Механика.Методика решения задач

Ƚɥɚɜɚ 1. Ʉɢɧɟɦɚɬɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɨɫɬɟɣɲɢɯ ɫɢɫɬɟɦ

11

rr(t);

­x	x(t),	(1.3)
°	y(t),
®y
°	z(t).
¯z

Ɍɪɚɟɤɬɨɪɢɹ ɞɜɢɠɟɧɢɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ – ɥɢɧɢɹ, ɨɩɢ-

ɫɵɜɚɟɦɚɹ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɤɨɧɰɨɦ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ.

ɍɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ ɡɚɞɚɟɬɫɹ ɫɨɜɨɤɭɩɧɨɫɬɶɸ ɞɜɭɯ ɭɪɚɜ-

ɧɟɧɢɣ

­F (x, y, z)	0,	(1.4)
® 1	0,
¯F2 (x, y, z)

ɤɨɬɨɪɵɟ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ, ɢɫɤɥɸɱɚɹ ɜɪɟɦɹ ɢɡ ɡɚɤɨɧɚ ɞɜɢɠɟɧɢɹ ɜ ɤɨɨɪɞɢɧɚɬɧɨɣ ɮɨɪɦɟ (1.3). Ɂɚɦɟɬɢɦ, ɱɬɨ ɫɚɦ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɜ ɤɨɨɪɞɢɧɚɬɧɨɣ ɮɨɪɦɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɭɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ, ɡɚɞɚɧɧɨɟ ɜ ɩɚɪɚɦɟɬɪɢɱɟɫɤɨɦ ɜɢɞɟ.

ɉɟɪɟɦɟɳɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ 'r(t) – ɢɡɦɟɧɟɧɢɟ ɪɚ-

ɞɢɭɫ-ɜɟɤɬɨɪɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɡɚ ɜɪɟɦɹ 't ɫ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ t

(ɪɢɫ. 1ɚ):

'r(t) r(t 't) r(t)
{x(t 't) x(t), y(t 't) y(t), z(t 't) z(t)} .	(1.5)

ɋɤɨɪɨɫɬɶ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ȣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɚɧɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ – ɮɢɡɢɱɟɫɤɚɹ ɜɟɥɢɱɢɧɚ, ɪɚɜɧɚɹ ɩɪɨɢɡɜɨɞɧɨɣ ɪɚɞɢ- ɭɫ-ɜɟɤɬɨɪɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɩɨ ɜɪɟɦɟɧɢ (ɩɪɨɢɡɜɨɞɧɚɹ ɛɟɪɟɬɫɹ ɩɪɢ ɩɨɫɬɨɹɧɧɵɯ ɨɪɬɚɯ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ, ɩɨɫɤɨɥɶɤɭ ɨɧɢ ɠɟɫɬɤɨ ɫɜɹɡɚɧɵ ɫ ɬɟɥɨɦ ɨɬɫɱɟɬɚ):

ȣ(t)	^ȣx (t), ȣy		(t), ȣz	(t)`{	d r(t)		(t) { lim		ǻr(t)
					d t	{ r			ǻt
								ǻt o0
									(1.6)
	^x(t), y(t), z(t)`,
ɝɞɟ Xx , Xy ,	Xz	– ɩɪɨɟɤɰɢɢ ɫɤɨɪɨɫɬɢ ȣ						ɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɨɫɢ

ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ. ɋɤɨɪɨɫɬɶ ȣ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ ɫɭɦɦɵ ɫɨɫɬɚɜɥɹɸɳɢɯ ɫɤɨɪɨɫɬɢ ɜɞɨɥɶ ɨɫɟɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ:

ȣ(t)				Xx	(t)i Xy	(t) j Xz (t)k .	(1.7)
	x(t)i y(t) j z(t)k
ɉɪɢ ɷɬɨɦ ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ȣ ɪɚɜɟɧ
X(t)		Xx2 Xy2 Xz2 .					(1.8)

12 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ

ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɨɩɪɟɞɟɥɟɧɢɟɦ ɫɤɨɪɨɫɬɶ ɜɫɟɝɞɚ ɧɚɩɪɚɜɥɟɧɚ ɩɨ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɬɪɚɟɤɬɨɪɢɢ (ɫɦ. ɪɢɫ. 1.1ɛ).

Ɂɧɚɹ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɫɤɨɪɨɫɬɢ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ȣ(t) ,

ɢ ɪɚɞɢɭɫ-ɜɟɤɬɨɪ r0 { r(t0 )	ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t0, ɦɨɠɧɨ
ɧɚɣɬɢ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ:
t
r(t) r0 ³r(t)dt .	(1.9)
t0

ɉɭɬɶ s(t), ɩɪɨɣɞɟɧɧɵɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɨɣ ɜɞɨɥɶ ɬɪɚɟɤɬɨɪɢɢ (ɞɥɢɧɚ ɬɪɚɟɤɬɨɪɢɢ) ɡɚ ɜɪɟɦɹ t, ɪɚɜɟɧ

t
s(t) ³X(t)dt ,							(1.10)
0
ɩɪɢ ɷɬɨɦ ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ X(t) ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɪɚɜɟɧ
X(t) {		ȣ(t)			d s(t)		(1.11)
					d t	s(t) .
ɍɫɤɨɪɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ a							ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɚɧɧɨɣ

ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ – ɮɢɡɢɱɟɫɤɚɹ ɜɟɥɢɱɢɧɚ, ɪɚɜɧɚɹ ɩɪɨɢɡɜɨɞɧɨɣ ɫɤɨɪɨɫɬɢ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɩɨ ɜɪɟɦɟɧɢ (ɩɪɢ ɩɨɫɬɨɹɧɧɵɯ ɨɪɬɚɯ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ):

				(t)`,	(1.12)
a(t) ^ax (t), a y (t), az (t)`{ ȣ(t)	^Xx	(t),Xy	(t),Xz
ɝɞɟ ax, ay, az – ɩɪɨɟɤɰɢɢ ɭɫɤɨɪɟɧɢɹ a	ɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɨɫɢ ɫɢɫ-

ɬɟɦɵ ɤɨɨɪɞɢɧɚɬ. ɍɫɤɨɪɟɧɢɟ a ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ ɫɭɦɦɵ

ɫɨɫɬɚɜɥɹɸɳɢɯ ɭɫɤɨɪɟɧɢɹ ɜɞɨɥɶ ɨɫɟɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ:
a(t)				ax (t)i ay	(t) j az (t)k .	(1.13)
	ȣx (t)i ȣy (t) j		ȣz (t)k
ɉɪɢ ɷɬɨɦ ɦɨɞɭɥɶ ɭɫɤɨɪɟɧɢɹ a ɪɚɜɟɧ
a(t)	ax2 ay2 az2		.			(1.14)
Ɂɧɚɹ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɭɫɤɨɪɟɧɢɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ a(t) , ɚ
ɬɚɤɠɟ ɫɤɨɪɨɫɬɶ		ȣ0 { ȣ(t0 )	ɢ ɪɚɞɢɭɫ-ɜɟɤɬɨɪ		r0 { r(t0 ) ɜ ɧɚɱɚɥɶɧɵɣ

ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t0, ɦɨɠɧɨ ɧɚɣɬɢ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɫɤɨɪɨɫɬɢ ɢ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ:

t
ȣ(t) ȣ0 ³a(t) d t ,	(1.15)
t0

Ƚɥɚɜɚ 1. Ʉɢɧɟɦɚɬɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɨɫɬɟɣɲɢɯ ɫɢɫɬɟɦ									13
			t t			t	§t ''	·
r(t)	r	ȣ		0			¨	a(t') d t'¸d t'' .	(1.16)
	0	0				³¨ ³		¸
						t0	©t0	¹

ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɥɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ – ɡɧɚɱɟɧɢɹ ɪɚ- ɞɢɭɫ-ɜɟɤɬɨɪɚ ɢ ɫɤɨɪɨɫɬɢ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t0 ɨɬɧɨɫɢɬɟɥɶɧɨ ɡɚɞɚɧɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ:

­r

(t

)

r ,

(1.17)

®

0

0

¯ȣ(t0 ) ȣ0 .

Ɍɚɧɝɟɧɰɢɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ aW

– ɫɨɫɬɚɜɥɹɸɳɚɹ ɭɫɤɨɪɟɧɢɹ

a ɜɞɨɥɶ ɧɚɩɪɚɜɥɟɧɢɹ ɫɤɨɪɨɫɬɢ IJ (ɫɦ. ɪɢɫ. 1.2):

IJ (t) {

ȣ(t)

d r

,

IJ (t)

1 , aW (t)

aW (t)IJ (t) ,

(1.18)

X(t)

d s

aW

(t)

dX(t)

(1.19)

d t

X(t) ,

ɝɞɟ aW (t)

– ɩɪɨɟɤɰɢɹ ɭɫɤɨɪɟɧɢɹ a ɧɚ ɧɚɩɪɚɜɥɟɧɢɟ ɫɤɨɪɨɫɬɢ IJ .

r(t)

M

S

an (t)

aW (t)

n(t

a(t)

IJ (t)

Ɋɢɫ. 1.2. ɍɫɤɨɪɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ a ɢ ɟɟ ɬɚɧɝɟɧɰɢɚɥɶɧɚɹ aW ɢ

ɧɨɪɦɚɥɶɧɚɹ an ɫɨɫɬɚɜɥɹɸɳɢɟ

Ⱦɜɢɠɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɩɪɢ

aW (t) ! 0 –

ɭɫɤɨɪɟɧɧɨɟ,

ɩɪɢ aW (t) 0 – ɡɚɦɟɞɥɟɧɧɨɟ, ɩɪɢ aW (t) 0

– ɪɚɜɧɨɦɟɪɧɨɟ, ɚ ɩɪɢ

aW (t) const z 0 – ɪɚɜɧɨɩɟɪɟɦɟɧɧɨɟ.

ɇɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ an –

ɫɨɫɬɚɜɥɹɸɳɚɹ ɭɫɤɨɪɟɧɢɹ a ,

ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɚɹ ɧɚɩɪɚɜɥɟɧɢɸ ɫɤɨɪɨɫɬɢ (ɪɢɫ. 1.2):

an (t) an (t)n(t) , n(t) A IJ (t) ,

n(t)

1,

(1.20)

ɝɞɟ an (t) – ɩɪɨɟɤɰɢɹ ɭɫɤɨɪɟɧɢɹ a ɧɚ ɧɚɩɪɚɜɥɟɧɢɟ n , ɩɟɪɩɟɧɞɢɤɭ-

ɥɹɪɧɨɟ ɫɤɨɪɨɫɬɢ ɢ ɧɚɩɪɚɜɥɟɧɧɨɟ ɤ ɰɟɧɬɪɭ ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ. ɇɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɜɫɟɝɞɚ ɧɚɩɪɚɜɥɟɧɨ ɤ ɰɟɧɬɪɭ ɤɪɢɜɢɡɧɵ

ɬɪɚɟɤɬɨɪɢɢ – ɰɟɧɬɪɭ ɨɤɪɭɠɧɨɫɬɢ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɪɚɞɢɭɫɚ (ɪɚɞɢɭɫɚ

14 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ

ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ), ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɬɪɚɟɤɬɨɪɢɢ ɜ ɞɚɧɧɨɣ ɬɨɱɤɟ, ɩɪɢ ɷɬɨɦ

		an (t)		X2 (t)	t 0 ,		(1.21)
				ȡ(t)
	ɝɞɟ	ȡ(t) {	d s	– ɪɚɞɢɭɫ ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ ɜ ɞɚɧɧɨɣ ɬɨɱɤɟ, ɚ
			d Į
	dĮ – ɭɝɨɥ ɦɟɠɞɭ ɫɤɨɪɨɫɬɹɦɢ ɜ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ t ɢ t + dt.
		ɍɫɤɨɪɟɧɢɟ a				ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ ɫɭɦɦɵ ɧɨɪɦɚɥɶɧɨɝɨ
	an	ɢ ɬɚɧɝɟɧɰɢɚɥɶɧɨɝɨ aW ɭɫɤɨɪɟɧɢɣ:
		a(t) an (t) aW (t) .					(1.22)
	ɉɪɢ ɷɬɨɦ ɦɨɞɭɥɶ ɭɫɤɨɪɟɧɢɹ a ɪɚɜɟɧ
		a(t)		a2 (t) a2 (t) .			(1.23)
				n		W

ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (1.21) ɢ (1.22) ɭɫɤɨɪɟɧɢɟ ɜɫɟɝɞɚ ɨɬɤɥɨɧɟɧɨ ɨɬ ɧɚɩɪɚɜɥɟɧɢɹ ɫɤɨɪɨɫɬɢ ɜ ɫɬɨɪɨɧɭ ɰɟɧɬɪɚ ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ ɜ ɞɚɧɧɨɣ ɬɨɱɤɟ, ɬɨ ɟɫɬɶ ɜɧɭɬɪɶ ɬɪɚɟɤɬɨɪɢɢ (ɫɦ. ɪɢɫ. 1.2).

Y	ȣ(t)		ȼ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ ɞɜɢɠɟɧɢɹ
			ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɩɨ ɨɤɪɭɠɧɨɫɬɢ,
dM	R		ɬ.ɟ. ɞɜɢɠɟɧɢɹ ɜ ɩɥɨɫɤɨɫɬɢ ɩɨ ɬɪɚɟɤ-
			ɬɨɪɢɢ ɫ ɩɨɫɬɨɹɧɧɵɦ ɪɚɞɢɭɫɨɦ ɤɪɢ-
	(t)
			ɜɢɡɧɵ – U(t) R (ɪɢɫ. 1.3),	ɦɨɠɧɨ
Z		X
			ɜɜɟɫɬɢ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ	Z(t) ɢ

Ɋɢɫ. 1.3. Ʉɢɧɟɦɚɬɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɩɪɢ ɟɟ ɞɜɢɠɟɧɢɢ ɩɨ ɨɤɪɭɠɧɨɫɬɢ

ɉɪɢ ɷɬɨɦ:

an (t)

aW (t)

	ȣ2 (t)		Z2 (t)R,
	R
Z(t)R.

	ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ E (t) :
	Z(t) {	dM(t)				X(t)		,
		dt				R
			{ M(t)
			X(t)			aW (t)		(1.24)
								.
	E (t) { Z(t)			R		R
								(1.25)

Ɇɟɯɚɧɢɱɟɫɤɚɹ ɫɢɫɬɟɦɚ – ɫɨɜɨɤɭɩɧɨɫɬɶ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɟɥ.

ɋɢɫɬɟɦɚ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ – ɫɨɜɨɤɭɩɧɨɫɬɶ ɬɟɥ, ɤɚɠɞɨɟ ɢɡ ɤɨɬɨɪɵɯ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɨɣ. Ⱦɚɥɟɟ ɛɭɞɟɦ ɫɱɢ-

Ƚɥɚɜɚ 1. Ʉɢɧɟɦɚɬɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɨɫɬɟɣɲɢɯ ɫɢɫɬɟɦ

15

ɬɚɬɶ, ɱɬɨ ɜɫɹɤɭɸ ɪɚɫɫɦɚɬɪɢɜɚɟɦɭɸ ɧɚɦɢ ɦɟɯɚɧɢɱɟɫɤɭɸ ɫɢɫɬɟɦɭ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɫɢɫɬɟɦɭ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ.

Ⱥɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɟ ɬɟɥɨ – ɬɟɥɨ (ɫɢɫɬɟɦɚ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ), ɪɚɫɫɬɨɹɧɢɹ ɦɟɠɞɭ ɞɜɭɦɹ ɥɸɛɵɦɢ ɦɚɬɟɪɢɚɥɶɧɵɦɢ ɬɨɱɤɚɦɢ ɤɨɬɨɪɨɝɨ ɧɟ ɦɟɧɹɸɬɫɹ ɜ ɭɫɥɨɜɢɹɯ ɞɚɧɧɨɣ ɡɚɞɚɱɢ.

ɉɨɫɬɭɩɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ – ɞɜɢɠɟɧɢɟ, ɩɪɢ ɤɨɬɨɪɨɦ ɩɪɹɦɚɹ, ɫɨɟɞɢɧɹɸɳɚɹ ɥɸɛɵɟ ɞɜɟ ɦɚɬɟɪɢɚɥɶɧɵɟ ɬɨɱɤɢ ɬɟɥɚ, ɩɟɪɟɦɟɳɚɟɬɫɹ ɩɚɪɚɥɥɟɥɶɧɨ ɫɚɦɨɣ ɫɟɛɟ.

ɉɪɢɧɰɢɩ ɫɭɩɟɪɩɨɡɢɰɢɢ ɞɜɢɠɟɧɢɣ – ɜ ɫɥɭɱɚɟ ɩɨɫɬɭɩɚɬɟɥɶ-

ɧɨɝɨ ɞɜɢɠɟɧɢɹ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ Sc ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ S (ɪɢɫ. 1.4) ɪɚɞɢɭɫ-ɜɟɤɬɨɪ (ɫɤɨɪɨɫɬɶ, ɭɫɤɨɪɟɧɢɟ) ɩɪɨɢɡɜɨɥɶɧɨɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ S ɪɚɜɟɧ ɫɭɦɦɟ ɪɚɞɢɭɫɜɟɤɬɨɪɨɜ (ɫɤɨɪɨɫɬɟɣ, ɭɫɤɨɪɟɧɢɣ) ɧɚɱɚɥɚ ɨɬɫɱɟɬɚ O' ɫɢɫɬɟɦɵ S' ɢ ɬɨɣ ɠɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ S':

r(t) ȣ(t) a(t)

Ɂɞɟɫɶ ȣOc ɜɟɧɧɨ.

c
rOc (t) r (t),
c	(1.26)
ȣOc (t) ȣ (t),

aOc (t) ac(t).

ɢ aOc – ɩɟɪɟɧɨɫɧɵɟ ɫɤɨɪɨɫɬɶ ɢ ɭɫɤɨɪɟɧɢɟ ɫɨɨɬɜɟɬɫɬ-

S'

S M rc(t)

rO' (t)

O'

O

Ɋɢɫ. 1.4. ɉɨɥɨɠɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ M ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɜɭɯ ɩɨɫɬɭɩɚɬɟɥɶɧɨ ɞɜɢɠɭɳɢɯɫɹ ɫɢɫɬɟɦ ɨɬɫɱɟɬɚ S ɢ Sc

ɍɪɚɜɧɟɧɢɹ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ – ɭɪɚɜɧɟɧɢɹ, ɫɜɹɡɵɜɚɸ-

ɳɢɟ ɤɢɧɟɦɚɬɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɪɚɡɥɢɱɧɵɯ ɬɟɥ ɫɢɫɬɟɦɵ: fr (r1, r2 ,..., rN ) 0,

fȣ (ȣ1, ȣ2 ,..., ȣN )	0,	(1.27)
fa (a1, a2 ,..., aN )	0.

16	ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ

ɋɭɳɟɫɬɜɭɸɬ ɞɜɚ ɫɩɨɫɨɛɚ ɧɚɯɨɠɞɟɧɢɹ ɭɪɚɜɧɟɧɢɣ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ.

ɋɩɨɫɨɛ 1. ɉɪɢɧɰɢɩ ɧɟɡɚɜɢɫɢɦɵɯ ɩɟɪɟɦɟɳɟɧɢɣ. ɉɟɪɟɦɟ-

ɳɟɧɢɟ ɤɚɤɨɝɨ-ɥɢɛɨ ɬɟɥɚ ɜ ɫɢɫɬɟɦɟ ɫɜɹɡɚɧɧɵɯ ɬɟɥ ɫɤɥɚɞɵɜɚɟɬɫɹ ɢɡ ɬɚɤ ɧɚɡɵɜɚɟɦɵɯ «ɧɟɡɚɜɢɫɢɦɵɯ» ɩɟɪɟɦɟɳɟɧɢɣ, ɤɚɠɞɨɟ ɢɡ ɤɨɬɨɪɵɯ ɨɛɭɫɥɨɜɥɟɧɨ (ɜɵɡɜɚɧɨ) ɩɟɪɟɦɟɳɟɧɢɟɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɞɪɭɝɨɝɨ ɬɟɥɚ ɫɢɫɬɟɦɵ ɩɪɢ ɩɨɤɨɹɳɢɯɫɹ ɨɫɬɚɥɶɧɵɯ ɬɟɥɚɯ:

ǻri ¦ǻrik .	(1.28)
k zi

ɋɩɨɫɨɛ 2. Ɂɚɩɢɫɚɬɶ ɜɟɥɢɱɢɧɵ ɩɨɫɬɨɹɧɧɵɯ ɤɢɧɟɦɚɬɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɷɥɟɦɟɧɬɨɜ ɫɜɹɡɟɣ (ɧɢɬɟɣ, ɲɬɚɧɝ, ɛɥɨɤɨɜ, ɩɨɜɟɪɯɧɨɫɬɟɣ ɢ ɬ.ɞ.) ɱɟɪɟɡ ɤɨɨɪɞɢɧɚɬɵ ɬɟɥ ɫɢɫɬɟɦɵ, ɢɫɩɨɥɶɡɭɹ ɫɜɨɣɫɬɜɚ ɷɬɢɯ ɷɥɟɦɟɧɬɨɜ (ɧɟɪɚɫɬɹɠɢɦɨɫɬɶ, ɧɟɩɨɞɜɢɠɧɨɫɬɶ, ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɧɨɫɬɶ), ɢ ɩɪɨɞɢɮɮɟɪɟɧɰɢɪɨɜɚɬɶ ɷɬɢ ɜɟɥɢɱɢɧɵ ɩɨ ɜɪɟɦɟɧɢ.

1.2. Ɉɫɧɨɜɧɵɟ ɬɢɩɵ ɡɚɞɚɱ ɢ ɦɟɬɨɞɵ ɢɯ ɪɟɲɟɧɢɹ

1.2.1. Ʉɥɚɫɫɢɮɢɤɚɰɢɹ ɡɚɞɚɱ ɤɢɧɟɦɚɬɢɤɢ

Ɉɫɧɨɜɧɨɣ ɡɚɞɚɱɟɣ ɤɢɧɟɦɚɬɢɤɢ ɹɜɥɹɟɬɫɹ ɨɩɪɟɞɟɥɟɧɢɟ ɤɢɧɟ-

ɦɚɬɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɬɟɥ, ɞɜɢɠɭɳɢɯɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɚɧɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ.

Ȼɨɥɶɲɢɧɫɬɜɨ ɡɚɞɚɱ ɤɢɧɟɦɚɬɢɤɢ ɦɨɠɧɨ ɭɫɥɨɜɧɨ ɨɬɧɟɫɬɢ ɤ ɫɥɟɞɭɸɳɢɦ ɬɢɩɚɦ ɡɚɞɚɱ ɢɥɢ ɢɯ ɤɨɦɛɢɧɚɰɢɹɦ:

1)ɤɢɧɟɦɚɬɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ,

2)ɩɪɢɧɰɢɩ ɫɭɩɟɪɩɨɡɢɰɢɢ ɞɜɢɠɟɧɢɣ,

3)ɭɪɚɜɧɟɧɢɹ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ,

4)ɤɢɧɟɦɚɬɢɤɚ ɩɪɨɫɬɟɣɲɢɯ ɦɟɯɚɧɢɱɟɫɤɢɯ ɫɢɫɬɟɦ.

Ʉɚɤ ɩɪɚɜɢɥɨ, ɨɞɢɧ ɢɡ ɬɢɩɨɜ ɡɚɞɚɱ ɢɦɟɟɬ ɨɫɧɨɜɧɨɟ, ɞɪɭɝɢɟ – ɩɨɞɱɢɧɟɧɧɨɟ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɡɧɚɱɟɧɢɟ.

1.2.2.Ɉɛɳɚɹ ɫɯɟɦɚ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɤɢɧɟɦɚɬɢɤɢ

I. Ɉɩɪɟɞɟɥɢɬɶɫɹ ɫ ɦɨɞɟɥɹɦɢ ɦɚɬɟɪɢɚɥɶɧɵɯ ɨɛɴɟɤɬɨɜ ɢ ɹɜɥɟɧɢɣ.

1.ɇɚɪɢɫɨɜɚɬɶ ɱɟɪɬɟɠ, ɧɚ ɤɨɬɨɪɨɦ ɢɡɨɛɪɚɡɢɬɶ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɬɟɥɚ.

2.ȼɵɛɪɚɬɶ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ ɢ ɢɡɨɛɪɚɡɢɬɶ ɧɚ ɱɟɪɬɟɠɟ ɟɟ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ (ɢɡ ɫɨɨɛɪɚɠɟɧɢɣ ɭɞɨɛɫɬɜɚ).

Ƚɥɚɜɚ 1. Ʉɢɧɟɦɚɬɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɨɫɬɟɣɲɢɯ ɫɢɫɬɟɦ

17

3.ɂɡɨɛɪɚɡɢɬɶ ɢ ɨɛɨɡɧɚɱɢɬɶ ɤɢɧɟɦɚɬɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɬɟɥ.

4.ȼɵɛɪɚɬɶ ɦɨɞɟɥɢ ɬɟɥ ɢ ɢɯ ɞɜɢɠɟɧɢɹ (ɟɫɥɢ ɷɬɨ ɧɟ ɫɞɟɥɚɧɨ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ).

II. Ɂɚɩɢɫɚɬɶ ɩɨɥɧɭɸ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɢɫɤɨɦɵɯ ɜɟɥɢɱɢɧ.

1.Ɂɚɩɢɫɚɬɶ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ ɤɨɨɪɞɢɧɚɬ:

ɚ) ɡɚɤɨɧɵ ɞɜɢɠɟɧɢɹ, ɛ) ɡɚɤɨɧɵ ɢɡɦɟɧɟɧɢɹ ɫɤɨɪɨɫɬɢ,

ɜ) ɡɚɤɨɧɵ ɢɡɦɟɧɟɧɢɹ ɭɫɤɨɪɟɧɢɹ.

