cui:

ϕ = ⋅ = =2 2

3003 0 020h

f rad.

f l cmV = ⋅ = ⋅ =ϕ 0 02 250 5.

1Il momento di inerzia I di una sezione circolare cava vale, indicando con D e d rispettivamente idiametri esterno ed interno:

( )I

D d=

⋅ −π 4 4

64

f

h lV

+=

+>

5

300 250

1

125

La colonna non è sufficientemente rigida. Si sceglie allora un tubo di diametro 244.5 mm (tabella) conmomento di inerzia1 pari a 3346 cm4. Si ottiene perciò:

f cm0 103= .

ϕ = ⋅ = ⋅ = ⋅ −2 2

3002 02 6 8 100

3

hf rad. .

f l cmV = ⋅ = ⋅ =ϕ 250 0 0135 17. .

f

h lcmV

+=

+= ⋅ −3 4

300 25031 10 3.. <

1

125

Le modificazioni della flangia e delle posizioni dei bulloni conseguenti alla maggiorazione della sezionedella colonna sono a favore della stabilità e non necessitano di ulteriore verifica

Verifica del braccio in mezzeria nel punto 2

2

1 La sezione resistente Wz della quota-parte di ala collaborante a flessione vale:

Wl t

l x Wx t

z z=⋅

= ⋅ =⋅ ⋅2 2

62 14

2 14

6 e posto si ha .

.

La tensione massima al punto 2 generata dall’azione locale della ruota del carrello assume il valore:

σ xz

z

M

W

P x

x t

P

t2 2 2

6 4

2 142 8

4= =

⋅ ⋅

⋅ ⋅= ⋅

/

..

/

Sotto l’azione del carico P/4 trasmesso dalla ruota, la semi-ala si comporta come una lastra incastratain corrispondenza dell’anima. La tensione locale massima nel punto 2 generata dal carico trasmesso dallaruota vale1:

σ x

P

tN mm2 2 2

22 84

2 81653

872= ⋅ = ⋅ =.

/. /

σ fz

M

WN mm= =

⋅=

4132500

73 1056 63

2. /

La tensione ideale vale:

σ σ σ σ σid x zf fz x N mm= + + ⋅ =22 2

22111 /

La tensione ideale è inferiore alla tensione ammissibile

Appendice A(P.Pierotti Meccanica vol. 2 Calderini)

Il nocciolo centrale di inerzia, per una sezione rettangolare,consiste in un parallelogramma avente per diagonali i terzimedi degli assi del rettangolo. Se si vuole che la sezionesopporti tensioni di un unico segno, il centro di sollecitazionedeve cadere all’interno del nocciolo centrale.

Nel caso di un plinto di fondazione è ovvio che il terreno potrà essere sottoposto solamente a tensionidi compressione essendo nulla la capacità reattiva del terreno a trazione.

Si considererà allora unasezione reagente ridotta, in modoche il centro della sollecitazionecada all’estremo del nocciolocentrale di tale sezione ridotta. Conriferimento alla figura a lato,affinché N cada all’estremo delnocciolo della sezione ridotta

occorre che rappresenti( )h e/ 2 −un terzo della nuova dimensione h’della sezione reagente.

hh

e' = ⋅ −

3

2

La parte di sezione tratteggiata in figura,teoricamente assoggettata a trazione, vieneconsiderata non reagente

Leonhard Euler

Born: 15 April 1707 in Basel, SwitzerlandDied: 18 Sept 1783 in St Petersburg, Russia

Leonard Euler's father was Paul Euler. Paul Euler had studied theology at the University of Baseland had attended Jacob Bernoulli's lectures there. In fact Paul Euler and Johann Bernoulli had bothlived in Jacob Bernoulli's house while undergraduates at Basel. Paul Euler became a Protestantminister and married Margaret Brucker, the daughter of another Protestant minister. Their sonLeonard Euler was born in Basel, but the family moved to Riehen when he was one year old and itwas in Riehen, not far from Basel, that Leonard was brought up. Paul Euler had, as we havementioned, some mathematical training and he was able to teach his son elementary mathematicsalong with other subjects.

