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 da
un 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’installazione
di un paranco, portato da un carrello scorrevole lungo il braccio orizzontale della gru. La massa
complessiva dell’equipaggio mobile, a gancio scarico, è pari a 25 kg ed il carico viene sollevato con
una accelerazione massima di 1 m/sec2.
Il candidato, analizzate le sollecitazioni indotte nella struttura e fissati opportunamente i dati
occorrenti, effettui il dimensionamento di massima della gru, eseguendone infine lo schizzo quotato.
Relazione di calcolo
Schizzo quotato
Durata massima della prova: 6 ore
Durante lo svolgimento della prova è consentito l’uso di tavole numeriche, manuali tecnici, del regolo calcolatore e di
calcolatrici tascabili.
Relazione di calcolo
Premessa
In questa relazione, pur facendo riferimento per quanto possibile alla normative vigenti (CNRUNI10011), si introdurranno delle semplificazioni di calcolo tali da consentire a studenti del quinto anno
di corso di un Istituto Tecnico un orientamento facilitato alle problematiche connesse con la
progettazione richiesta.
La relazione perciò, anche se corretta dal punto di vista sostanziale, non è conforme alle norme di legge
e 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 le
tensioni ammissibili sono da assumersi pari a:
1125
. ⋅ σ amm e 1.125 ⋅ τ amm
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 I
Cumula sulla struttura le azioni principali nel modo più sfavorevole
Condizione di carico II
Cumula sulla struttura le azioni principali e complementari nel modo più sfavorevole
Vista l’entità del carico massimo sollevabile ci si orienta su una gru a bandiera di tipo leggero con
colonna realizzata con un tubo di acciaio cilindrico1 e con braccio in profilato a doppio T e rinforzato
con un tirante. ( tabella )
Analisi dei carichi2
Carico massimo sollevabile
Peso dell’equipaggio mobile
Forza di inerzia di sollevamento3
PC
PM
FI
5000
250
500
N
N
N
Carico Totale
PT
5750
N
Carico di riferimento4
P = PT ⋅ ψ = 5750 ⋅ 115
. = 6612 N
Dimensionamento della colonna
Indicata con h l’altezza della colonna e con l la lunghezza del braccio, il momento flettente al
piede della colonna vale: ( figura )
M F = P ⋅ l = 6612 ⋅ 2.5 = 16530 Nm
La sollecitazione di compressione, sempre al piede della colonna, vale:
P = 6612 N
La lunghezza libera h0 della colonna, considerata come una trave incastrata al piede, risulta pari a:
h0 = 2 ⋅ h = 6 m
Fissata, in prima battuta, una snellezza massima pari a 100, la colonna potrebbe essere realizzata con un
tubo in acciaio Fe 360 con le seguenti caratteristiche geometriche ( tabella ):
1
In genere tuttavia, le gru a bandiera di tipo leggero hanno colonne a sezione pentagonale irregolare
con spigoli smussati.
2
Nell’analisi dei carichi si è trascurato sia l’effetto del vento sia la forza d’inerzia longitudinale
conseguente 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 carico
massimo sollevabile, la forza di inerzia FI dovuta al sollevamento del carico vale:
P
5000
FI = C ⋅ a ≅
⋅ 1 = 500 N
g
10
4
Appunto perchè si è trascurato sia l’effetto del vento, sia l’effetto dinamico dovuto alla frenatura si è
pensato arbitrariamente di maggiorare il carico del 15%
Diametro esterno
Spessore
Sezione
Momento d’inerzia1
Momento resistente
Raggio giratorio
De
s
S
I
W
D
191
4.5
26.4
1147
120
6.59
mm
mm
cm2
cm4
cm3
cm
La snellezza 8 massima, con riferimento alla sezione sopra definita, vale:
λ=
h0
600
=
= 91
ρ min 6.59
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
σ =
MF
ωP
1.46 ⋅ 6612
16530000
+
=
+
= 144 N / mm2
. ⋅ 6612 
15
S  15
2640

. P
 1−
 ⋅ 120000
1−
W

250 ⋅ 26400 
 σ ES
dove T
è il coefficiente di amplificazione del carico relativo alla snellezza massima dell’asta
(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 presso
flessione è:
ω⋅P
M
+
S

ν⋅P 
1 −
 ⋅ψ ⋅ W
 σE ⋅S
dove
.