2.Ɂɚɩɢɫɚɬɶ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ.

3.Ɂɚɩɢɫɚɬɶ ɭɪɚɜɧɟɧɢɹ ɤɢɧɟɦɚɬɢɱɟɫɤɢɯ ɫɜɹɡɟɣ.

4.ɂɫɩɨɥɶɡɨɜɚɬɶ ɪɟɡɭɥɶɬɚɬɵ ɪɚɧɟɟ ɪɟɲɟɧɧɵɯ ɡɚɞɚɱ ɢ ɨɫɨɛɵɟ ɭɫɥɨɜɢɹ ɡɚɞɚɱɢ (ɧɚɩɪɢɦɟɪ, ɡɚɞɚɧɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ ɦɟɠɞɭ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ ɫɢɫɬɟɦɵ).

III. ɉɨɥɭɱɢɬɶ ɢɫɤɨɦɵɣ ɪɟɡɭɥɶɬɚɬ ɜ ɚɧɚɥɢɬɢɱɟɫɤɨɦ ɢ ɱɢɫɥɟɧɧɨɦ ɜɢɞɚɯ.

1.Ɋɟɲɢɬɶ ɫɢɫɬɟɦɭ ɩɨɥɭɱɟɧɧɵɯ ɭɪɚɜɧɟɧɢɣ.

2.ɉɪɨɜɟɫɬɢ ɚɧɚɥɢɡ ɪɟɲɟɧɢɹ (ɩɪɨɜɟɪɢɬɶ ɪɚɡɦɟɪɧɨɫɬɶ ɢ ɥɢɲɧɢɟ ɤɨɪɧɢ, ɪɚɫɫɦɨɬɪɟɬɶ ɯɚɪɚɤɬɟɪɧɵɟ ɫɥɭɱɚɢ, ɭɫɬɚɧɨɜɢɬɶ ɨɛɥɚɫɬɶ ɩɪɢɦɟɧɢɦɨɫɬɢ).

3.ɉɨɥɭɱɢɬɶ ɱɢɫɥɟɧɧɵɣ ɪɟɡɭɥɶɬɚɬ.

ɉɪɢɦɟɱɚɧɢɹ.

ȼɫɥɭɱɚɟ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɧɚ ɤɢɧɟɦɚɬɢɤɭ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɩɩ. I.3 – II.2 ɪɟɱɶ ɢɞɟɬ ɨ ɤɢɧɟɦɚɬɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɯ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ, ɚ ɩ. II.3 ɧɚɞɨ ɨɩɭɫɬɢɬɶ.

ȼɫɥɭɱɚɟ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɧɚ ɤɢɧɟɦɚɬɢɤɭ ɩɪɨɫɬɟɣɲɢɯ ɦɟɯɚɧɢɱɟɫɤɢɯ ɫɢɫɬɟɦ ɜ ɩɩ. I.3 – II.2 ɪɟɱɶ ɢɞɟɬ ɨ ɤɢɧɟɦɚɬɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɯ ɬɟɥ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɫɢɫɬɟɦɵ.

ɉɭɧɤɬɵ II.1 – II.3 (ɜ ɬɨɦ ɱɢɫɥɟ II.2.a – II.2.ɜ) ɦɨɠɧɨ ɜɵɩɨɥɧɹɬɶ ɜ ɬɨɣ ɢɥɢ ɢɧɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɬɢɩɚ ɡɚɞɚɱɢ.

1.3. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ

Ɂɚɞɚɱɚ 1.1

(Ʉɢɧɟɦɚɬɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ)

ɋɤɨɪɨɫɬɶ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɡɚɜɢɫɢɬ ɨɬ ɟɟ ɩɨɥɨɠɟɧɢɹ ɜ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: ȣ ci bxj , ɝɞɟ

18 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ

c ɢ b – ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɩɨɫɬɨɹɧɧɵɟ ɜɟɥɢɱɢɧɵ. ȼ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɪɚɞɢɭɫ-ɜɟɤɬɨɪ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɪɚɜɟɧ ɧɭɥɸ: r(0) 0 .

Ɉɩɪɟɞɟɥɢɬɶ:

ɚ) ɡɚɤɨɧɵ ɞɜɢɠɟɧɢɹ r(t) , ɢɡɦɟɧɟɧɢɹ ɫɤɨɪɨɫɬɢ ȣ(t) ɢ ɭɫɤɨɪɟɧɢɹ a(t) , ɬɚɧɝɟɧɰɢɚɥɶɧɭɸ aW (t) ɢ ɧɨɪɦɚɥɶɧɭɸ an (t) ɩɪɨɟɤɰɢɢ ɭɫ-

ɤɨɪɟɧɢɹ;

ɛ) ɭɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ y(x) ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ; ɜ) ɪɚɞɢɭɫ ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ U(t) ;

ɝ) ɭɝɨɥ M(t) ɦɟɠɞɭ ɫɤɨɪɨɫɬɶɸ ȣ(t) ɢ ɭɫɤɨɪɟɧɢɟɦ a(t) .