Leonard was sent to school in Basel and during this time he lived with his grandmother on hismother's side. This school was a rather poor one, by all accounts, and Euler learnt no mathematics atall from the school. However his interest in mathematics had certainly been sparked by his father'steaching, and he read mathematics texts on his own and took some private lessons. Euler's fatherwanted his son to follow him into the church and sent him to the University of Basel to prepare for theministry. He entered the University in 1720, at the age of 14, first to obtain a general education beforegoing on to more advanced studies. Johann Bernoulli soon discovered Euler's great potential formathematics in private tuition that Euler himself engineered. Euler's own account given in hisunpublished autobiographical writings, see [1], is as follows:-

... I soon found an opportunity to be introduced to a famous professor Johann Bernoulli.

... True, he was very busy and so refused flatly to give me private lessons; but he gaveme much more valuable advice to start reading more difficult mathematical books on myown and to study them as diligently as I could; if I came across some obstacle ordifficulty, I was given permission to visit him freely every Sunday afternoon and hekindly explained to me everything I could not understand ...

In 1723 Euler completed his Master's degree in philosophy having compared and contrasted thephilosophical ideas of Descartes and Newton. He began his study of theology in the autumn of 1723,following his father's wishes, but, although he was to be a devout Christian all his life, he could notfind the enthusiasm for the study of theology, Greek and Hebrew that he found in mathematics. Eulerobtained his father's consent to change to mathematics after Johann Bernoulli had used his persuasion.The fact that Euler's father had been a friend of Johann Bernoulli's in their undergraduate daysundoubtedly made the task of persuasion much easier.

Euler completed his studies at the University of Basel in 1726. He had studied many mathematicalworks during his time in Basel, and Calinger [23] has reconstructed many of the works that Euler readwith the advice of Johann Bernoulli. They include works by Varignon, Descartes, Newton, Galileo,

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von Schooten, Jacob Bernoulli, Hermann, Taylor and Wallis. By 1726 Euler had already a paper inprint, a short article on isochronous curves in a resisting medium. In 1727 he published another articleon reciprocal trajectories and submitted an entry for the 1727 Grand Prize of the Paris Academy onthe best arrangement of masts on a ship.

The Prize of 1727 went to Bouguer, an expert on mathematics relating to ships, but Euler's essay wonhim second place which was a fine achievement for the young graduate. However, Euler now had tofind himself an academic appointment and when Nicolaus(II) Bernoulli died in St Petersburg in July1726 creating a vacancy there, Euler was offered the post which would involve him in teachingapplications of mathematics and mechanics to physiology. He accepted the post in November 1726but stated that he did not want to travel to Russia until the spring of the following year. He had tworeasons to delay. He wanted time to study the topics relating to his new post but also he had a chanceof a post at the University of Basel since the professor of physics there had died. Euler wrote anarticle on acoustics, which went on to become a classic, in his bid for selection to the post but he wasnor chosen to go forward to the stage where lots were drawn to make the final decision on whowould fill the chair. Almost certainly his youth (he was 19 at the time) was against him. HoweverCalinger [23] suggests:-

This decision ultimately benefited Euler, because it forced him to move from a smallrepublic into a setting more adequate for his brilliant research and technological work.

As soon as he knew he would not be appointed to the chair of physics, Euler left Basel on 5 April1727. He travelled down the Rhine by boat, crossed the German states by post wagon, then by boatfrom Lübeck arriving in St Petersburg on 17 May 1727. He had joined the St. Petersburg Academy ofScience two years after it had been founded by Catherine I the wife of Peter the Great. Through therequests of Daniel Bernoulli and Jakob Hermann, Euler was appointed to the mathematical-physicaldivision of the Academy rather than to the physiology post he had originally been offered. At StPetersburg Euler had many colleagues who would provide an exceptional environment for him [1]:-

Nowhere else could he have been surrounded by such a group of eminent scientists,including the analyst, geometer Jakob Hermann, a relative; Daniel Bernoulli, withwhom Euler was connected not only by personal friendship but also by common interestsin the field of applied mathematics; the versatile scholar Christian Goldbach, withwhom Euler discussed numerous problems of analysis and the theory of numbers; FMaier, working in trigonometry; and the astronomer and geographer J-N Delisle.

Euler served as a medical lieutenant in the Russian navy from 1727 to 1730. In St Petersburg he livedwith Daniel Bernoulli who, already unhappy in Russia, had requested that Euler bring him tea, coffee,brandy and other delicacies from Switzerland. Euler became professor of physics at the academy in1730 and, since this allowed him to became a full member of the Academy, he was able to give up hisRussian navy post.