15
v=
1.5 / 1.125
I condizione di carico 
 tensioni ammissibili
II condizione di carico
ν =1
stati limite
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, non
vale il principio di sovrapposizione degli effetti. Per cui la deformazione dovuta a più carichi non è uguale alla
somma delle deformazioni dovute ai carichi singoli. Anche i teoremi di Maxwell e di Castigliano, fondati sulla
proporzionalità suddetta, non valgono in questo caso (O.Belluzzi Scienza delle Costruzioni Vol.1 Zanichelli)
FE
S
W
è la tensione critica euleriana per la snellezza relativa al piano di flessione (tabella)
sezione resistente
momento resistente
Se M varia lungo l’asta si assume Mi =1.3 Mm , essendo Mm il valore medio di M con la
limitazione:
0.75 M max ≤ Mi ≤ M max
La tensione trovata di 144 N/mm2 è inferiore alla tensione ammissibile relativa ad un acciaio
Fe360 (160 N/mm2).
Dimensionamento del braccio
Il dimensionamento si conduce considerando il braccio come facente parte di una struttura
reticolare 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= P
l
2.5
= 6612 ⋅
= 25550 N
a
0.6
Fissando, in prima battuta, la snellezza massima del puntone 120, la sezione in oggetto
dovrebbe 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 esigenze
imposte.
Verifica a carico di punta (materiale Fe 360 con Famm = 160 N/mm2 )
la snellezza del puntone vale
1
8
250/2.51=99
La lunghezza libera del braccio è 2.5 m (asta vincolata agli estremi da due cerniere)
σ =
. ⋅ 25550
ω ⋅ F 181
=
= 22 N / mm2
2120
A
la tensione è inferiore alla tensione ammissibile
Carico applicato in mezzeria del braccio
F 6612
=
= 3306 N
2
2
l P
H B = ⋅ = 13775 N
a 2
V A = VB =
Il momento flettente in mezzeria vale:
M f = VA ⋅
l
2500
= 3306 ⋅
= 4132.5 Nm
2
2
La verifica a presso flessione è condotta con la seguente relazione:
σ=
M if
ωH A
181
0.75 ⋅ 4132500
. ⋅ 13775
+
=
+
≅ 55 N / mm 2
. ⋅ 13775 
A
2120
 15
 15
. HA 
1 −
 ⋅ 73000
1 −
 ⋅W
 557 ⋅ 2120 
 σ E ⋅ A
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 profilato
scelto per il braccio ( HE 100 A )
Si può scegliere come primo tentativo un profilato avente le dimensioni 80x45x5 con area resistente A
pari a 1070 mm2
Verifica a trazione
σ =
V A2 + H A2
=
A
33062 + 137752
≅ 13 N / mm2
1070
La tensione è inferiore a quella ammissibile
Progetto della fondazione
Si ipotizzi un plinto delle seguenti dimensioni in cm:
Base quadrata di lato
altezza
a
h
80
60
cm
cm
Peso del Plinto
Forza totale di riferimento
Peso della struttura
Momento flettente alla base
PP
P
PS
M
9216
6612
1000
4132.5
N
N
N
Nm
(stimata)
L’eccentricità e vale:
e=
M
= 24.5 mm
PP + P + PS
Poiché e > a/6 la sezione risulta parzializzata.
La massima tensione di compressione sul terreno vale (vedi Appendice A):
σ ct =
2 ⋅ ( Pt + P + PS )
2 ⋅ 16828
=
= 9 N / cm2
a 
 80

3 ⋅  − e ⋅ a
3 ⋅  − 24.5 ⋅ 80
2 
 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
sabbia umida compatta
sabbia e argilla compatta
sabbia asciutta compatta, argilla asciutta,
ghiaia a strati con tracce sabbiose
ghiaia grossa compatta
rocce sane
1÷5
10÷15
15÷20
25÷30
40
50÷100÷200
Verifica al ribaltamento
Il momento ribaltante rispetto allo spigolo (A) vale:
0.8 
 a

M R = P ⋅  l −  = 6612 ⋅  2.5 −
 = 13885 Nm

 2
2
Il momento stabilizzante, rispetto allo stesso spigolo:
M S = PP ⋅
a
= 9126 ⋅ 0.4 = 3686 Nm
2
La gru si ribalta !!!
