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α
n
β
n
(n = 0, 1, 2, 3, . . .)
α
n
=
2
l
Z
l
0
f(x) cos(
πnx
l
)dx,
β
n
=
2
vπ
Z
l
0
F (x) cos(
πnx
l
)dx.
l
x = 0 x = l
f(x) = u(x, 0) =
x
2
h
, 0 ≤ x ≤
l
4
(x−
l
2
)
2
h
,
l
4
≤ x ≤
l
2
0,
l
2
≤ x ≤ l
,
F (x) = u
′
t
(x, 0) = 0.
α
n
=
2
l
Z
l/4
0
x
2
h
cos(
πnx
l
)dx +
2
l
Z
l/2
l/4
(x −
l
2
)
2
h
cos(
πnx
l
)dx
=
8l
2
hπ
3
n
3
cos(
πn
4
)
n
πn
4
− sin(
πn
4
)
o
,
α
0
=
l
2
48h
,
β
n
=
2
vπ
Z
l
0
F (x) cos(
πnx
l
)dx = 0.
u(x, t) =
l
2
96h
+
8l
2
hπ
3
∞
X
n=1
1
n
3
cos(
πn
4
)
n
πn
4
− sin(
πn
4
)
o
0.25 0.5 0.75 1
x
- 1
1
2
u
t= 0 cek
0.25 0.5 0.75 1
x
- 1
1
2
u
t= 0.05 cek
0.25 0.5 0.75 1
x
- 1
1
2
u
t= 0.1 cek
0.25 0.5 0.75 1
x
- 1
1
2
u
t= 0.15 cek
0.25 0.5 0.75 1
x
- 1
1
2
u
t= 0.2 cek
0.25 0.5 0.75 1
x
- 1
1
2
u
t= 0.25 cek
0.25 0.5 0.75 1
x
- 1
1
2
u
t= 0.3 cek
0.25 0.5 0.75 1
x
- 1
1
2
u
t= 0.35 cek
0.25 0.5 0.75 1
x
- 1
1
2
u
t= 0.4 cek
0.25 0.5 0.75 1
x
- 1
1
2
u
t= 0.45 cek
0.25 0.5 0.75 1
x
- 1
1
2
u
t= 0.5 cek
v = 2 / , l = 1 , h = (4 − π)/π
3
× cos(ω
n
t) cos(
πnx
l
).
t
t = 1
ν = 1
u u u
ãäå αn è βn íàõîäÿòñÿ êàê êîý èöèåíòû ðàçëîæåíèÿ â ðÿä Ôóðüå
ïî êîñèíóñàì (n = 0, 1, 2, 3, . . .)
2 2 2
1 1 1
l
2 πnx
Z
x x x
αn = f (x) cos( )dx, (12a) 0.25 0.5 0.75 1 0.25 0.5 0.75 1 0.25 0.5 0.75 1
l 0 l -1 t=0 cek -1 t=0.05 cek -1 t=0.1 cek
l u u u
2 πnx
Z
βn = F (x) cos( )dx. (12b) 2 2 2
vπ 0 l
1 1 1
Ïðèìåð 2. àññìîòðèì ñòðóíó äëèíîé l , êàñàòåëüíàÿ ê êîòî- x x x
0.25 0.5 0.75 1 0.25 0.5 0.75 1 0.25 0.5 0.75 1
ðîé â òî÷êàõ x = 0 è x = l ðàâíà íóëþ, è ðàññìîòðèì ñëåäóþùèå -1 t=0.15 cek -1 t=0.2 cek -1 t=0.25 cek
íà÷àëüíûå óñëîâèÿ: u u u
2 2 2
x2
h ,2 0 ≤ x ≤ 4l 1 1 1
l
f (x) = u(x, 0) = (x− 2 )
h , 4l ≤ x ≤ 2l , 0.25 0.5 0.75 1
x 0.25 0.5 0.75 1
x 0.25 0.5 0.75 1
x
l
0, 2 ≤x ≤l
-1 t=0.3 cek -1 t=0.35 cek -1 t=0.4 cek
u u
F (x) = u′t (x, 0) = 0.
2 2
Íà÷àëüíûå óñëîâèÿ äîëæíû áûòü òàêèìè, ÷òîáû êàñàòåëüíàÿ íà 1 1
êîíöàõ ñòðóíû áûëà ðàâíà íóëþ è â íà÷àëüíûé ìîìåíò âðåìåíè. 0.25 0.5 0.75 1
x x
0.25 0.5 0.75 1
Íà÷àëüíàÿ îðìà ñòðóíû èçîáðàæåíà íà èñ. 3. Ïî îðìóëàì (12) -1 t=0.45 cek -1 t=0.5 cek
ïîëó÷àåì:
èñ. 4: Íà ðèñóíêàõ èçîáðàæåíà îðìà ñòðóíû â ðàçëè÷íûå ìîìåíòû âðåìåíè.
2 l/4
x2 πnx 2 l/2 (x − 2l )2 πnx
Z Z
Êîíöû ñòðóíû ñâîáîäíî äâèæóòñÿ, ïðè÷åì óãîë íàêëîíà êîíöîâ ñòðóíû ðàâåí
αn = cos( )dx + cos( )dx íóëþ âî âñå ìîìåíòû âðåìåíè; âíåøíèå ñèëû îòñóòñòâóþò. Âûáðàíû ñëåäóþùèå
l 0 h l l l/4 h l
çíà÷åíèÿ ïàðàìåòðîâ: v = 2ì/ñåê, l = 1ì, h = (4 − π)/π 3 ì.
8l2 πn n πn πn o
= cos( ) − sin( ) ,
hπ 3 n3 4 4 4 πnx
l2 × cos(ωn t) cos( ).
α0 = , l
48h
Z l Íà ðèñóíêå 4 èçîáðàæåíî äâèæåíèå ñòðóíû â òå÷åíèè ïîëóïåðèî-
2 πnx
βn = F (x) cos( )dx = 0. äà. Ïðè äàëüíåéøåì óâåëè÷åíèè t îðìà ñòðóíû áóäåò ìåíÿòüñÿ â
vπ 0 l îáðàòíîì ïîðÿäêå, è â ìîìåíò t = 1ñåê ñòðóíà âåðíåòñÿ â èñõîäíîå
Òàêèì îáðàçîì, ïîäñòàâëÿÿ ïîëó÷åííûå âûðàæåíèÿ â (11), ïîëó÷àåì ïîëîæåíèå. Ëèíåéíàÿ ÷àñòîòà êîëåáàíèé òàêîé ñòðóíû ν = 1 ö.
çàêîí êîëåáàíèÿ ñòðóíû ñ òàêèìè íà÷àëüíûìè äàííûìè Èç ðèñóíêà âèäíî, ÷òî óãîë íàêëîíà êàñàòåëüíîé íà êîíöàõ ñòðó-
íû ðàâåí íóëþ âî âñå ìîìåíòû âðåìåíè. Âèä êîëåáàíèé ñòðóíû ñ
l2
∞
8l2 X 1 πn n πn πn o òàêèìè ãðàíè÷íûìè óñëîâèÿìè îòëè÷àåòñÿ îò êîëåáàíèé ñòðóíû ñ
u(x, t) = + cos( ) − sin( ) çàêðåïëåííûìè êîíöàìè. Ñòðóíà â ïðîöåññå êîëåáàíèé âñåãäà îñòà-
96h hπ 3 n=1 n3 4 4 4
11 12
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