Now Daniel Bernoulli held the senior chair in mathematics at the Academy but when he left StPetersburg to return to Basel in 1733 it was Euler who was appointed to this senior chair ofmathematics. The financial improvement which came from this appointment allowed Euler to marrywhich he did on 7 January 1734, marrying Katharina Gsell, the daughter of a painter from the StPetersburg Gymnasium. Katharina, like Euler, was from a Swiss family. They had 13 childrenaltogether although only five survived their infancy. Euler claimed that he made some of his greatestmathematical discoveries while holding a baby in his arms with other children playing round his feet.

We will examine Euler's mathematical achievements later in this article but at this stage it is worthsummarising Euler's work in this period of his career. This is done in [23] as follows:-

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... after 1730 he carried out state projects dealing with cartography, science education,magnetism, fire engines, machines, and ship building. ... The core of his researchprogram was now set in place: number theory; infinitary analysis including its emergingbranches, differential equations and the calculus of variations; and rational mechanics.He viewed these three fields as intimately interconnected. Studies of number theory werevital to the foundations of calculus, and special functions and differential equationswere essential to rational mechanics, which supplied concrete problems.

The publication of many articles and his book Mechanica (1736-37), which extensively presentedNewtonian dynamics in the form of mathematical analysis for the first time, started Euler on the wayto major mathematical work.

Euler's health problems began in 1735 when he had a severe fever and almost lost his life. However,he kept this news from his parents and members of the Bernoulli family back in Basel until he hadrecovered. In his autobiographical writings Euler says that his eyesight problems began in 1738 withoverstrain due to his cartographic work and that by 1740 he had [23]:-

... lost an eye and [the other] currently may be in the same danger.

However, Calinger in [23] argues that Euler's eyesight problems almost certainly started earlier andthat the severe fever of 1735 was a symptom of the eyestrain. He also argues that a portrait of Eulerfrom 1753 suggests that by that stage the sight of his left eye was still good while that of his right eyewas poor but not completely blind. Calinger suggests that Euler's left eye became blind from a latercataract rather than eyestrain.

By 1740 Euler had a very high reputation, having won the Grand Prize of the Paris Academy in 1738and 1740. On both occasions he shared the first prize with others. Euler's reputation was to bring anoffer to go to Berlin, but at first he preferred to remain in St Petersburg. However political turmoil inRussia made the position of foreigners particularly difficult and contributed to Euler changing hismind. Accepting an improved offer Euler, at the invitation of Frederick the Great, went to Berlinwhere an Academy of Science was planned to replace the Society of Sciences. He left St Petersburgon 19 June 1741, arriving in Berlin on 25 July. In a letter to a friend Euler wrote:-

I can do just what I wish [in my research] ... The king calls me his professor, and I thinkI am the happiest man in the world.

Even while in Berlin Euler continued to receive part of his salary from Russia. For this remunerationhe bought books and instruments for the St Petersburg Academy, he continued to write scientificreports for them, and he educated young Russians.

Maupertuis was the president of the Berlin Academy when it was founded in 1744 with Euler asdirector of mathematics. He deputised for Maupertuis in his absence and the two became greatfriends. Euler undertook an unbelievable amount of work for the Academy [1]:-

... he supervised the observatory and the botanical gardens; selected the personnel;oversaw various financial matters; and, in particular, managed the publication ofvarious calendars and geographical maps, the sale of which was a source of income forthe Academy. The king also charged Euler with practical problems, such as the projectin 1749 of correcting the level of the Finow Canal ... At that time he also supervised thework on pumps and pipes of the hydraulic system at Sans Souci, the royal summerresidence.

This was not the limit of his duties by any means. He served on the committee of the Academy dealing

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with the library and of scientific publications. He served as an advisor to the government on statelotteries, insurance, annuities and pensions and artillery. On top of this his scientific output during thisperiod was phenomenal.

During the twenty-five years spent in Berlin, Euler wrote around 380 articles. He wrote books on thecalculus of variations; on the calculation of planetary orbits; on artillery and ballistics (extending thebook by Robins); on analysis; on shipbuilding and navigation; on the motion of the moon; lectures onthe differential calculus; and a popular scientific publication Letters to a Princess of Germany (3 vols.,1768-72).