Occorre aumentare il peso del plinto
lato plinto
altezza
Peso plinto
a
h
PP
Momento ribaltante
12
. 

M R = 6612 ⋅  2.5 −  = 12563 Nm

2
Nuovo Momento stabilizzante
M S = 34560 ⋅ 0.6 = 20736 Nm
120 cm
100 cm
34560 N
Occorre, a questo punto riverificare la pressione massima sul terreno. Con riferimento ad un plinto
quadrato di lato 120 cm e altezza 100 cm, l’eccentricità e vale:
e = 9.7 cm
La sezione non è parzializzata.
Tensione dovuta a flessione
σf =
6 ⋅ M 6 ⋅ 1653000
=
= 5.7 N / cm2
3
3
a
120
Tensione dovuta a compressione
σc =
(P + P
S
2
a
+ PP )
=
6612 + 1000 + 34560
= 3 N / cm2
2
120
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 trazione
NS nel gambo della vite
N S = 0.8 ⋅ f kN ⋅ ARe s = 0.8 ⋅ 700 ⋅ 157 = 87920 N
fkN
ARes
resistenza caratteristica di un bullone a trazione in N/mm2
area della sezione resistente della vite
mm2
Il momento massimo trasmissibile dalla flangia vale:
c
2
M F = 2 ⋅ 87920 ⋅ 0.4 = 70336 Nm
MF = 2 ⋅ NS ⋅
Nettamente superiore al momento massimo insistente alla base della
colonna (M=6612 Nm)
(tabella)
(tabella)
Verifiche della rigidezza della struttura
Verifica della freccia fV
La deformabilità della struttura viene valutata in base all’entità del rapporto
fV
1
1
≅
÷
( h + l ) 125 800
tra la freccia verticale fV e la somma dell’altezza h della colonna e della lunghezza del braccio l dello
sbraccio.
Essendo il braccio della gru irrigidito, si può supporre con buona approssimazione che la freccia verticale
sia dovuta esclusivamente alla rotazione n dell’estremo B della colonna, considerata come una mensola
di lunghezza h incastrata in A e sollecitata in B da un momento Fl
M ⋅ h2
1653000 ⋅ 3002
f0 =
=
= 3 cm
2 EI
2 ⋅ 20600000 ⋅ 1147
ϕ =
M ⋅h
EI
da cui:
ϕ =
2
2
⋅ f0 =
3 = 0.02 rad
h
300
fV = ϕ ⋅ l = 0.02 ⋅ 250 = 5 cm
fV
5
1
=
>
h + l 300 + 250 125
La colonna non è sufficientemente rigida. Si sceglie allora un tubo di diametro 244.5 mm (tabella) con
momento di inerzia1 pari a 3346 cm4. Si ottiene perciò:
f 0 = 103
. cm
2
2
ϕ = ⋅ f0 =
⋅ 2.02 = 6.8 ⋅ 10−3 rad
h
300
fV = l ⋅ ϕ = 250 ⋅ 0.0135 = 17
. cm
1
fV
3.4
=
= 31
. ⋅ 10−3 cm <
125
h + l 300 + 250
Le modificazioni della flangia e delle posizioni dei bulloni conseguenti alla maggiorazione della sezione
della colonna sono a favore della stabilità e non necessitano di ulteriore verifica
Verifica del braccio in mezzeria nel punto 2
2
1
Il momento di inerzia I di una sezione circolare cava vale, indicando con D e d rispettivamente i
diametri esterno ed interno:
I=
(
π ⋅ D4 − d 4
64
)
Sotto l’azione del carico P/4 trasmesso dalla ruota, la semi-ala si comporta come una lastra incastrata
in corrispondenza dell’anima. La tensione locale massima nel punto 2 generata dal carico trasmesso dalla
ruota vale1:
σ x 2 = 2.8 ⋅
σ fz =
1653
P/4
= 2.8 ⋅ 2 = 72 N / mm2
2
8
t
M 4132500
2
=
3 = 56.6 N / mm
W
73 ⋅ 10
La tensione ideale vale:
σ id = σ x22 + σ zf2 + σ fz ⋅ σ x 2 = 111 N / mm2
La tensione ideale è inferiore alla tensione ammissibile
1
La sezione resistente Wz della quota-parte di ala collaborante a flessione vale:
Wz =
l ⋅t2
6
e posto l = 2.14 ⋅ x si ha Wz =
2.14 ⋅ x ⋅ t 2
6
La tensione massima al punto 2 generata dall’azione locale della ruota del carrello assume il valore:
σ x2 =
Mz
6⋅ P / 4 ⋅ x
P/4
=
= 2.8 ⋅ 2
2
Wz
2.14 ⋅ x ⋅ t
t
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 terzi
medi degli assi del rettangolo. Se si vuole che la sezione
sopporti tensioni di un unico segno, il centro di sollecitazione
deve cadere all’interno del nocciolo centrale.