Ɋɟɲɟɧɢɟ

ɋɥɟɞɭɟɦ ɨɛɳɟɣ ɫɯɟɦɟ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɤɢɧɟɦɚɬɢɤɢ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɨɫɬɟɣɲɢɯ ɫɢɫɬɟɦ.

I. ɉɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɞɜɢɠɟɧɢɟ ɩɪɨɢɫɯɨɞɢɬ ɜ ɩɥɨɫɤɨɫɬɢ XY, ɨɛɪɚɡɨɜɚɧɧɨɣ ɤɨɨɪɞɢɧɚɬɧɵɦɢ ɨɫɹɦɢ, ɧɚɩɪɚɜɥɟɧɢɹ ɤɨɬɨɪɵɯ ɡɚɞɚɧɵ ɨɪɬɚɦɢ i ɢ j .

II. Ɂɚɩɢɲɟɦ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɢ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɫɤɨɪɨɫɬɢ

ɬɟɥɚ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ:
­x(0)	0, y(0)					0,	(1.29)
®		c, Xy (0)				b 0 0,
¯Xx (0)
­		d x			c,
°Xx (t)
		d t					(1.30)
®
			d y
°					bx(t).
° Xy (t)
			d t
¯

III. Ɂɚɩɢɫɚɧɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɨɪɞɢɧɚɬ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ (1.29) ɫ ɭɱɟɬɨɦ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ (1.29) ɩɨɡɜɨɥɹɸɬ ɧɚɣɬɢ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ ɤɨɨɪɞɢɧɚɬ ɢ ɡɚɜɢɫɢɦɨɫɬɶ ɨɬ ɜɪɟɦɟɧɢ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚ

r(t) :
­x(t)	ct,
°			2	(1.31)
®		cbt
°y(t)				,
	2
¯

cbt2

r(t) cti j . (1.32) 2

Ƚɥɚɜɚ 1. Ʉɢɧɟɦɚɬɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɨɫɬɟɣɲɢɯ ɫɢɫɬɟɦ

19

ɂɫɩɨɥɶɡɭɹ ɧɚɣɞɟɧɧɭɸ ɡɚɜɢɫɢɦɨɫɬɶ x(t) (1.31), ɨɩɪɟɞɟɥɢɦ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɫɤɨɪɨɫɬɢ ȣ(t) ci bx(t) j ɢ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɭɫɤɨ-

ɪɟɧɢɹ a(t) :

ȣ(t)

ci cbtj ,

(1.33)

a(t)

d ȣ

cbj .

(1.34)

d t

ɍɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ ɧɚɯɨɞɢɬɫɹ ɢɡ ɡɚɤɨɧɚ ɞɜɢɠɟɧɢɹ ɦɚɬɟ-

ɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɩɭɬɟɦ ɢɫɤɥɸɱɟɧɢɹ ɢɡ (1.31) ɜɪɟɦɟɧɢ t:

y(x)

cb

x2

b

x

2

.

(1.35)

2

c2

2c

Ɉɫɬɚɥɶɧɵɟ ɢɫɤɨɦɵɟ ɜɟɥɢɱɢɧɵ ɨɩɪɟɞɟɥɹɸɬɫɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ

ɫ ɮɨɪɦɭɥɚɦɢ, ɩɪɢɜɟɞɟɧɧɵɦɢ ɜ ɩ. 1 ɞɚɧɧɨɣ Ƚɥɚɜɵ.

Ɇɨɞɭɥɶ ɫɤɨɪɨɫɬɢ (1.7) ɪɚɜɟɧ:

X(t)

Xx2 Xy2

c2 c2b2t 2 .

(1.36)

ɉɪɨɟɤɰɢɢ ɭɫɤɨɪɟɧɢɹ aW (t) ɢ an (t)

(1.19, 1.23) ɩɨɥɭɱɢɦ ɜ ɜɢ-

ɞɟ:

­

dX

c2b2t

cb2t

°aW

(t)

,

d t

c2 c2b2t 2

1 b

°

2t 2

(1.37)

®

c2b4t 2

cb

°a

n

(t)

a2 a2

c2b2

.

2

2

°

W

t

2

t

2

¯

1 b

1 b

Ɋɚɞɢɭɫ ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ (1.21) ɪɚɜɟɧ:

U(t)

X2

c2 c2b2t 2

1 b2t 2

c

1 b2t 2

3 / 2 .

(1.38)

an

cb

b

ɍɝɨɥ M(t) ɦɟɠɞɭ ɫɤɨɪɨɫɬɶɸ ȣ(t) ɢ ɭɫɤɨɪɟɧɢɟɦ a(t) ɨɩɪɟɞɟ-

ɥɹɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ:

tgM(t)

an

cb

1 b2t 2

cb

1

.

(1.39)

aW

1 b2t 2

cb2t

cb2t

bt

Ɂɚɦɟɬɢɦ, ɱɬɨ ɦɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ ɞɜɢɠɟɬɫɹ ɩɨ ɩɚɪɚɛɨɥɢɱɟɫɤɨɣ ɬɪɚɟɤɬɨɪɢɢ (1.35) ɫ ɩɨɫɬɨɹɧɧɵɦ ɭɫɤɨɪɟɧɢɟɦ, ɧɚɩɪɚɜɥɟɧɧɵɦ ɜɞɨɥɶ ɨɫɢ Y (1.34). ɇɚ ɪɢɫ. 1.5 ɫɯɟɦɚɬɢɱɧɨ ɢɡɨɛɪɚɠɟɧɚ ɬɪɚɟɤɬɨɪɢɹ ɞɜɢɠɟɧɢɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɢɡɨɛɪɚɠɟɧɵ ɜɟɤɬɨɪɵ ɭɫɤɨɪɟɧɢɹ ɢ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ.

