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ɉɪɢɦɟɪ 2 (ɪɢɫ.8.6) [5]. ɉɨ ɬɨɧɤɨɫɬɟɧɧɨɣ ɬɪɭɛɤɟ (ɬɨɥɳɢɧɚ t, ɞɢɚɦɟɬɪ D) ɩɪɨɬɟɤɚɟɬ ɠɢɞɤɨɫɬɶ ɩɥɨɬɧɨɫɬɢ U. Ɍɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɤɪɢɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ ɫɤɨɪɨɫɬɢ w*, ɩɪɢ ɤɨɬɨɪɨɦ ɜɨɡɦɨɠɧɚ ɫɦɟɠɧɚɹ (ɢɫɤɪɢɜɥɟɧɧɚɹ) ɮɨɪɦɚ ɪɚɜɧɨɜɟɫɢɹ ɬɪɭɛɤɢ.

Ɋɟɲɟɧɢɟ. ɉɪɢɦɟɧɢɦ ɦɟɬɨɞ ɗɣɥɟɪɚ. Ɋɚɫɫɦɨɬɪɢɦ ɢɫɤɪɢɜɥɟɧɧɨɟ ɫɨɫɬɨɹɧɢɟ ɬɪɭɛɤɢ (ɪɢɫ.8.7). ɉɪɢ ɩɪɨɬɟɤɚɧɢɢ ɠɢɞɤɨɫɬɢ ɩɨ ɢɡɨɝɧɭɬɨɦɭ ɭɱɚɫɬɤɭ ɬɪɭɛɤɢ, ɤɪɢɜɢɡɧɚ ɤɨɬɨɪɨɝɨ ɪɚɜɧɚ F= 1/R (R – ɪɚɞɢɭɫ ɤɪɢɜɢɡɧɵ), ɧɚ ɷɥɟɦɟɧɬ ɞɥɢɧɵ ɬɪɭɛɤɢ dz

t	W
D
	l
	Ɋɢɫ.8.6

y	dPi
z	dz	z
	Ɋɢɫ. 8.7

ɫɨ ɫɬɨɪɨɧɵ ɠɢɞɤɨɫɬɢ ɞɟɣɫɬɜɭɟɬ ɢɧɟɪɰɢɨɧɧɚɹ ɫɢɥɚ

dPi = – dma,

(8.2)

ɝɞɟ a – ɜɟɤɬɨɪ ɭɫɤɨɪɟɧɢɹ ɱɚɫɬɢ ɠɢɞɤɨɫɬɢ ɦɚɫɫɨɣ dm = USdz, (S – ɩɥɨɳɚɞɶ ɫɟɱɟɧɢɹ ɩɨɬɨɤɚ). ɉɨɥɚɝɚɹ (ɩɪɢ ɦɚɥɨɦ ɢɫɤɪɢɜɥɟɧɢɢ), ɱɬɨ ɥɢɧɟɣɧɚɹ ɫɤɨɪɨɫɬɶ ɩɨɬɨɤɚ ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ, ɧɚɣɞɟɦ, ɱɬɨ ɭɫɤɨɪɟɧɢɟ ɧɚɩɪɚɜɥɟɧɨ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɨɫɢ ɬɪɭɛɤɢ, ɚ ɜɟɥɢɱɢɧɚ ɟɝɨ a= w2/R=Fw2. Ɉɬɫɸɞɚ ɧɚɯɨɞɢɬɫɹ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɩɨ ɞɥɢɧɟ ɬɪɭɛɤɢ ɰɟɧɬɪɨɛɟɠɧɨɣ ɫɢɥɵ

qi=dPi /dz = USw2F .

ɍɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ qi= d2M/dz2 ɫ ɭɱɟɬɨɦ ɡɚɤɨɧɚ Ƚɭɤɚ F = M/(EI) ɩɪɢɧɢɦɚɟɬ

ɜɢɞ

X 4+k2Xcc= 0,

(8.3)

ɝɞɟ X – ɩɪɨɝɢɛ ɬɪɭɛɤɢ, ɲɬɪɢɯɨɦ ɨɛɨɡɧɚɱɟɧɚ ɩɪɨɢɡɜɨɞɧɚɹ ɩɨ ɞɥɢɧɟ, k2=USw2/(EI). Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (8.3) (ɩɪɢ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɹɯ z = 0,l: X = 0, Xcc = 0)

ɢɦɟɟɬ ɜɢɞ

X=Asin kz,

A z 0 ɩɪɢ kl=S.

(8.4)

Ɉɬɫɸɞɚ ɧɚɣɞɟɦ, ɱɬɨ ɤɪɢɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ ɫɤɨɪɨɫɬɢ, ɩɪɢ ɤɨɬɨɪɨɦ ɢɡɨɝɧɭɬɚɹ ɬɪɭɛɤɚ (A z 0) ɬɚɤ ɠɟ ɧɚɯɨɞɢɬɫɹ ɜ ɪɚɜɧɨɜɟɫɢɢ, ɤɚɤ ɢ ɩɪɹɦɚɹ (A = 0), ɪɚɜɧɚ

w* = S / l(EI/(US))0.5.

(8.5)

8.3. ɉɪɢɛɥɢɠɟɧɧɵɣ ɷɧɟɪɝɟɬɢɱɟɫɤɢɣ ɦɟɬɨɞ

ɉɪɢ ɪɟɲɟɧɢɢ ɦɧɨɝɢɯ ɡɚɞɚɱ – ɦɟɬɨɞɨɦ ɗɣɥɟɪɚ ɢɥɢ ɷɧɟɪɝɟɬɢɱɟɫɤɢɦ ɦɟɬɨɞɨɦ

– ɜɨɡɧɢɤɚɸɬ ɡɧɚɱɢɬɟɥɶɧɵɟ ɬɪɭɞɧɨɫɬɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɯɚɪɚɤɬɟɪɚ, ɫɜɹɡɚɧɧɵɟ ɫ ɩɨɥɭɱɟɧɢɟɦ ɜɢɞɚ ɮɭɧɤɰɢɢ X(z). ɗɬɢ ɬɪɭɞɧɨɫɬɢ ɦɨɠɧɨ ɨɛɨɣɬɢ, ɡɚɞɚɜɚɹɫɶ ɜɢɞɨɦ ɮɭɧɤɰɢɢ X(z) ɩɪɢɛɥɢɠɟɧɧɨ, ɫɬɚɪɚɹɫɶ ɥɢɲɶ ɭɞɨɜɥɟɬɜɨɪɢɬɶ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ. ɉɨɥɭɱɟɧɧɚɹ ɤɪɢɬɢɱɟɫɤɚɹ ɧɚɝɪɭɡɤɚ ɛɭɞɟɬ ɢɦɟɬɶ ɩɪɢ ɷɬɨɦ ɡɚɜɵɲɟɧɧɨɟ ɡɧɚɱɟɧɢɟ:

ɭɩɪɭɝɢɣ ɫɬɟɪɠɟɧɶ ɫ ɛɟɫɤɨɧɟɱɧɵɦ ɱɢɫɥɨɦ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ ɡɚɦɟɧɹɟɬɫɹ ɫɢɫɬɟɦɨɣ ɫ ɨɞɧɨɣ ɫɬɟɩɟɧɶɸ ɫɜɨɛɨɞɵ, ɤɨɬɨɪɨɣ ɪɚɡɪɟɲɚɟɬɫɹ ɢɡɝɢɛɚɬɶɫɹ ɬɨɥɶɤɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɜɩɨɥɧɟ ɨɩɪɟɞɟɥɟɧɧɨɣ ɤɪɢɜɨɣ X(z). ɇɚ ɫɬɟɪɠɟɧɶ ɤɚɤ ɛɵ ɧɚɤɥɚɞɵɜɚɟɬɫɹ ɤɨɪɫɟɬ. Ⱦɨɩɨɥɧɢɬɟɥɶɧɵɟ ɫɜɹɡɢ (ɤɨɬɨɪɵɟ ɜ ɞɟɣɫɬɜɢɬɟɥɶɧɨɫɬɢ ɨɬɫɭɬɫɬɜɭɸɬ) ɩɪɢɜɨɞɹɬ ɤ ɭɜɟɥɢɱɟɧɢɸ ɡɧɚɱɟɧɢɹ ɤɪɢɬɢɱɟɫɤɨɣ ɧɚɝɪɭɡɤɢ.

dz

z

y

ɉɪɢɦɟɪ 3 (ɪɢɫ.8.8). Ɉɩɪɟɞɟɥɢɬɶ ɤɪɢɬɢɱɟɫɤɭɸ ɞɥɢɧɭ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɨɣ ɫɬɨɣɤɢ, ɧɚɝɪɭɠɟɧɧɨɣ ɫɨɛɫɬɜɟɧɧɵɦ ɜɟɫɨɦ (ɩɨɝɨɧɧɵɣ ɜɟɫ – q).