Nel caso di un plinto di fondazione è ovvio che il terreno potrà essere sottoposto solamente a tensioni
di compressione essendo nulla la capacità reattiva del terreno a trazione.
Si considererà allora una
sezione reagente ridotta, in modo
che il centro della sollecitazione
cada all’estremo del nocciolo
centrale di tale sezione ridotta. Con
riferimento alla figura a lato,
affinché N cada all’estremo del
nocciolo della sezione ridotta
occorre che ( h / 2 − e) rappresenti
un terzo della nuova dimensione h’
della sezione reagente.
h 
h ' = 3 ⋅  − e
2 
La parte di sezione tratteggiata in figura,
teoricamente assoggettata a trazione, viene
considerata non reagente
Euler
file:///D|/ZIP05/Esercizi/Esame90/Euler.html
Leonhard Euler
Born: 15 April 1707 in Basel, Switzerland
Died: 18 Sept 1783 in St Petersburg, Russia
Leonard Euler's father was Paul Euler. Paul Euler had studied theology at the University of Basel
and had attended Jacob Bernoulli's lectures there. In fact Paul Euler and Johann Bernoulli had both
lived in Jacob Bernoulli's house while undergraduates at Basel. Paul Euler became a Protestant
minister and married Margaret Brucker, the daughter of another Protestant minister. Their son
Leonard Euler was born in Basel, but the family moved to Riehen when he was one year old and it
was in Riehen, not far from Basel, that Leonard was brought up. Paul Euler had, as we have
mentioned, some mathematical training and he was able to teach his son elementary mathematics
along with other subjects.
Leonard was sent to school in Basel and during this time he lived with his grandmother on his
mother's side. This school was a rather poor one, by all accounts, and Euler learnt no mathematics at
all from the school. However his interest in mathematics had certainly been sparked by his father's
teaching, and he read mathematics texts on his own and took some private lessons. Euler's father
wanted his son to follow him into the church and sent him to the University of Basel to prepare for the
ministry. He entered the University in 1720, at the age of 14, first to obtain a general education before
going on to more advanced studies. Johann Bernoulli soon discovered Euler's great potential for
mathematics in private tuition that Euler himself engineered. Euler's own account given in his
unpublished 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 gave
me much more valuable advice to start reading more difficult mathematical books on my
own and to study them as diligently as I could; if I came across some obstacle or
difficulty, I was given permission to visit him freely every Sunday afternoon and he
kindly explained to me everything I could not understand ...
In 1723 Euler completed his Master's degree in philosophy having compared and contrasted the
philosophical 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 not
find the enthusiasm for the study of theology, Greek and Hebrew that he found in mathematics. Euler
obtained 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 days
undoubtedly made the task of persuasion much easier.