In 1759 Maupertuis died and Euler assumed the leadership of the Berlin Academy, although not thetitle of President. The king was in overall charge and Euler was not now on good terms withFrederick despite the early good favour. Euler, who had argued with d'Alembert on scientific matters,was disturbed when Frederick offered d'Alembert the presidency of the Academy in 1763. Howeverd'Alembert refused to move to Berlin but Frederick's continued interference with the running of theAcademy made Euler decide that the time had come to leave.

In 1766 Euler returned to St Petersburg and Frederick was greatly angered at his departure. Soonafter his return to Russia, Euler became almost entirely blind after an illness. In 1771 his home wasdestroyed by fire and he was able to save only himself and his mathematical manuscripts. A cataractoperation shortly after the fire, still in 1771, restored his sight for a few days but Euler seems to havefailed to take the necessary care of himself and he became totally blind. Because of his remarkablememory was able to continue with his work on optics, algebra, and lunar motion. Amazingly after hisreturn to St Petersburg (when Euler was 59) he produced almost half his total works despite thetotally blindness.

Euler of course did not achieve this remarkable level of output without help. He was helped by hissons, Johann Albrecht Euler who was appointed to the chair of physics at the Academy in StPetersburg in 1766 (becoming its secretary in 1769) and Christoph Euler who had a military career.Euler was also helped by two other members of the Academy, W L Krafft and A J Lexell, and theyoung mathematician N Fuss who was invited to the Academy from Switzerland in 1772. Fuss, whowas Euler's grandson-in-law, became his assistant in 1776. Yushkevich writes in [1]:-

.. the scientists assisting Euler were not mere secretaries; he discussed the generalscheme of the works with them, and they developed his ideas, calculating tables, andsometimes compiled examples.

For example Euler credits Albrecht, Krafft and Lexell for their help with his 775 page work on themotion of the moon, published in 1772. Fuss helped Euler prepare over 250 articles for publicationover a period on about seven years in which he acted as Euler's assistant, including an important workon insurance which was published in 1776.

Yushkevich describes the day of Euler's death in [1]:-

On 18 September 1783 Euler spent the first half of the day as usual. He gave amathematics lesson to one of his grandchildren, did some calculations with chalk on twoboards on the motion of balloons; then discussed with Lexell and Fuss the recentlydiscovered planet Uranus. About five o'clock in the afternoon he suffered a brainhaemorrhage and uttered only "I am dying" before he lost consciousness. He died abouteleven o'clock in the evening.

After his death in 1783 the St Petersburg Academy continued to publish Euler's unpublished work fornearly 50 more years.

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Euler's work in mathematics is so vast that an article of this nature cannot but give a very superficialaccount of it. He was the most prolific writer of mathematics of all time. He made large boundsforward in the study of modern analytic geometry and trigonometry where he was the first to considersin, cos etc. as functions rather than as chords as Ptolemy had done.

He made decisive and formative contributions to geometry, calculus and number theory. He integratedLeibniz's differential calculus and Newton's method of fluxions into mathematical analysis. Heintroduced beta and gamma functions, and integrating factors for differential equations. He studiedcontinuum mechanics, lunar theory with Clairaut, the three body problem, elasticity, acoustics, thewave theory of light, hydraulics, and music. He laid the foundation of analytical mechanics, especiallyin his Theory of the Motions of Rigid Bodies (1765).

We owe to Euler the notation f(x) for a function (1734), e for the base of natural logs (1727), i for thesquare root of -1 (1777), for pi, for summation (1755), the notation for finite differences y and

y and many others.

Let us examine in a little more detail some of Euler's work. Firstly his work in number theory seems tohave been stimulated by Goldbach but probably originally came from the interest that the Bernoullishad in that topic. Goldbach asked Euler, in 1729, if he knew of Fermat's conjecture that the numbers 2 + 1 were always prime if n is a power of 2. Euler verified this for n = 1, 2, 4, 8 and 16 and, by 1732

at the latest, showed that the next case 2 + 1 = 4294967297 is divisible by 641 and so is not prime.Euler also studied other unproved results of Fermat and in so doing introduced the Euler phi function

(n), the number of integers k with 1 k n and k coprime to n. He proved another of Fermat'sassertions, namely that if a and b are coprime then a + b has no divisor of the form 4n+1, in 1749.