20	ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ

ɇɟɬɪɭɞɧɨ ɜɢɞɟɬɶ, ɱɬɨ ɩɪɢ t 0 ɪɟɲɟɧɢɹ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ ɡɚɞɚɱɢ. ɉɪɢ ɷɬɨɦ ɬɚɧɝɟɧɰɢɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɜ ɭɤɚɡɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɪɚɜɧɨ ɧɭɥɸ, ɪɚɞɢɭɫ ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ

	U	c	, ɚ ɭɝɨɥ ɦɟɠɞɭ ɫɤɨɪɨɫɬɶɸ ɢ ɭɫɤɨ-
		b

ɪɟɧɢɟɦ M S / 2 .

Y

y(x)

a

O	ȣ0	X
	Ɋɢɫ. 1.5

ɉɪɢ t o f ɡɧɚɱɟɧɢɹ ɤɨɨɪɞɢɧɚɬ ɬɨɱɤɢ ɢ ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ, ɤɚɤ ɢ ɫɥɟɞɨɜɚɥɨ ɨɠɢɞɚɬɶ, ɧɟɨɝɪɚɧɢɱɟɧɧɨ ɜɨɡɪɚɫɬɚɸɬ, ɧɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɢ ɭɝɨɥ ɦɟɠɞɭ ɫɤɨɪɨɫɬɶɸ ɢ ɭɫɤɨɪɟɧɢɟɦ ɫɬɪɟɦɹɬɫɹ ɤ ɧɭɥɸ, ɚ ɪɚɞɢɭɫ ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ – ɤ ɛɟɫɤɨɧɟɱɧɨɫɬɢ.

Ɂɚɞɚɱɚ 1.2

(Ʉɢɧɟɦɚɬɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ)

ɇɚɯɨɞɹɳɟɟɫɹ ɧɚ ɜɵɫɨɬɟ H ɧɚɞ Ɂɟɦɥɟɣ ɬɟɥɨ ɛɪɨɫɢɥɢ ɝɨɪɢɡɨɧɬɚɥɶɧɨ ɫ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɶɸ ȣ0 . ɇɚɣɬɢ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɬɟɥɚ,

ɭɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ, ɡɚɤɨɧɵ ɢɡɦɟɧɟɧɢɹ ɫɤɨɪɨɫɬɢ ɢ ɭɫɤɨɪɟɧɢɹ, ɚ ɬɚɤɠɟ ɧɨɪɦɚɥɶɧɭɸ ɢ ɬɚɧɝɟɧɰɢɚɥɶɧɭɸ ɩɪɨɟɤɰɢɢ ɭɫɤɨɪɟɧɢɹ ɢ ɪɚɞɢɭɫ ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ ɜ ɩɪɨɢɡɜɨɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ.

Ɋɟɲɟɧɢɟ

I. ɇɚɪɢɫɭɟɦ ɱɟɪɬɟɠ ɢ ɢɡɨɛɪɚɡɢɦ ɧɚ ɧɟɦ ɡɚɞɚɧɧɭɸ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɫɤɨɪɨɫɬɶ ɬɟɥɚ ȣ0 ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ (t = 0) ɢ

ɩɪɟɞɩɨɥɚɝɚɟɦɭɸ ɬɪɚɟɤɬɨɪɢɸ ɞɜɢɠɟɧɢɹ ɬɟɥɚ (ɪɢɫ. 1.6).

ȼɵɛɟɪɟɦ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ,	Y
ɫɜɹɡɚɧɧɭɸ ɫ Ɂɟɦɥɟɣ. Ɉɫɶ X ɞɟɤɚɪ-		ȣ0
ɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɧɚɩɪɚ-	H
ɜɢɦ ɝɨɪɢɡɨɧɬɚɥɶɧɨ ɜɞɨɥɶ ɩɨɜɟɪɯ-
ɧɨɫɬɢ Ɂɟɦɥɢ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɧɚ-
ɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ȣ0 , ɚ ɨɫɶ Y –
ɜɟɪɬɢɤɚɥɶɧɨ ɜɜɟɪɯ ɧɚ ɩɨɥɨɠɟɧɢɟ
ɬɟɥɚ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟ-	O
ɧɢ. Ȼɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɬɟɥɨ ɹɜɥɹ-
ɟɬɫɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɨɣ, ɚ ɞɜɢ-		Ɋɢɫ. 1.6
ɠɟɧɢɟ ɬɟɥɚ ɭ ɩɨɜɟɪɯɧɨɫɬɢ Ɂɟɦɥɢ

ɩɪɨɢɫɯɨɞɢɬ ɫ ɩɨɫɬɨɹɧɧɵɦ ɭɫɤɨɪɟɧɢɟɦ ɫɜɨɛɨɞɧɨɝɨ ɩɚɞɟɧɢɹ g .

g

X