Euler completed his studies at the University of Basel in 1726. He had studied many mathematical
works during his time in Basel, and Calinger [23] has reconstructed many of the works that Euler read
with 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 in
print, a short article on isochronous curves in a resisting medium. In 1727 he published another article
on reciprocal trajectories and submitted an entry for the 1727 Grand Prize of the Paris Academy on
the 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 won
him second place which was a fine achievement for the young graduate. However, Euler now had to
find himself an academic appointment and when Nicolaus(II) Bernoulli died in St Petersburg in July
1726 creating a vacancy there, Euler was offered the post which would involve him in teaching
applications of mathematics and mechanics to physiology. He accepted the post in November 1726
but stated that he did not want to travel to Russia until the spring of the following year. He had two
reasons to delay. He wanted time to study the topics relating to his new post but also he had a chance
of a post at the University of Basel since the professor of physics there had died. Euler wrote an
article on acoustics, which went on to become a classic, in his bid for selection to the post but he was
nor chosen to go forward to the stage where lots were drawn to make the final decision on who
would fill the chair. Almost certainly his youth (he was 19 at the time) was against him. However
Calinger [23] suggests:This decision ultimately benefited Euler, because it forced him to move from a small
republic 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 April
1727. He travelled down the Rhine by boat, crossed the German states by post wagon, then by boat
from Lübeck arriving in St Petersburg on 17 May 1727. He had joined the St. Petersburg Academy of
Science two years after it had been founded by Catherine I the wife of Peter the Great. Through the
requests of Daniel Bernoulli and Jakob Hermann, Euler was appointed to the mathematical-physical
division of the Academy rather than to the physiology post he had originally been offered. At St
Petersburg 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, with
whom Euler was connected not only by personal friendship but also by common interests
in the field of applied mathematics; the versatile scholar Christian Goldbach, with
whom Euler discussed numerous problems of analysis and the theory of numbers; F
Maier, 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 lived
with 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 in
1730 and, since this allowed him to became a full member of the Academy, he was able to give up his
Russian navy post.
Now Daniel Bernoulli held the senior chair in mathematics at the Academy but when he left St
Petersburg to return to Basel in 1733 it was Euler who was appointed to this senior chair of
mathematics. The financial improvement which came from this appointment allowed Euler to marry
which he did on 7 January 1734, marrying Katharina Gsell, the daughter of a painter from the St
Petersburg Gymnasium. Katharina, like Euler, was from a Swiss family. They had 13 children
altogether although only five survived their infancy. Euler claimed that he made some of his greatest
mathematical 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 worth
summarising 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 research
program was now set in place: number theory; infinitary analysis including its emerging
branches, differential equations and the calculus of variations; and rational mechanics.
He viewed these three fields as intimately interconnected. Studies of number theory were
vital to the foundations of calculus, and special functions and differential equations
were essential to rational mechanics, which supplied concrete problems.
The publication of many articles and his book Mechanica (1736-37), which extensively presented
Newtonian dynamics in the form of mathematical analysis for the first time, started Euler on the way
to 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 had
recovered. In his autobiographical writings Euler says that his eyesight problems began in 1738 with
overstrain 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 and
that the severe fever of 1735 was a symptom of the eyestrain. He also argues that a portrait of Euler
from 1753 suggests that by that stage the sight of his left eye was still good while that of his right eye
was poor but not completely blind. Calinger suggests that Euler's left eye became blind from a later
cataract rather than eyestrain.
By 1740 Euler had a very high reputation, having won the Grand Prize of the Paris Academy in 1738
and 1740. On both occasions he shared the first prize with others. Euler's reputation was to bring an
offer to go to Berlin, but at first he preferred to remain in St Petersburg. However political turmoil in
Russia made the position of foreigners particularly difficult and contributed to Euler changing his
mind. Accepting an improved offer Euler, at the invitation of Frederick the Great, went to Berlin
where an Academy of Science was planned to replace the Society of Sciences. He left St Petersburg
on 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 think
I am the happiest man in the world.
Even while in Berlin Euler continued to receive part of his salary from Russia. For this remuneration
he bought books and instruments for the St Petersburg Academy, he continued to write scientific
reports for them, and he educated young Russians.
Maupertuis was the president of the Berlin Academy when it was founded in 1744 with Euler as
director of mathematics. He deputised for Maupertuis in his absence and the two became great
friends. 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 of
various calendars and geographical maps, the sale of which was a source of income for
the Academy. The king also charged Euler with practical problems, such as the project
in 1749 of correcting the level of the Finow Canal ... At that time he also supervised the
work on pumps and pipes of the hydraulic system at Sans Souci, the royal summer
residence.
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 state
lotteries, insurance, annuities and pensions and artillery. On top of this his scientific output during this
period was phenomenal.