Ɋɟɲɟɧɢɟ. ɇɚ ɪɢɫɭɧɤɟ ɩɨɤɚɡɚɧɨ ɫɦɟɠɧɨɟ ɩɨɥɨɠɟɧɢɟ ɫɬɨɣɤɢ, ɤɨ- q ɬɨɪɚɹ ɜɧɚɱɚɥɟ ɛɵɥɚ ɩɪɹɦɨɣ. ɉɪɢ ɤɪɢɬɢɱɟɫɤɨɦ ɡɧɚɱɟɧɢɢ ɧɚɝɪɭɡɤɢ ɜɟ-

				ɥɢɱɢɧɚ ɉ ɩɪɢ ɩɟɪɟɯɨɞɟ ɜ ɫɦɟɠɧɨɟ ɩɨɥɨɠɟɧɢɟ ɧɟ ɦɟɧɹɟɬɫɹ. ɉɨɷɬɨɦɭ
	l			ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɫɥɟɞɭɟɬ ɩɪɢɪɚɜɧɹɬɶ ɢɡɦɟɧɟɧɢɟ ɩɨɬɟɧɰɢɚɥɶɧɨɣ
				ɷɧɟɪɝɢɢ ɞɟɮɨɪɦɚɰɢɢ ɢɡɦɟɧɟɧɢɸ ɩɨɬɟɧɰɢɚɥɚ ɜɧɟɲɧɢɯ ɫɢɥ (ɫɢɥ ɜɟɫɚ).
	EI			ɉɪɢɦɟɦ, ɱɬɨ ɭɩɪɭɝɚɹ ɥɢɧɢɹ ɫɬɨɣɤɢ ɩɪɟɞɫɬɚɜɥɹɟɬ ɩɨɥɭɜɨɥɧɭ ɫɢ-
				ɧɭɫɨɢɞɵ X = b sin(Sz/l), ɝɞɟ ɚɦɩɥɢɬɭɞɚ b ɧɟɢɡɜɟɫɬɧɚ. Ɍɨɝɞɚ ɩɨɬɟɧɰɢ-
				ɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɞɟɮɨɪɦɚɰɢɢ ɢɡɝɢɛɚ ɪɚɜɧɚ
				l

Ɋɢɫ. 8.8	U=0.5	³EIX	cc2	dz=0.25EIb	2	S	4	/l	3	(8.6)
		0
ɍɦɟɧɶɲɟɧɢɟ ɩɨɬɟɧɰɢɚɥɚ ɫɢɥɵ ɜɟɫɚ ɩɪɢ ɩɟɪɟɯɨɞɟ ɜ ɷɬɨ ɫɦɟɠɧɨɟ ɩɨɥɨɠɟɧɢɟ
ɨɩɪɟɞɟɥɢɬɫɹ ɜɵɪɚɠɟɧɢɟɦ
	l
	Uc=³qO(z)dz,									(8.7)
	0

ɝɞɟ O(z) – ɜɟɪɬɢɤɚɥɶɧɨɟ ɩɟɪɟɦɟɳɟɧɢɟ ɷɥɟɦɟɧɬɚ ɞɥɢɧɨɣ dz ɜɫɥɟɞɫɬɜɢɟ ɢɡɝɢɛɚ ɫɬɟɪɠɧɹ:

z	z
O(z)=³(1– cosT)dz|1/2³Xc 2dz=1/4b2(S/l)2(z+l/(2S)sin(2Sz/l)).		(8.8)
0	0

Ɂɞɟɫɶ T = Xc – ɭɝɨɥ ɩɨɜɨɪɨɬɚ ɷɥɟɦɟɧɬɚ dz. ɉɨɞɫɬɚɜɥɹɹ (8.8) ɜ (8.7) ɢ ɩɪɢɪɚɜɧɢɜɚɹ ɩɨɥɭɱɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɭ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ (8.6), ɧɚɣɞɟɦ ɤɪɢɬɢɱɟɫɤɭɸ ɞɥɢɧɭ ɫɬɨɣɤɢ

l* = (2S2EI/q)1/3.

(8.9)

Ɋɚɫɫɦɨɬɪɟɧɧɵɟ ɦɟɬɨɞɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɭɫɬɨɣɱɢɜɨɫɬɢ ɩɨɡɜɨɥɹɸɬ ɧɚɣɬɢ ɤɪɢɬɢɱɟɫɤɢɟ ɡɧɚɱɟɧɢɹ ɜɧɟɲɧɢɯ ɜɨɡɞɟɣɫɬɜɢɣ ɜ ɛɨɥɶɲɢɧɫɬɜɟ ɫɥɭɱɚɟɜ. Ɉɞɧɚɤɨ ɫɭɳɟɫɬɜɭɟɬ ɤɥɚɫɫ ɡɚɞɚɱ, ɜ ɤɨɬɨɪɵɯ ɩɪɢ ɞɨɫɬɢɠɟɧɢɢ ɫɢɥɨɣ ɧɟɤɨɬɨɪɨɝɨ ɡɧɚɱɟɧɢɹ (ɤɪɢɬɢɱɟɫɤɨɝɨ) ɩɪɨɢɫɯɨɞɢɬ ɩɟɪɟɯɨɞ ɧɟ ɤ ɧɨɜɨɣ ɮɨɪɦɟ ɪɚɜɧɨɜɟɫɢɹ, ɚ ɤ ɨɩɪɟɞɟɥɟɧɧɨɣ ɮɨɪɦɟ ɞɜɢɠɟɧɢɹ. ȼ ɷɬɢɯ ɫɥɭɱɚɹɯ ɫɥɟɞɭɟɬ ɡɚɩɢɫɵɜɚɬɶ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɫɢɫɬɟɦɵ, ɬ.ɟ. ɩɪɢɦɟɧɹɬɶ ɫɬɚɬɢɱɟɫɤɢɣ ɦɟɬɨɞ ɗɣɥɟɪɚ, ɧɨ ɜ ɪɚɫɫɦɨɬɪɟɧɢɟ ɜɜɨɞɢɬɶ Ⱦ`Ⱥɥɚɦɛɟɪɨɜɵ ɫɢɥɵ ɢɧɟɪɰɢɢ.

ɉɪɢɦɟɪ 4 (ɪɢɫ.8.9) [5]. ɇɚɣɬɢ ɤɪɢɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ "ɫɥɟɞɹɳɟɣ" ɫɢɥɵ Ɋ (ɫɢɥɚ ɜɫɟɝɞɚ ɧɚɩɪɚɜɥɟɧɚ ɩɨ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɢɡɨɝɧɭɬɨɣ ɨɫɢ ɧɚ ɫɜɨɛɨɞɧɨɦ ɤɨɧɰɟ ɫɬɨɣɤɢ).

Ɋɟɲɟɧɢɟ. ȼɵɪɟɠɟɦ ɢɡ ɫɬɨɣɤɢ, ɧɚɯɨɞɹɳɟɣɫɹ ɜ ɫɦɟɠɧɨɦ (ɢɡɨɝɧɭɬɨɦ) ɩɨɥɨɠɟ-
ɧɢɢ, ɷɥɟɦɟɧɬ dz (ɪɢɫ.8.10). ɉɪɨɟɰɢɪɭɹ ɜɫɟ ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɧɟɝɨ ɫɢɥɵ (dPi=USw2X
/w t2dz – ɫɢɥɚ ɢɧɟɪɰɢɢ, S – ɩɥɨɳɚɞɶ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ), ɧɚ ɨɫɶ y ɢ ɫɱɢɬɚɹ ɧɨɪ-
ɦɚɥɶɧɭɸ ɫɢɥɭ, ɜɜɢɞɭ ɩɪɟɞɩɨɥɚɝɚɟɦɨɣ ɦɚɥɨɫɬɢ ɩɟɪɟɦɟɳɟɧɢɣ, ɪɚɜɧɨɣ P, ɩɨɥɭɱɢɦ
ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɫɬɟɪɠɧɹ:
		EIw 4X /w z4 + Pw 2 X / w z2 + US w 2X /w t2 = 0,			(8.10)
ɪɟɲɟɧɢɟ ɤɨɬɨɪɨɝɨ ɫɥɟɞɭɟɬ ɢɫɤɚɬɶ ɜ ɜɢɞɟ
		/+d / Q+dQ		X =Y(z)·e irt ,
P	P	N+dN		z ɝɞɟ r – ɧɟɢɡɜɟɫɬɧɚɹ,	ɜɨɨɛɳɟ ɝɨɜɨɪɹ, ɤɨɦ-
			dPi	ɩɥɟɤɫɧɚɹ ɩɨɫɬɨɹɧɧɚɹ. Ɉɬ ɟɟ ɡɧɚɱɟɧɢɹ ɡɚɜɢ-
				ɫɢɬ ɨɛɳɢɣ ɯɚɪɚɤɬɟɪ ɞɜɢɠɟɧɢɹ ɫɬɟɪɠɧɹ.
		M+dM
dz				ȿɫɥɢ ɨɤɚɠɟɬɫɹ, ɱɬɨ	r – ɞɟɣɫɬɜɢɬɟɥɶɧɨɟ
		Q		ɱɢɫɥɨ, ɬɨ ɪɟɲɟɧɢɟ ɛɭɞɟɬ ɫɨɞɟɪɠɚɬɶ ɫɥɚ-
z		M	N	ɝɚɟɦɵɟ ɬɢɩɚ e i: t ɢ e – i: t, ɤɨɬɨɪɵɟ ɜ ɫɭɦɦɟ
y			/	ɨɩɪɟɞɟɥɹɸɬ ɝɚɪɦɨɧɢɱɟɫɤɢɟ ɤɨɥɟɛɚɧɢɹ. ȼ
				ɷɬɨɦ ɫɥɭɱɚɟ ɫɢɫɬɟɦɭ ɩɪɢɧɹɬɨ ɫɱɢɬɚɬɶ ɭɫ-
	Ɋɢɫ.8.9	Ɋɢɫ.8.10		ɬɨɣɱɢɜɨɣ. Ɉɞɧɚɤɨ,	ɟɫɥɢ : ɛɭɞɟɬ ɤɨɦ-
				ɩɥɟɤɫɧɵɦ ɢɥɢ ɱɢɫɬɨ ɦɧɢɦɵɦ ɱɢɫɥɨɦ, ɬɨ ɜ
ɪɟɲɟɧɢɢ ɩɨɹɜɹɬɫɹ ɫɥɚɝɚɟɦɵɟ, ɫɨɞɟɪɠɚɳɢɟ ɦɧɨɠɢɬɟɥɢ e at ɢ e – at (ɚ – ɞɟɣɫɬɜɢ-
ɬɟɥɶɧɨɟ ɱɢɫɥɨ). ɗɬɨ ɨɡɧɚɱɚɟɬ ɩɟɪɢɨɞɢɱɟɫɤɨɟ ɞɜɢɠɟɧɢɟ ɫ ɜɨɡɪɚɫɬɚɸɳɢɦ ɪɚɡɦɚɯɨɦ,
ɬɨ ɟɫɬɶ ɧɟɭɤɥɨɧɧɵɣ ɭɯɨɞ ɫɢɫɬɟɦɵ ɨɬ ɩɨɥɨɠɟɧɢɹ ɢɫɯɨɞɧɨɝɨ ɪɚɜɧɨɜɟɫɢɹ (ɧɟɭɫɬɨɣ-
ɱɢɜɨɟ ɩɨɜɟɞɟɧɢɟ ɫɬɨɣɤɢ).
ȼɜɟɞɹ ɨɛɨɡɧɚɱɟɧɢɹ k2			= Pl2/(EI), Z = : l2(US/(EI))0.5 , [ = z /l ɢ ɪɟɲɚɹ ɫɨ-
ɜɦɟɫɬɧɨ (8.10) ɢ (8.11), ɧɚɣɞɟɦ:

d4Y/d[ 4 + k2d2Y/d[2 – Z2Y= 0.	(8.12)
Ɋɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ:
Y= Asin s1z + Bcos s1z + Csh s2z + Dch s2z,	(8.13)
ɝɞɟ s12 = k2/2 + (k4/4 + Z2)0.5, s22 = s12– k2.
Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ:

Y = 0; Yc = 0 ɩɪɢ z = 0; Ycc = 0 ɢ Yccc = 0 ɩɪɢ z = l ([=1).

ɉɪɢ ɩɨɞɫɬɚɧɨɜɤɟ ɷɬɢɯ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɜ ɜɵɪɚɠɟɧɢɟ (8.13) ɩɪɢɯɨɞɢɦ ɤ ɫɢɫɬɟɦɟ 4-ɯ ɨɞɧɨɪɨɞɧɵɯ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɨɬɧɨɫɢɬɟɥɶɧɨ A, B, C ɢ D. Ɉɩɪɟɞɟɥɢɬɟɥɶ ɷɬɨɣ ɫɢɫɬɟɦɵ ɞɨɥɠɟɧ ɪɚɜɧɹɬɶɫɹ ɧɭɥɸ, ɟɫɥɢ ɨɧɚ ɢɦɟɟɬ ɨɬɥɢɱɧɵɟ ɨɬ ɧɭɥɹ ɪɟɲɟɧɢɹ; ɨɬɫɸɞɚ ɫɥɟɞɭɟɬ ɭɪɚɜɧɟɧɢɟ, ɫɜɹɡɵɜɚɸɳɟɟ ɩɚɪɚɦɟɬɪ ɧɚɝɪɭɡɤɢ k ɢ ɱɚɫɬɨɬɭ ɤɨɥɟɛɚɧɢɣ Z

k4+ 2Z2+ k2Z·sin s1 sh s2+ 2Z2·cos s1 ch s2 = 0.

(8.14)

k 2	k*2	A	Ƚɪɚɮɢɤ	ɩɨɥɭɱɟɧɧɨɣ	ɡɚɜɢɫɢɦɨɫɬɢ	ɩɨɤɚɡɚɧ	ɧɚ
20			ɪɢɫ.8.11. ȼɢɞɧɨ, ɱɬɨ ɩɪɢ ɤɚɠɞɨɦ ɡɧɚɱɟɧɢɢ ɧɚɝɪɭɡɤɢ (ɞɨ
			ɤɪɢɬɢɱɟɫɤɨɝɨ) ɢɦɟɸɬɫɹ ɞɜɟ ɫɨɛɫɬɜɟɧɧɵɟ ɱɚɫɬɨɬɵ (Z1 ɢ
10	1		Z2). ɉɪɢ ɜɨɡɪɚɫɬɚɧɢɢ ɧɚɝɪɭɡɤɢ ɱɚɫɬɨɬɵ			ɫɛɥɢɠɚɸɬɫɹ.
			2
			Ʉɪɢɜɵɟ ɡɚɜɢɫɢɦɨɫɬɟɣ k2(Z1) ɢ k2(Z2) ɩɟɪɟɫɟɤɚɸɬɫɹ ɜ
			ɬɨɱɤɟ A, ɬ.ɟ. ɩɪɢ ɷɬɨɦ ɡɧɚɱɟɧɢɢ ɫɢɥɵ P ɤɨɪɧɢ ɭɪɚɜɧɟɧɢɹ
			(8.14) ɫɬɚɧɨɜɹɬɫɹ ɤɪɚɬɧɵɦɢ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɞɜɟ ɮɨɪ-
0	10	*	ɦɵ ɤɨɥɟɛɚɧɢɣ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɨɞɧɨɣ ɱɚɫɬɨɬɟ, ɤɨɥɟɛɚɧɢɹ
		20	ɫɬɚɧɨɜɹɬɫɹ ɧɟɩɟɪɢɨɞɢɱɟɫɤɢɦɢ. ɉɪɢ ɧɚɝɪɭɠɟɧɢɢ ɫɢɥɨɣ
	Ɋɢɫ.8.11		ɜɵɲɟ ɬɨɣ, ɤɨɬɨɪɨɣ ɨɬɜɟɱɚɟɬ ɬɨɱɤɚ A, ɜ ɪɟɲɟɧɢɢ ɩɨɹɜ-
			ɥɹɸɬɫɹ ɤɨɦɩɥɟɤɫɧɵɟ ɤɨɪɧɢ, ɬɨ ɟɫɬɶ ɞɜɢɠɟɧɢɟ ɫɬɟɪɠɧɹ
ɩɪɨɢɫɯɨɞɢɬ ɫ ɜɨɡɪɚɫɬɚɸɳɟɣ ɚɦɩɥɢɬɭɞɨɣ. ɇɚɣɞɟɧɨ:

P*=20,05EI/l2.

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɪɟɲɟɧɢɟ ɦɟɬɨɞɨɦ ɗɣɥɟɪɚ ɫ ɭɱɟɬɨɦ ɢɧɟɪɰɢɨɧɧɵɯ ɫɢɥ ɫɜɨɞɢɬɫɹ ɤ ɪɟɲɟɧɢɸ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ, ɞɨɫɬɚɬɨɱɧɨ ɫɥɨɠɧɵɯ ɞɚɠɟ ɜ ɷɬɨɦ ɩɪɨɫɬɨɦ ɩɪɢɦɟɪɟ.

Ɋɟɲɢɬɟ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ.

1. Ⱥɛɫɨɥɸɬɧɨ ɠɟɫɬɤɢɣ ɫɬɟɪɠɟɧɶ ɩɨɫɬɨɹɧɧɨɝɨ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɢɦɟɟɬ ɩɨɝɨɧɧɵɣ ɜɟɫ q ɢ ɩɪɢɜɚɪɟɧ ɤ ɛɚɥɤɟ ɞɥɢɧɨɣ l,

			ɠɟɫɬɤɨɫɬɶɸ EI (ɪɢɫ.8.12). Ɉɩɪɟɞɟɥɢɬɶ, ɩɪɢ
q	L	EI l	ɤɚɤɨɣ	ɞɥɢɧɟ L ɫɢɫɬɟɦɚ ɩɨɬɟɪɹɟɬ ɭɫɬɨɣɱɢ-
l/2		l/2	ɜɨɫɬɶ.
			2.	ɋɬɟɪɠɟɧɶ ɞɥɢɧɨɣ l, ɠɟɫɬɤɨɫɬɶɸ EI
	EI	P
				ɧɟɫɟɬ ɩɨ ɤɨɧɰɚɦ ɦɚɫɫɵ m/2 ɢ ɧɚɝɪɭ-
Ɋɢɫ.8.12		Ɋɢɫ.8.13		ɠɟɧ ɫɥɟɞɹɳɟɣ ɫɢɥɨɣ P (ɪɢɫ.8.13).
				ɉɪɟɧɟɛɪɟɝɚɹ ɦɚɫɫɨɣ ɫɬɟɪɠɧɹ, ɨɩɪɟ-

ɞɟɥɢɬɶ, ɩɪɢ ɤɚɤɨɦ ɡɧɚɱɟɧɢɢ ɫɢɥɵ P ɫɬɟɪɠɟɧɶ ɩɨɬɟɪɹɟɬ ɭɫɬɨɣɱɢɜɨɫɬɶ.