Perhaps the result that brought Euler the most fame in his young days was his solution of what hadbecome known as the Basel problem. This was to find a closed form for the sum of the infiniteseries (2) = (1/n ), a problem which had defeated many of the top mathematicians including JacobBernoulli, Johann Bernoulli and Daniel Bernoulli. The problem had also been studied unsuccessfullyby Leibniz, Stirling, de Moivre and others. Euler showed in 1735 that (2) = /6 but he went on toprove much more, namely that (4) = /90, (6) = /945, (8) = /9450, (10) = /93555and (12) = 691 /638512875. In 1737 he proved the connection of the zeta function with the seriesof prime numbers giving the famous relation

(s) = (1/n ) = (1 - p )

Here the sum is over all natural numbers n while the product is over all prime numbers.

By 1739 Euler had found the rational coefficients C in (2n) = C in terms of the Bernoulli numbers.

Other work done by Euler on infinite series included the introduction of his famous Euler'sconstant , in 1735, which he showed to be the limit of

1/1 + 1/2 + 1/3 + ... + 1/n - ln n

as n tends to infinity. He calculated the constant to 16 decimal places. Euler also studied Fourierseries and in 1744 he was the first to express an algebraic function by such a series when he gave theresult

/2 - x/2 = sin x + (sin 2x)/2 + (sin 3x)/3 + ...

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in a letter to Goldbach. Like most of Euler's work there was a fair time delay before the results werepublished; this result was not published until 1755.

Euler wrote to James Stirling on 8 June 1736 telling him about his results on summing reciprocals ofpowers, the harmonic series and Euler's constant and other results on series. In particular he wrote[59]:-

Concerning the summation of very slowly converging series, in the past year I havelectured to our Academy on a special method of which I have given the sums of verymany series sufficiently accurately and with very little effort.

He then goes on to describe what is now called the Euler-Maclaurin summation formula. Two yearslater Stirling replied telling Euler that Maclaurin:-

... will be publishing a book on fluxions. ... he has two theorems for summing series bymeans of derivatives of the terms, one of which is the self-same result that you sent me.

Euler replied:-

... I have very little desire for anything to be detracted from the fame of the celebratedMr Maclaurin since he probably came upon the same theorem for summing seriesbefore me, and consequently deserves to be named as its first discoverer. For I foundthat theorem about four years ago, at which time I also described its proof andapplication in greater detail to our Academy.

Some of Euler's number theory results have been mentioned above. Further important results innumber theory by Euler included his proof of Fermat's Last Theorem for the case of n = 3. Perhapsmore significant than the result here was the fact that he introduced a proof involving numbers of theform a + b -3 for integers a and b. Although there were problems with his approach this eventuallyled to Kummer's major work on Fermat's Last Theorem and to the introduction of the concept of aring.

One could claim that mathematical analysis began with Euler. In 1748 in Introductio in analysininfinitorum Euler made ideas of Johann Bernoulli more precise in defining a function, and he statedthat mathematical analysis was the study of functions. This work bases the calculus on the theoryelementary functions rather than on geometric curves, as had been done previously. Also in this workEuler gave the formula

e = cos x + i sin x.

In Introductio in analysin infinitorum Euler dealt with logarithms of a variable taking only positivevalues although he had discovered the formula

ln(-1) = i

in 1727. He published his full theory of logarithms of complex numbers in 1751.

Analytic functions of a complex variable were investigated by Euler in a number of different contexts,including the study of orthogonal trajectories and cartography. He discovered the Cauchy-Riemannequations in 1777, although d'Alembert had discovered them in 1752 while investigatinghydrodynamics.

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In 1755 Euler published Institutiones calculi differentialis which begins with a study of the calculusof finite differences. The work makes a thorough investigation of how differentiation behaves undersubstitutions.

In Institutiones calculi integralis (1768-70) Euler made a thorough investigation of integrals whichcan be expressed in terms of elementary functions. He also studied beta and gamma functions, whichhe had introduced first in 1729. Legendre called these 'Eulerian integrals of the first and second kind'respectively while they were given the names beta function and gamma function by Binet and Gaussrespectively. As well as investigating double integrals, Euler considered ordinary and partialdifferential equations in this work.