During the twenty-five years spent in Berlin, Euler wrote around 380 articles. He wrote books on the
calculus of variations; on the calculation of planetary orbits; on artillery and ballistics (extending the
book by Robins); on analysis; on shipbuilding and navigation; on the motion of the moon; lectures on
the 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 the
title of President. The king was in overall charge and Euler was not now on good terms with
Frederick 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. However
d'Alembert refused to move to Berlin but Frederick's continued interference with the running of the
Academy 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. Soon
after his return to Russia, Euler became almost entirely blind after an illness. In 1771 his home was
destroyed by fire and he was able to save only himself and his mathematical manuscripts. A cataract
operation shortly after the fire, still in 1771, restored his sight for a few days but Euler seems to have
failed to take the necessary care of himself and he became totally blind. Because of his remarkable
memory was able to continue with his work on optics, algebra, and lunar motion. Amazingly after his
return to St Petersburg (when Euler was 59) he produced almost half his total works despite the
totally blindness.
Euler of course did not achieve this remarkable level of output without help. He was helped by his
sons, Johann Albrecht Euler who was appointed to the chair of physics at the Academy in St
Petersburg 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 the
young mathematician N Fuss who was invited to the Academy from Switzerland in 1772. Fuss, who
was 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 general
scheme of the works with them, and they developed his ideas, calculating tables, and
sometimes compiled examples.
For example Euler credits Albrecht, Krafft and Lexell for their help with his 775 page work on the
motion of the moon, published in 1772. Fuss helped Euler prepare over 250 articles for publication
over a period on about seven years in which he acted as Euler's assistant, including an important work
on 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 a
mathematics lesson to one of his grandchildren, did some calculations with chalk on two
boards on the motion of balloons; then discussed with Lexell and Fuss the recently
discovered planet Uranus. About five o'clock in the afternoon he suffered a brain
haemorrhage and uttered only "I am dying" before he lost consciousness. He died about
eleven o'clock in the evening.
After his death in 1783 the St Petersburg Academy continued to publish Euler's unpublished work for
nearly 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 superficial
account of it. He was the most prolific writer of mathematics of all time. He made large bounds
forward in the study of modern analytic geometry and trigonometry where he was the first to consider
sin, 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 integrated
Leibniz's differential calculus and Newton's method of fluxions into mathematical analysis. He
introduced beta and gamma functions, and integrating factors for differential equations. He studied
continuum mechanics, lunar theory with Clairaut, the three body problem, elasticity, acoustics, the
wave theory of light, hydraulics, and music. He laid the foundation of analytical mechanics, especially
in 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 the
square 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 to
have been stimulated by Goldbach but probably originally came from the interest that the Bernoullis
had 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's
assertions, 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 had
become known as the Basel problem. This was to find a closed form for the sum of the infinite
series (2) = (1/n ), a problem which had defeated many of the top mathematicians including Jacob
Bernoulli, Johann Bernoulli and Daniel Bernoulli. The problem had also been studied unsuccessfully
by Leibniz, Stirling, de Moivre and others. Euler showed in 1735 that (2) = /6 but he went on to
prove much more, namely that (4) = /90, (6) = /945, (8) = /9450, (10) = /93555
and (12) = 691 /638512875. In 1737 he proved the connection of the zeta function with the series
of 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's
constant , 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 Fourier
series and in 1744 he was the first to express an algebraic function by such a series when he gave the
result
/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 were
published; this result was not published until 1755.
Euler wrote to James Stirling on 8 June 1736 telling him about his results on summing reciprocals of
powers, 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 have
lectured to our Academy on a special method of which I have given the sums of very
many series sufficiently accurately and with very little effort.
He then goes on to describe what is now called the Euler-Maclaurin summation formula. Two years
later Stirling replied telling Euler that Maclaurin:... will be publishing a book on fluxions. ... he has two theorems for summing series by
means 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 celebrated
Mr Maclaurin since he probably came upon the same theorem for summing series
before me, and consequently deserves to be named as its first discoverer. For I found
that theorem about four years ago, at which time I also described its proof and
application in greater detail to our Academy.
Some of Euler's number theory results have been mentioned above. Further important results in
number theory by Euler included his proof of Fermat's Last Theorem for the case of n = 3. Perhaps
more significant than the result here was the fact that he introduced a proof involving numbers of the
form a + b -3 for integers a and b. Although there were problems with his approach this eventually
led to Kummer's major work on Fermat's Last Theorem and to the introduction of the concept of a
ring.