9.ɇȺɉɊəɀȿɇɇɈ-ȾȿɎɈɊɆɂɊɈȼȺɇɇɈȿ ɋɈɋɌɈəɇɂȿ

ȼɌɈɑɄȿ ɌȿɅȺ

9.1. ɇɚɩɪɹɠɟɧɧɨɟ ɫɨɫɬɨɹɧɢɟ

ɐɟɧɬɪɚɥɶɧɵɦ ɷɥɟɦɟɧɬɨɦ ɬɟɨɪɢɢ ɧɚɩɪɹɠɟɧɢɣ ɹɜɥɹɟɬɫɹ ɜɟɤɬɨɪ ɧɚɩɪɹɠɟɧɢɣ p

– ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɫɢɥɵ ɞɟɣɫɬɜɢɹ ɨɞɧɨɣ ɱɚɫɬɢ ɬɟɥɚ ɧɚ ɞɪɭɝɭɸ ɱɟɪɟɡ ɪɚɡɞɟɥɹɸɳɭɸ ɢɯ (ɦɵɫɥɟɧɧɨ) ɩɨɜɟɪɯɧɨɫɬɶ. ɗɬɨɬ ɜɟɤɬɨɪ ɦɨɠɧɨ ɩɪɨɟɰɢɪɨɜɚɬɶ ɧɚ ɨɫɢ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ, ɩɨɥɭɱɚɹ ɦɚɬɪɢɰɭ-ɫɬɨɥɛɟɰ ɩɪɨɟɤɰɢɣ [p] = [px, py, pz]Ɍ (ɡɧɚɤ «T» ɨɡɧɚɱɚɟɬ ɬɪɚɧɫɩɨɧɢɪɨɜɚɧɢɟ ɦɚɬɪɢɰɵ), ɢɥɢ ɩɪɨɟɰɢɪɨɜɚɬɶ ɧɚ ɧɨɪɦɚɥɶ n ɤ ɩɥɨɳɚɞɤɟ ɢ ɤɚɫɚɬɟɥɶɧɭɸ, ɩɨɥɭɱɚɹ ɧɨɪɦɚɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ V ɢ ɤɚɫɚɬɟɥɶɧɨɟ W. ɇɚɩɪɹɠɟɧɧɵɦ ɫɨɫɬɨɹɧɢɟɦ ɜ ɬɨɱɤɟ ɬɟɥɚ ɧɚɡɵɜɚɸɬ ɫɨɜɨɤɭɩɧɨɫɬɶ ɜɟɤɬɨɪɨɜ p ɧɚ ɛɟɫɱɢɫɥɟɧɧɨɦ ɤɨɥɢɱɟɫɬɜɟ ɩɥɨɳɚɞɨɤ n, ɤɨɬɨɪɵɟ ɦɨɠɧɨ ɩɪɨɜɟɫɬɢ ɱɟɪɟɡ ɷɬɭ ɬɨɱɤɭ. ɗɬɨ ɦɧɨɠɟɫɬɜɨ ɭɩɨɪɹɞɨɱɟɧɧɨ: ɪɚɫɫɦɨɬɪɟɜ ɪɚɜɧɨɜɟɫɢɟ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɬɟɬɪɚɷɞɪɚ, ɜɵɪɟɡɚɧɧɨɝɨ ɢɡ ɬɟɥɚ ɜ ɦɚɥɨɣ ɨɤɪɟɫɬɧɨɫɬɢ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɬɨɱɤɢ, ɥɟɝɤɨ ɭɜɢɞɟɬɶ, ɱɬɨ ɞɨɫɬɚɬɨɱɧɨ ɡɧɚɬɶ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɬɪɟɯ ɩɥɨɳɚɞɤɚɯ, ɱɬɨɛɵ ɧɚɣɬɢ ɧɚɩɪɹɠɟɧɢɟ ɧɚ ɥɸɛɨɣ ɱɟɬɜɟɪɬɨɣ. ɗɬɢ ɬɪɢ ɩɥɨɳɚɞɤɢ ɭɞɨɛɧɨ ɛɪɚɬɶ ɨɪɬɨɝɨɧɚɥɶɧɵɦɢ (ɩɭɫɬɶ ɧɨɪɦɚɥɢ ɤ ɧɢɦ i, j, k ɧɚɩɪɚɜɥɟɧɵ ɜɞɨɥɶ ɞɟɤɚɪɬɨɜɵɯ ɨɫɟɣ x, y, z), ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɧɢɯ – ɜɟɤɬɨɪɵ px, py, pz. ɂɡ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ (ɩɨ ɫɢɥɚɦ) ɩɨɥɭɱɢɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɧɚɩɪɹɠɟɧɢɹ p ɧɚ ɥɸɛɨɣ ɩɥɨɳɚɞɤɟ, ɧɨɪɦɚɥɶ ɤɨɬɨɪɨɣ ɨɛɨɡɧɚɱɟɧɚ n:

p(n)=px cos(i, n)+py cos(j, n)+pz cos(k, n),	(9.1)
ɢɥɢ, ɜ ɦɚɬɪɢɱɧɨɣ ɮɨɪɦɟ,
>p@=>TV@ >n@.	(9.2)

Ɂɞɟɫɶ >TV@ – ɦɚɬɪɢɰɚ ɧɚɩɪɹɠɟɧɢɣ, ɟɟ ɫɬɨɥɛɰɵ ɩɪɟɞɫɬɚɜɥɹɸɬ ɤɨɨɪɞɢɧɚɬɵ ɜɟɤɬɨɪɨɜ px, py, pz ɜ ɛɚɡɢɫɟ i, j, k; >n@ – ɦɚɬɪɢɰɚ ɤɨɫɢɧɭɫɨɜ, ɢɥɢ ɤɨɨɪɞɢɧɚɬ ɜɟɤɬɨɪɚ n ɜ ɬɨɦ ɠɟ ɛɚɡɢɫɟ. Ɇɚɬɪɢɰɚ >TV@ ɨɞɧɨɡɧɚɱɧɨ ɨɩɪɟɞɟɥɹɟɬ ɧɚɩɪɹɠɟɧɧɨɟ ɫɨɫɬɨɹɧɢɟ ɜ ɬɨɱɤɟ ɬɟɥɚ.

	Ⱦɨɩɨɥɧɢɬɟɥɶɧɨɟ ɭɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ ɩɨ ɦɨɦɟɧɬɚɦ ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɦɚɬɪɢɰɚ
	z						>TV@ ɫɢɦɦɟɬɪɢɱɧɚ (ɩɚɪɧɨɫɬɶ ɤɚɫɚɬɟɥɶɧɵɯ ɧɚɩɪɹɠɟɧɢɣ).
		ɋ					ɉɪɢɦɟɪ 1. ɇɚ ɪɢɫ.9.1 ɩɨɤɚɡɚɧɵ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɝɪɚɧɹɯ
ȼ							ɤɭɛɢɤɚ (ɜ Ɇɉɚ). Ɍɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɧɨɪɦɚɥɶɧɨɟ ɧɚɩɪɹɠɟ-
					60
							ɧɢɟ V ɧɚ ɩɥɨɳɚɞɤɟ ABC, ɪɚɜɧɨɧɚɤɥɨɧɟɧɧɨɣ ɤ ɞɜɭɦ ɝɪɚɧɹɦ
30							ɤɭɛɢɤɚ.
			30		x
		Ⱥ					Ⱦɥɹ ɪɟɲɟɧɢɹ ɜɜɨɞɢɦ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ (x, y, z), ɮɨɪ-

ɭɦɢɪɭɟɦ ɦɚɬɪɢɰɵ >TV@ ɢ >n@, ɧɚɯɨɞɢɦ ɫɧɚɱɚɥɚ ɩɨɥɧɨɟ ɧɚɩɪɹ-

Ɋɢɫ.9.1	ɠɟɧɢɟ ɧɚ ɩɥɨɳɚɞɤɟ ABC (ɜɵɪɚɠɟɧɢɟ (9.2))
		60;	0;	30		1			30
	>p@=							1 =		1 Ɇɉɚ.
		0;	30;	0	Ɇɉɚ	0			0
		30;	0;	0		1		2	30	2

ɇɨɪɦɚɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ ɧɚɯɨɞɢɬɫɹ ɩɪɨɟɰɢɪɨɜɚɧɢɟɦ ɜɟɤɬɨɪɚ ɩɨɥɧɨɝɨ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɜɟɤɬɨɪ ɧɨɪɦɚɥɢ n. ȼ ɦɚɬɪɢɱɧɨɣ ɮɨɪɦɟ ɷɬɨ ɨɡɧɚɱɚɟɬ ɩɟɪɟɦɧɨɠɟɧɢɟ ɞɜɭɯ ɦɚɬɪɢɰ

1

V = [p]Ɍ >n@ = [30 0 – 30] 1 Ɇɉɚ 0 1 =0.

2 2

1

Ɂɚɞɚɱɚ. ɇɚɣɞɢɬɟ ɜɟɥɢɱɢɧɭ ɤɚɫɚɬɟɥɶɧɨɝɨ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɷɬɨɣ ɩɥɨɳɚɞɤɟ.