The calculus of variations is another area in which Euler made fundamental discoveries. His workMethodus inveniendi lineas curvas ... published in 1740 began the proper study of the calculus ofvariations. In [11] it is noted that Carathéodory considered this as:-

... one of the most beautiful mathematical works ever written.

Problems in mathematical physics had led Euler to a wide study of differential equations. Heconsidered linear equations with constant coefficients, second order differential equations withvariable coefficients, power series solutions of differential equations, a method of variation ofconstants, integrating factors, a method of approximating solutions, and many others. Whenconsidering vibrating membranes, Euler was led to the Bessel equation which he solved byintroducing Bessel functions.

Euler made substantial contributions to differential geometry, investigating the theory of surfaces andcurvature of surfaces. Many unpublished results by Euler in this area were rediscovered by Gauss.Other geometric investigations led him to fundamental ideas in topology such as the Eulercharacteristic of a polyhedron.

In 1736 Euler published Mechanica which provided a major advance in mechanics. As Yushkevichwrites in [1]:-

The distinguishing feature of Euler's investigations in mechanics as compared to thoseof his predecessors is the systematic and successful application of analysis. Previouslythe methods of mechanics had been mostly synthetic and geometrical; they demandedtoo individual an approach to separate problems. Euler was the first to appreciate theimportance of introducing uniform analytic methods into mechanics, thus enabling itsproblems to be solved in a clear and direct way.

In Mechanica Euler considered the motion of a point mass both in a vacuum and in a resistingmedium. He analysed the motion of a point mass under a central force and also considered the motionof a point mass on a surface. In this latter topic he had to solve various problems of differentialgeometry and geodesics.

Mechanica was followed by another important work in rational mechanics, this time Euler's twovolume work on naval science. It is described in [23] as:-

Outstanding in both theoretical and applied mechanics, it addresses Euler's intenseoccupation with the problem of ship propulsion. It applies variational principles todetermine the optimal ship design and first establish the principles of hydrostatics ...Euler here also begins developing the kinematics and dynamics of rigid bodies,introducing in part the differential equations for their motion.

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Of course hydrostatics had been studied since Archimedes, but Euler gave a definitive version.

In 1765 Euler published another major work on mechanics Theoria motus corporum solidorum inwhich he decomposed the motion of a solid into a rectilinear motion and a rotational motion. Heconsidered the Euler angles and studied rotational problems which were motivated by the problem ofthe precession of the equinoxes.

Euler's work on fluid mechanics is also quite remarkable. He published a number of major pieces ofwork through the 1750s setting up the main formulas for the topic, the continuity equation, theLaplace velocity potential equation, and the Euler equations for the motion of an inviscidincompressible fluid. In 1752 he wrote:-

However sublime are the researches on fluids which we owe to Messrs Bernoulli,Clairaut and d'Alembert, they flow so naturally from my two general formulae that onecannot sufficiently admire this accord of their profound meditations with the simplicityof the principles from which I have drawn my two equations ...

Euler contributed to knowledge in many other areas, and in all of them he employed his mathematicalknowledge and skill. He did important work in astronomy including [1]:-

... determination of the orbits of comets and planets by a few observations, methods ofcalculation of the parallax of the sun, the theory of refraction, consideration of thephysical nature of comets, .... His most outstanding works, for which he won many prizesfrom the Paris Académie des Sciences, are concerned with celestial mechanics, whichespecially attracted scientists at that time.

In fact Euler's lunar theory was used Tobias Mayer in constructing his tables the moon. In 1765Tobias Mayer's widow received £3000 from Britain for the contribution the tables made to theproblem of the determination of the longitude, while Euler received £300 from the British governmentfor his theoretical contribution to the work.

Euler also published on the theory of music, in particular he published Tentamen novae theoriaemusicae in 1739 in which he tried to make music:-

... part of mathematics and deduce in an orderly manner, from correct principles,everything which can make a fitting together and mingling of tones pleasing.

However, according to [7] the work was:-

... for musicians too advanced in its mathematics and for mathematicians too musical.

Cartography was another area that Euler became involved in when he was appointed director of the StPetersburg Academy's geography section in 1735. He had the specific task of helping Delisle preparea map of the whole of the Russian Empire. The Russian Atlas was the result of this collaboration andit appeared in 1745, consisting of 20 maps. Euler, in Berlin by the time of its publication, proudlyremarked that this work put the Russians well ahead of the Germans in the art of cartography.

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Euler.html

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