One could claim that mathematical analysis began with Euler. In 1748 in Introductio in analysin
infinitorum Euler made ideas of Johann Bernoulli more precise in defining a function, and he stated
that mathematical analysis was the study of functions. This work bases the calculus on the theory
elementary functions rather than on geometric curves, as had been done previously. Also in this work
Euler gave the formula
e = cos x + i sin x.
In Introductio in analysin infinitorum Euler dealt with logarithms of a variable taking only positive
values 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-Riemann
equations in 1777, although d'Alembert had discovered them in 1752 while investigating
hydrodynamics.
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In 1755 Euler published Institutiones calculi differentialis which begins with a study of the calculus
of finite differences. The work makes a thorough investigation of how differentiation behaves under
substitutions.
In Institutiones calculi integralis (1768-70) Euler made a thorough investigation of integrals which
can be expressed in terms of elementary functions. He also studied beta and gamma functions, which
he 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 Gauss
respectively. As well as investigating double integrals, Euler considered ordinary and partial
differential equations in this work.
The calculus of variations is another area in which Euler made fundamental discoveries. His work
Methodus inveniendi lineas curvas ... published in 1740 began the proper study of the calculus of
variations. 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. He
considered linear equations with constant coefficients, second order differential equations with
variable coefficients, power series solutions of differential equations, a method of variation of
constants, integrating factors, a method of approximating solutions, and many others. When
considering vibrating membranes, Euler was led to the Bessel equation which he solved by
introducing Bessel functions.
Euler made substantial contributions to differential geometry, investigating the theory of surfaces and
curvature 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 Euler
characteristic of a polyhedron.
In 1736 Euler published Mechanica which provided a major advance in mechanics. As Yushkevich
writes in [1]:The distinguishing feature of Euler's investigations in mechanics as compared to those
of his predecessors is the systematic and successful application of analysis. Previously
the methods of mechanics had been mostly synthetic and geometrical; they demanded
too individual an approach to separate problems. Euler was the first to appreciate the
importance of introducing uniform analytic methods into mechanics, thus enabling its
problems 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 resisting
medium. He analysed the motion of a point mass under a central force and also considered the motion
of a point mass on a surface. In this latter topic he had to solve various problems of differential
geometry and geodesics.
Mechanica was followed by another important work in rational mechanics, this time Euler's two
volume work on naval science. It is described in [23] as:Outstanding in both theoretical and applied mechanics, it addresses Euler's intense
occupation with the problem of ship propulsion. It applies variational principles to
determine 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 in
which he decomposed the motion of a solid into a rectilinear motion and a rotational motion. He
considered the Euler angles and studied rotational problems which were motivated by the problem of
the precession of the equinoxes.
Euler's work on fluid mechanics is also quite remarkable. He published a number of major pieces of
work through the 1750s setting up the main formulas for the topic, the continuity equation, the
Laplace velocity potential equation, and the Euler equations for the motion of an inviscid
incompressible 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 one
cannot sufficiently admire this accord of their profound meditations with the simplicity
of 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 mathematical
knowledge and skill. He did important work in astronomy including [1]:... determination of the orbits of comets and planets by a few observations, methods of
calculation of the parallax of the sun, the theory of refraction, consideration of the
physical nature of comets, .... His most outstanding works, for which he won many prizes
from the Paris Académie des Sciences, are concerned with celestial mechanics, which
especially attracted scientists at that time.
In fact Euler's lunar theory was used Tobias Mayer in constructing his tables the moon. In 1765
Tobias Mayer's widow received £3000 from Britain for the contribution the tables made to the
problem of the determination of the longitude, while Euler received £300 from the British government
for his theoretical contribution to the work.
Euler also published on the theory of music, in particular he published Tentamen novae theoriae
musicae 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 St
Petersburg Academy's geography section in 1735. He had the specific task of helping Delisle prepare
a map of the whole of the Russian Empire. The Russian Atlas was the result of this collaboration and
it appeared in 1745, consisting of 20 maps. Euler, in Berlin by the time of its publication, proudly
remarked 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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