9.2. Ʉɪɭɝ Ɇɨɪɚ

ȼɵɪɚɠɟɧɢɟ (9.2) ɭɞɨɛɧɨ ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɬɢɩɚ ɩɨɫɥɟɞɧɟɝɨ ɩɪɢɦɟɪɚ, ɧɨ ɧɟ ɧɚɝɥɹɞɧɨ, ɧɟ ɩɨɡɜɨɥɹɟɬ ɭɜɢɞɟɬɶ ɜɫɟ ɦɧɨɠɟɫɬɜɨ ɧɚɩɪɹɠɟɧɢɣ ɜ ɩɥɨɳɚɞɤɚɯ, ɩɪɨɯɨɞɹɳɢɯ ɱɟɪɟɡ ɪɚɫɫɦɚɬɪɢɜɚɟɦɭɸ ɬɨɱɤɭ. ȼ ɡɧɚɱɢɬɟɥɶɧɨɣ ɦɟɪɟ ɩɨɫɥɟɞɧɟɝɨ ɭɞɚɟɬɫɹ ɞɨɫɬɢɱɶ ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɩɥɨɫɤɨɫɬɢ ^V,W`. ɇɚ ɷɬɨɣ ɩɥɨɫɤɨɫɬɢ ɨɬɤɥɚɞɵɜɚɸɬɫɹ ɬɨɱɤɢ >V,W@, ɤɚɠɞɚɹ ɢɡ ɤɨɬɨɪɵɯ ɩɨɤɚɡɵɜɚɟɬ ɡɧɚɱɟɧɢɹ ɧɨɪɦɚɥɶɧɨɝɨ ɢ ɤɚɫɚɬɟɥɶɧɨɝɨ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɧɟɤɨɬɨɪɨɣ ɩɥɨɳɚɞɤɟ ɩɪɢ ɡɚɞɚɧɧɨɦ ɦɚɬɪɢɰɟɣ >TV@ ɧɚɩɪɹɠɟɧɧɨɦ ɫɨɫɬɨɹɧɢɢ. ȼ ɫɥɭɱɚɟ ɩɪɨɫɬɟɣɲɟɝɨ ɧɚɩɪɹɠɟɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ – ɪɚɫɬɹɠɟɧɢɹ (ɫɠɚɬɢɹ) – ɥɟɝɤɨ ɧɚɣɬɢ, ɱɬɨ ɩɚɪɵ ɡɧɚɱɟɧɢɣ V ɢ W ɞɥɹ ɜɫɟɯ ɩɥɨɳɚɞɨɤ ɨɛɪɚɡɭɸɬ ɧɚ ɷɬɨɣ ɩɥɨɫɤɨɫɬɢ ɬɨɱɤɢ, ɥɟɠɚɳɢɟ ɧɚ ɨɞɧɨɦ ɤɪɭɝɟ. ɋ ɪɨɫɬɨɦ ɭɝɥɚ ɦɟɠɞɭ ɨɫɶɸ ɪɚɫɬɹɠɟɧɢɹ ɢ ɧɨɪɦɚɥɶɸ n ɤ ɩɥɨɳɚɞɤɟ ɬɨɱɤɚ >V, W@ ɛɟɠɢɬ ɩɨ ɤɪɭɝɭ ɫɨ ɫɤɨɪɨɫɬɶɸ (ɜɨɤɪɭɝ ɰɟɧɬɪɚ ɤɪɭɝɚ), ɜɞɜɨɟ ɛɨɥɶɲɟɣ, ɱɟɦ ɫɤɨɪɨɫɬɶ ɪɨɫɬɚ ɭɝɥɚ.

ɋɥɟɞɭɟɬ ɨɫɨɛɨ ɨɝɨɜɨɪɢɬɶ ɡɧɚɤ ɤɚɫɚɬɟɥɶɧɵɯ ɧɚɩɪɹɠɟɧɢɣ ɞɥɹ ɞɚɧɧɨɣ ɤɪɭɝɨɜɨɣ ɞɢɚɝɪɚɦɦɵ. ɉɪɚɜɢɥɨ ɡɧɚɤɨɜ ɡɞɟɫɶ ɩɪɢɧɢɦɚɸɬ ɬɚɤɢɦ ɠɟ, ɤɚɤ ɞɥɹ ɩɨɩɟɪɟɱɧɨɣ ɫɢɥɵ (ɧɚ ɪɢɫ. 9.2 ɤɚɫɚɬɟɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ ɧɚ ɩɥɨɳɚɞɤɟ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɨɫɢ x ɪɚɜɧɨ +50 Ɇɉɚ, ɧɚ ɩɥɨɳɚɞɤɟ y – ɪɚɜɧɨ –50 Ɇɉɚ).

	ɭ								ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɤɚɪɬɢɧɚ ɧɟɫɤɨɥɶɤɨ ɫɥɨɠɧɟɣ: ɬɨɱɤɚ-
			50						ɦɢ >V,W@ ɧɚ ɩɥɨɫɤɨɫɬɢ ^V,W` ɨɯɜɚɬɵɜɚɟɬɫɹ ɨɛɥɚɫɬɶ ɦɟɠɞɭ
					200				ɬɪɟɦɹ ɤɚɫɚɸɳɢɦɢɫɹ ɤɪɭɝɚɦɢ, ɰɟɧɬɪɵ ɤɨɬɨɪɵɯ ɥɟɠɚɬ ɧɚ
									ɨɫɢ V. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɧɚ ɨɫɶ V ɩɨɩɚɞɚɸɬ ɥɢɲɶ ɬɪɢ ɬɨɱ-
10000									ɤɢ, ɨɬɜɟɱɚɸɳɢɟ ɬɪɟɦ ɝɥɚɜɧɵɦ ɩɥɨɳɚɞɤɚɦ (ɫɥɨɜɨ “ɝɥɚɜ-
				50			x
									ɧɵɣ“ ɨɛɨɡɧɚɱɚɟɬ ɨɬɫɭɬɫɬɜɢɟ ɤɚɫɚɬɟɥɶɧɨɝɨ ɧɚɩɪɹɠɟɧɢɹ ɧɚ
z
									ɬɚɤɨɣ ɩɥɨɳɚɞɤɟ). Ƚɥɚɜɧɵɟ ɩɥɨɳɚɞɤɢ ɜɡɚɢɦɧɨ ɨɪɬɨɝɨ-
	Ɋɢɫ.9.2								ɧɚɥɶɧɵ. Ƚɥɚɜɧɵɟ ɧɨɪɦɚɥɶɧɵɟ ɧɚɩɪɹɠɟɧɢɹ ɨɛɨɡɧɚɱɚɸɬ
									V3, V2, V1 – ɜ ɩɨɪɹɞɤɟ ɜɨɡɪɚɫɬɚɧɢɹ ɧɚɩɪɹɠɟɧɢɹ, ɫ ɭɱɟɬɨɦ

ɡɧɚɤɚ. Ʉɚɠɞɵɣ ɤɪɭɝ ɨɬɜɟɱɚɟɬ ɩɥɨɳɚɞɤɚɦ, ɩɚɪɚɥɥɟɥɶɧɵɦ ɨɞɧɨɦɭ ɢɡ ɝɥɚɜɧɵɯ ɧɚɩɪɹɠɟɧɢɣ.

Ɉɬɦɟɬɢɦ, ɱɬɨ ɟɫɥɢ ɜɫɟ ɬɪɢ ɝɥɚɜɧɵɟ ɧɚɩɪɹɠɟɧɢɹ ɧɟ ɪɚɜɧɵ ɧɭɥɸ, ɬɨ ɧɚɩɪɹɠɟɧɧɨɟ ɫɨɫɬɨɹɧɢɟ (ɇɋ) ɧɚɡɵɜɚɸɬ ɨɛɴɟɦɧɵɦ. ȿɫɥɢ ɨɞɧɨ ɢɡ ɝɥɚɜɧɵɯ ɧɚɩɪɹɠɟɧɢɢ ɪɚɜɧɨ ɧɭɥɸ – ɷɬɨ ɩɥɨɫɤɨɟ ɇɋ. ɉɪɢ ɧɟɧɭɥɟɜɨɦ ɡɧɚɱɟɧɢɢ ɬɨɥɶɤɨ ɨɞɧɨɝɨ ɝɥɚɜɧɨɝɨ ɧɚɩɪɹɠɟɧɢɹ ɇɋ ɧɚɡɵɜɚɸɬ ɥɢɧɟɣɧɵɦ.

	ɉɪɢɦɟɪ 2. ɇɚ ɪɢɫ.9.2 ɩɨɤɚɡɚɧɵ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɬɪɟɯ ɝɪɚɧɹɯ ɤɭɛɢɤɚ (ɜ Ɇɉɚ);
ɜɢɞɧɨ, ɱɬɨ ɩɥɨɳɚɞɤɚ ɫ ɫɚɦɵɦ ɛɨɥɶɲɢɦ ɧɚɩɪɹɠɟɧɢɟɦ – ɝɥɚɜɧɚɹ. Ɍɪɟɛɭɟɬɫɹ ɩɨ-
							ɫɬɪɨɢɬɶ ɤɪɭɝ ɧɚɩɪɹɠɟɧɢɣ ɞɥɹ ɜɫɟɯ ɨɪɬɨɝɨ-
							ɧɚɥɶɧɵɯ ɟɣ ɩɥɨɳɚɞɨɤ (ɜ ɱɚɫɬɧɨɫɬɢ, ɬɚɤɢɦɢ
					x		ɩɥɨɳɚɞɤɚɦɢ ɹɜɥɹɸɬɫɹ ɩɥɨɳɚɞɤɢ x ɢ y).
							Ɍɨɱɤɢ x ɢ y ɧɚ ɪɢɫ.9.3 ɩɨɤɚɡɵɜɚɸɬ ɧɨɪ-
					50		ɦɚɥɶɧɵɟ ɢ ɤɚɫɚɬɟɥɶɧɵɟ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɩɥɨ-
		ȼ	.			Ⱥ	ɳɚɞɤɚɯ x ɢ y (ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɪɢɫ.9.2); ɨɬ-
0							ɪɟɡɤɢ ɩɪɹɦɵɯ ɫɨ ɲɬɪɢɯɨɜɤɨɣ ɫɢɦɜɨɥɢɡɢɪɭɸɬ
		50			200
			125				ɪɚɫɩɨɥɨɠɟɧɢɟ ɷɬɢɯ ɩɥɨɳɚɞɨɤ ɧɚ ɩɥɨɫɤɨɫɬɢ
	y	50					{x, y}: x – ɜɟɪɬɢɤɚɥɶɧɚɹ, y – ɝɨɪɢɡɨɧɬɚɥɶɧɚɹ
							ɩɥɨɳɚɞɤɚ. ɍɝɨɥ ɦɟɠɞɭ ɩɥɨɳɚɞɤɚɦɢ ɪɚɜɟɧ
							900, ɡɧɚɱɢɬ, ɰɟɧɬɪɚɥɶɧɵɣ ɭɝɨɥ ɦɟɠɞɭ ɬɨɱɤɚ-
			Ɋɢɫ. 9.3				ɦɢ ɯ ɢ y ɧɚ ɤɪɭɝɟ ɪɚɜɟɧ 2 900 = =1800. ɗɬɨ
							ɨɞɧɨɡɧɚɱɧɨ ɨɩɪɟɞɟɥɹɟɬ ɤɪɭɝ Ɇɨɪɚ ɢ, ɜ ɱɚɫɬ-
ɧɨɫɬɢ, ɞɜɚ ɝɥɚɜɧɵɯ ɧɚɩɪɹɠɟɧɢɹ – ɬɨɱɤɢ A ɢ B. ȼɟɥɢɱɢɧɵ ɷɬɢɯ ɧɚɩɪɹɠɟɧɢɣ ɧɚɯɨ-
ɞɹɬ ɢɡ ɝɟɨɦɟɬɪɢɱɟɫɤɢɯ ɫɨɨɛɪɚɠɟɧɢɣ; ɜ ɧɚɲɟɦ ɫɥɭɱɚɟ ɨɧɢ ɪɚɜɧɵ 215 ɢ 35 Ɇɉɚ.
Ɂɚɦɟɬɢɦ, ɱɬɨ ɜɟɥɢɱɢɧɚ ɝɥɚɜɧɨɝɨ ɧɚɩɪɹɠɟɧɢɹ 10000 Ɇɉɚ (ɩɟɪɜɨɟ ɝɥɚɜɧɨɟ ɧɚɩɪɹ-
ɠɟɧɢɟ) ɧɢɤɚɤ ɧɟ ɩɨɜɥɢɹɥɨ ɧɚ ɩɨɥɭɱɟɧɧɵɣ ɤɪɭɝ.
				ɩɪ.			ɍɞɨɛɧɚ ɫɥɟɞɭɸɳɚɹ ɝɪɚɮɢɱɟɫɤɚɹ ɢɧɬɟɪ-
			ɚ
			ɧ				ɩɪɟɬɚɰɢɹ [6]: ɦɨɠɧɨ ɜɵɞɟɥɢɬɶ ɩɨɥɸɫ (ɬɨɱɤɚ
		Ɋ			x		Ɋ ɧɚ ɪɢɫ.9.4) – ɧɚ ɩɟɪɟɫɟɱɟɧɢɢ ɥɢɧɢɣ, ɨɬɜɟ-
					50		ɱɚɸɳɢɦ ɧɨɪɦɚɥɹɦ ɤ ɩɨɤɚɡɚɧɧɵɦ ɩɥɨɳɚɞɤɚɦ
			.				ɯ ɢ y. ȿɫɥɢ ɢɡ ɬɨɱɤɢ Ɋ ɩɪɨɜɟɫɬɢ ɥɭɱ, ɩɚɪɚɥ-
		ȼ				Ⱥ	ɥɟɥɶɧɵɣ ɧɨɪɦɚɥɢ ɤ ɥɸɛɨɣ ɞɪɭɝɨɣ ɩɥɨɳɚɞɤɟ,
0					200		ɬɨ ɟɝɨ ɩɟɪɟɫɟɱɟɧɢɟ ɫ ɤɪɭɝɨɦ ɞɚɟɬ ɬɨɱɤɭ, ɨɬ-
		50	125
							ɜɟɱɚɸɳɭɸ ɧɚɩɪɹɠɟɧɢɹɦ ɧɚ ɷɬɨɣ ɩɥɨɳɚɞɤɟ.

y		50	ɉɪɢɦɟɪ 3. Ɉɩɪɟɞɟɥɢɬɶ ɜɟɥɢɱɢɧɭ ɢ ɧɚ-
			ɩɪɚɜɥɟɧɢɟ ɝɥɚɜɧɵɯ ɧɚɩɪɹɠɟɧɢɣ ɞɥɹ ɷɥɟɦɟɧ-

		ɬɚ, ɢɡɨɛɪɚɠɟɧɧɨɝɨ ɧɚ ɪɢɫ.9.5 (ɧɚɩɪɹɠɟɧɢɹ
	Ɋɢɫ. 9.4	ɞɚɧɵ ɜ Ɇɉɚ).
		Ɋɟɲɟɧɢɟ. ɋɬɪɨɢɦ ɤɪɭɝ Ɇɨɪɚ ɞɥɹ ɩɥɨ-

ɳɚɞɨɤ ɩɚɪɚɥɥɟɥɶɧɵɯ ɨɫɢ z (ɪɢɫ.9.6).

Ɉɬɜɟɬ: V1=95 Ɇɉɚ, V2=50 Ɇɉɚ, V3=5.2 Ɇɉɚ, D=320.

Ɂɞɟɫɶ ɢɦɟɟɬ ɦɟɫɬɨ ɨɛɴɟɦɧɨɟ ɧɚɩɪɹɠɟɧɧɨɟ ɫɨɫɬɨɹɧɢɟ, ɬɚɤ ɤɚɤ ɜɫɟ ɬɪɢ ɝɥɚɜɧɵɯ ɧɚɩɪɹɠɟɧɢɹ ɨɬɥɢɱɧɵ ɨɬ ɧɭɥɹ.

ɉɪɢɦɟɪ 4. Ⱦɥɹ ɷɥɟɦɟɧɬɚ, ɢɡɨɛɪɚɠɟɧɧɨɝɨ ɧɚ ɪɢɫ.9.5, ɧɚɣɬɢ ɜɟɥɢɱɢɧɭ ɢ ɧɚɩɪɚɜɥɟɧɢɟ ɧɚɩɪɹɠɟɧɢɣ ɜ ɩɥɨɳɚɞɤɚɯ, ɝɞɟ ɞɟɣɫɬɜɭɸɬ Wmax.

Ɋɟɲɟɧɢɟ. ɂɫɩɨɥɶɡɭɹ ɤɪɭɝ Ɇɨɪɚ, ɢɡɨɛɪɚɠɟɧɧɵɣ ɧɚ ɪɢɫ.9.6, ɜɢɞɢɦ ɱɬɨ ɦɚɤɫɢɦɚɥɶɧɵɦ ɤɚɫɚɬɟɥɶɧɵɦ ɧɚɩɪɹɠɟɧɢɹɦ ɨɬɜɟɱɚɸɬ ɬɨɱɤɢ Ⱥ ɢ ȼ ɧɚ ɤɪɭɝɟ. ȼ ɷɬɢɯ ɩɥɨ-

ɳɚɞɤɚɯ ɞɟɣɫɬɜɭɸɬ Wmax=45Ɇɉɚ, VD=50 Ɇɉɚ, ɚ ɭɝɨɥ D= =320+450=770. ɋɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɩɨɥɨɠɟɧɢɟ ɷɥɟɦɟɧɬɚ ɩɨɤɚɡɚɧɨ ɫɥɟɜɚ ɜɧɢɡɭ ɧɚ ɪɢɫ.9.6.

	ɭ		30					ɉɪɢɦɟɪ 5. ɉɭɫɬɶ ɡɚɞɚɧɨ ɨɛɴɟɦɧɨɟ ɧɚɩɪɹɠɟɧɧɨɟ ɫɨɫɬɨɹ-
								ɧɢɟ: V1=90 Ɇɉɚ, V2= =30 Ɇɉɚ, V3= –10 Ɇɉɚ. Ɍɪɟɛɭɟɬɫɹ ɨɩ-
					70			ɪɟɞɟɥɢɬɶ ɞɢɚɩɚɡɨɧ ɢɡɦɟɧɟɧɢɹ ɤɚɫɚɬɟɥɶɧɵɯ ɧɚɩɪɹɠɟɧɢɣ ɜɨ
								ɜɫɟɯ ɬɟɯ ɩɥɨɳɚɞɤɚɯ, ɝɞɟ ɧɨɪɦɚɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ ɪɚɜɧɨ V2.
50
				40		x		Ɋɟɲɟɧɢɟ. ɋɬɪɨɢɦ ɤɪɭɝ Ɇɨɪɚ (ɪɢɫ.9.7). Ɋɚɫɫɦɨɬɪɟɜ ɬɪɟ-
								ɭɝɨɥɶɧɢɤ Ɉ2ȼɋ, ɧɚɯɨɞɢɦ, ɱɬɨ ɤɚɫɚɬɟɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ ɩɨ

zɦɨɞɭɥɸ ɧɟ ɩɪɟɜɵɲɚɟɬ 49 Ɇɉɚ (ɱɬɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɥɢɧɢɢ Ⱥɋ

Ɋɢɫ. 9.5		ɧɚ ɤɪɭɝɟ Ɇɨɪɚ).
				V	D
				.		Ⱥ
				ɪ
	95			ɩ
5.2				ɚ		V1
				ɧ			ɋ
						.
						ɪ
						ɩ
						ɚ
	= 320					ɧ
50						Ɉ
50		770
						ȼ
			Ɋ	ȼ		0 Ɉ1	Ɉ2	Ɉ3
	50
45
	50	ɧɚɩɪ.	3
						2	Ⱥ
		Ɋɢɫ. 9.6					Ɋɢɫ.9.7
9.3. Ɍɟɨɪɢɹ ɞɟɮɨɪɦɚɰɢɣ

Ɍɟɨɪɢɸ ɞɟɮɨɪɦɚɰɢɣ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɨɛɨɛɳɟɧɢɟ ɢɡɜɟɫɬɧɨɝɨ ɨɩɪɟɞɟɥɟɧɢɹ H='l/l0 , ɜ ɤɨɬɨɪɨɦ 'l{l – l0 , ɝɞɟ l0, l – ɞɥɢɧɚ ɧɟɤɨɬɨɪɨɝɨ ɦɚɬɟɪɢɚɥɶɧɨɝɨ ɜɨɥɨɤɧɚ ɜ ɧɚɱɚɥɶɧɨɦ (ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɧɨɦ) ɢ ɜ ɬɟɤɭɳɟɦ (ɞɟɮɨɪɦɢɪɨɜɚɧɧɨɦ) ɫɨɫɬɨɹɧɢɹɯ, H – ɥɢɧɟɣɧɚɹ ɞɟɮɨɪɦɚɰɢɹ ɷɬɨɝɨ ɜɨɥɨɤɧɚ. Ɂɚɩɢɲɟɦ ɷɬɨ ɜɵɪɚɠɟɧɢɟ ɩɨɞɪɭɝɨɦɭ:

'l { l – l0=H l0 .

(9.3)

ȿɫɥɢ ɪɚɫɫɦɚɬɪɢɜɚɬɶ l0 ɤɚɤ ɚɪɝɭɦɟɧɬ, ɚ 'l – ɤɚɤ ɮɭɧɤɰɢɸ, ɬɨ ɜ ɜɵɪɚɠɟɧɢɢ (9.3) ɦɨɠɧɨ ɭɜɢɞɟɬɶ ɥɢɧɟɣɧɭɸ ɡɚɜɢɫɢɦɨɫɬɶ 'l(l0) ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨ ɜɵɛɢɪɚɟɦɵɯ ɞɥɢɧ l0. Ɉɩɟɪɚɬɨɪɨɦ ɷɬɨɣ ɡɚɜɢɫɢɦɨɫɬɢ ɹɜɥɹɟɬɫɹ ɞɟɮɨɪɦɚɰɢɹ.

Ɉɛɨɛɳɟɧɢɟ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɚɪɝɭɦɟɧɬɨɦ ɫɱɢɬɚɟɬɫɹ ɜɟɤɬɨɪ l0, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɣ ɧɟ ɬɨɥɶɤɨ ɞɥɢɧɭ, ɧɨ ɢ ɧɚɩɪɚɜɥɟɧɢɟ ɜɨɥɨɤɧɚ ɜ ɦɚɥɨɣ ɨɤɪɟɫɬɧɨɫɬɢ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɬɨɱɤɢ ɬɟɥɚ, ɚ ɮɭɧɤɰɢɟɣ ɹɜɥɹɟɬɫɹ ɜɟɤɬɨɪ ɪɚɡɧɨɫɬɢ

'l { l – l0='l(l0)

(9.4)

(l – ɜɟɤɬɨɪ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɣ ɬɟɤɭɳɢɟ ɞɥɢɧɭ ɢ ɧɚɩɪɚɜɥɟɧɢɟ ɬɨɝɨ ɠɟ ɜɨɥɨɤɧɚ). ɗɬɨɬ ɜɟɤɬɨɪ ɨɩɪɟɞɟɥɹɟɬ ɧɟ ɬɨɥɶɤɨ ɢɡɦɟɧɟɧɢɟ ɞɥɢɧɵ, ɧɨ ɢ ɩɨɜɨɪɨɬ ɜɨɥɨɤɧɚ, ɤɨɬɨɪɵɣ ɜ ɧɚɱɚɥɶɧɨɦ ɫɨɫɬɨɹɧɢɢ ɯɚɪɚɤɬɟɪɢɡɨɜɚɥɫɹ ɜɟɤɬɨɪɨɦ l0 (ɪɢɫ.9.8). ɇɚ ɪɢɫɭɧɤɟ

ɩɨɤɚɡɚɧɵ ɞɜɟ ɦɚɬɟɪɢɚɥɶɧɵɟ ɬɨɱɤɢ – ɜ ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɧɨɦ (A,B) ɢ ɜ ɞɟɮɨɪɦɢɪɨ-

			ȼ'	ɜɚɧɧɨɦ (Ac,Bc) ɫɨɫɬɨɹɧɢɹɯ. ɇɢɠɟ ɩɨɤɚɡɚɧɨ “ɢɡ-
	ȼ	l		ɦɟɧɟɧɢɟ” ɜɨɥɨɤɧɚ (9.4). ɇɚ ɪɢɫɭɧɤɟ ɜɟɥɢɱɢɧɚ 'l
				ɭɬɪɢɪɨɜɚɧɚ; ɜ ɞɟɣɫɬɜɢɬɟɥɶɧɨɫɬɢ ɞɟɮɨɪɦɚɰɢɹ ɢ
	l0
			l	ɩɨɜɨɪɨɬ ɨɛɵɱɧɨ ɜɟɫɶɦɚ ɦɚɥɵ. Ƚɢɩɨɬɟɬɢɱɟɫɤɢ ɢɯ
Ⱥ	Ⱥ'
				ɫɱɢɬɚɸɬ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɦɢ, ɬɨɝɞɚ ɩɪɨɟɤɰɢɹ 'l
	l		l0
				ɧɚ ɧɚɩɪɚɜɥɟɧɢɟ l0 (ɫɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ 'l n,
	t		l0	ɝɞɟ n – ɟɞɢɧɢɱɧɵɣ ɜɟɤɬɨɪ ɜɞɨɥɶ l0) ɨɩɪɟɞɟɥɹɟɬ
				ɞɟɮɨɪɦɚɰɢɸ ɜɨɥɨɤɧɚ, ɚ ɩɪɨɟɤɰɢɹ ɧɚ ɜɟɤɬɨɪ t,
	n
		l0		ɨɪɬɨɝɨɧɚɥɶɧɵɣ n – ɭɝɨɥ ɩɨɜɨɪɨɬɚ M :
	Ɉ
				'l n =H l0, 'l t=M l0.
	Ɋɢɫ. 9.8

ɂɫɯɨɞɹ ɢɡ ɨɛɵɱɧɵɯ ɫɨɨɛɪɚɠɟɧɢɣ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɨɫɬɢ ɩɨɥɟɣ ɫɦɟɳɟɧɢɣ ɞɨɤɚɡɵɜɚɸɬ, ɱɬɨ ɞɥɹ ɛɟɫɤɨɧɟɱɧɨ ɤɨɪɨɬɤɢɯ ɜɨɥɨɤɨɧ l0 ɮɭɧɤɰɢɹ (9.4) ɥɢɧɟɣɧɚ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɟɫɥɢ ɜɡɹɬɶ ɜɞɜɨɟ ɛɨɥɟɟ ɞɥɢɧɧɨɟ ɜɨɥɨɤɧɨ, ɬɨ ɢ ɟɝɨ ɢɡɦɟɧɟɧɢɟ ɛɭɞɟɬ ɜɞɜɨɟ ɛɨɥɶɲɢɦ. ɉɨɷɬɨɦɭ ɞɨɫɬɚɬɨɱɧɨ ɪɚɫɫɦɨɬɪɟɬɶ ɜɨɥɨɤɧɚ ɨɩɪɟɞɟɥɟɧɧɨɣ ɞɥɢɧɵ – ɧɚɩɪɢɦɟɪ, ɟɞɢɧɢɱɧɨɣ. ȼ ɱɚɫɬɧɨɫɬɢ, ɢɡɦɟɧɟɧɢɟ 'n ɟɞɢɧɢɱɧɨɝɨ ɜɨɥɨɤɧɚ n (ɪɢɫ.9.8) ɩɪɢ ɩɪɨɟɰɢɪɨɜɚɧɢɢ ɧɚ ɨɫɢ n ɢ t ɫɪɚɡɭ ɨɩɪɟɞɟɥɹɟɬ ɬɟ ɠɟ, ɱɬɨ ɢ ɜ (9.5) ɞɟɮɨɪɦɚɰɢɸ ɢ ɩɨɜɨɪɨɬ:

('n) n =H, ('n) t =M . (9.5')

ȿɫɬɟɫɬɜɟɧɧɨ, ɱɬɨ ɞɥɹ ɪɚɡɥɢɱɧɨ ɨɪɢɟɧɬɢɪɨɜɚɧɧɵɯ ɜɨɥɨɤɨɧ ɷɬɢ ɜɟɥɢɱɢɧɵ ɪɚɡɥɢɱɧɵ. ȿɫɥɢ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɢɧɬɟɪɟɫɭɸɳɟɣ ɧɚɫ ɬɨɱɤɢ ɬɟɥɚ ɪɚɫɫɦɨɬɪɟɬɶ ɩɭɱɨɤ ɟɞɢ-

ɧɢɱɧɵɯ ɜɨɥɨɤɨɧ ɫ ɨɛɳɢɦ ɧɚɱɚɥɨɦ, ɬɨ ɢɯ ɤɨɧɰɵ ɥɹɝɭɬ ɧɚ ɨɤɪɭɠɧɨɫɬɶ (ɦɵ ɩɨɤɚ ɨɝ-
2		ɪɚɧɢɱɢɦɫɹ ɞɟɮɨɪɦɢɪɨɜɚɧɢɟɦ ɜ ɨɞɧɨɣ ɩɥɨɫɤɨ-
		ɫɬɢ) ɪɚɞɢɭɫɚ 1. ɂɡ ɥɢɧɟɣɧɨɫɬɢ ɮɭɧɤɰɢɢ (9.4)
	2( 1+ 2)
		ɫɥɟɞɭɟɬ, ɱɬɨ ɜ ɞɟɮɨɪɦɢɪɨɜɚɧɧɨɦ ɫɨɫɬɨɹɧɢɢ
		ɧɚɱɚɥɨ ɷɬɢɯ ɜɨɥɨɤɨɧ ɩɟɪɟɣɞɟɬ, ɜɨɡɦɨɠɧɨ, ɜ
		ɧɨɜɭɸ ɬɨɱɤɭ, ɚ ɤɨɧɰɵ ɨɛɪɚɡɭɸɬ ɷɥɥɢɩɫ
	2(1+ 1)	(ɪɢɫ.9.9). ɉɨɥɭɨɫɢ ɷɥɥɢɩɫɚ ɩɨɤɚɡɵɜɚɸɬ, ɤɚɤɢɟ
		ɜɨɥɨɤɧɚ ɩɨɥɭɱɢɥɢ ɧɚɢɛɨɥɶɲɭɸ ɢ ɧɚɢɦɟɧɶ-
n	n + n	ɲɭɸ ɞɟɮɨɪɦɚɰɢɢ ɢ ɤɚɤɨɜɵ ɢɦɟɧɧɨ ɷɬɢ ɡɧɚɱɟ-
		ɧɢɹ (H1,H2). ɗɬɢ ɧɚɩɪɚɜɥɟɧɢɹ ɢ ɞɟɮɨɪɦɚɰɢɢ ɧɚ-

Ɋɢɫ. 9.9	ɡɵɜɚɸɬ ɝɥɚɜɧɵɦɢ.

ȼɜɟɞɟɦ ɞɟɤɚɪɬɨɜɵ ɤɨɨɪɞɢɧɚɬɵ x, y ɢ ɛɭɞɟɦ ɨɬɨɛɪɚɠɚɬɶ ɜɟɤɬɨɪɵ ɦɚɬɪɢɰɚɦɢɫɬɨɥɛɰɚɦɢ ɤɨɨɪɞɢɧɚɬ (ɧɚɩɪɢɦɟɪ, n o >nx, ny@T ). Ɍɨɝɞɚ ɮɭɧɤɰɢɹ (9.4), ɤɚɤ ɢ ɜɫɹɤɚɹ ɥɢɧɟɣɧɚɹ ɜɟɤɬɨɪ-ɮɭɧɤɰɢɹ ɜɟɤɬɨɪɚ, ɨɬɨɛɪɚɡɢɬɫɹ ɦɚɬɪɢɰɟɣ

ª'lx º		ªDxx	Dxy º ªlox º		(9.4')
«	»	«	» «	».
¬'ly ¼		¬Dyx	Dyy ¼ ¬loy ¼

Ɇɚɬɪɢɰɭ D ɧɚɡɵɜɚɸɬ ɦɚɬɪɢɰɟɣ ɞɢɫɬɨɪɫɢɢ. ɉɨɥɭɱɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ, ɫ ɨɞɧɨɣ ɢ ɬɨɣ ɠɟ ɦɚɬɪɢɰɟɣ ɞɢɫɬɨɪɫɢɢ, ɫɩɪɚɜɟɞɥɢɜɨ ɞɥɹ ɥɸɛɵɯ (ɛɟɫɤɨɧɟɱɧɨ ɤɨɪɨɬɤɢɯ) ɜɟɤɬɨɪɨɜ

l0 ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɬɨɱɤɢ ɬɟɥɚ. ȼ ɱɚɫɬɧɨɫɬɢ, ɞɥɹ ɟɞɢɧɢɱɧɨɝɨ ɜɨ-
ɥɨɤɧɚ ɜɞɨɥɶ ɨɫɢ x (ɜɟɤɬɨɪ i, ɤɨɨɪɞɢɧɚɬɵ: 1, 0) ɩɨɥɭɱɢɦ
				ª1º	ªDxx º
		>'i@=>D@ « »			«	D		».
				0	¬		yx	¼
				¬ ¼
ɂɡ ɪɢɫ.9.10 ɜɢɞɧɨ, ɱɬɨ ɩɟɪɜɚɹ ɩɪɨɟɤɰɢɹ ɜɟɤɬɨɪɚ 'i, – Dxx – ɩɪɟɞɫɬɚɜɥɹɟɬ ɞɟɮɨɪɦɚ-
ɰɢɸ ɜɨɥɨɤɧɚ i, ɚ ɜɬɨɪɚɹ – ɩɨɜɨɪɨɬ (ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ y, ɬɨ ɟɫɬɶ ɩɪɨɬɢɜ ɱɚɫɨɜɨɣ
	j			ɫɬɪɟɥɤɢ). Ⱥɧɚɥɨɝɢɱɧɨ, ɜɬɨɪɨɣ ɫɬɨɥɛɟɰ ɦɚɬɪɢɰɵ ɞɢɫɬɨɪɫɢɢ
y	Dyy			ɩɪɟɞɫɬɚɜɥɹɟɬ ɩɨɜɨɪɨɬ ɟɞɢɧɢɱɧɨɝɨ ɜɨɥɨɤɧɚ j ɜ ɧɚɩɪɚɜɥɟɧɢɢ
				ɨɫɢ x (ɬɨ ɟɫɬɶ ɩɨ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɟ) ɢ ɞɟɮɨɪɦɚɰɢɸ ɷɬɨɝɨ ɜɨ-
j	Dxy
		i		ɥɨɤɧɚ (ɪɢɫ.9.10). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɡɧɚɹ ɢɡɦɟɧɟɧɢɟ ɜɫɟɝɨ ɞɜɭɯ
		Dyx		ɜɨɥɨɤɨɧ, ɦɵ ɢɦɟɟɦ ɜɫɸ ɦɚɬɪɢɰɭ ɞɢɫɬɨɪɫɢɢ ɢ ɜɨɡɦɨɠɧɨɫɬɶ
	i	Dxx	x	ɧɚɣɬɢ ɢɡɦɟɧɟɧɢɹ ɥɸɛɵɯ ɜɨɥɨɤɨɧ ɢɡ ɜɵɪɚɠɟɧɢɹ (9.4c).
	Ɋɢɫ. 9.10			ɉɪɢɦɟɪ 6. Ɍɨɱɤɢ O, A, B, C ɬɟɥɚ ɩɨɫɥɟ ɞɟɮɨɪɦɢɪɨɜɚɧɢɹ
				ɩɨɩɚɥɢ ɜ ɩɨɥɨɠɟɧɢɹ O, Ac, Bc, C (ɪɢɫ.9.11). ɇɚɣɬɢ ɦɚɬɪɢɰɭ

ɞɢɫɬɨɪɫɢɢ.

Ɋɟɲɟɧɢɟ. ȼɨɥɨɤɧɚ, ɨɪɢɟɧɬɢɪɨɜɚɧɧɵɟ ɩɚɪɚɥɥɟɥɶɧɨ ɨɫɢ x, (ɧɚɩɪɢɦɟɪ, OC, AB) ɧɟ ɞɟɮɨɪɦɢɪɭɸɬɫɹ ɢ ɧɟ ɩɨɜɨɪɚɱɢɜɚɸɬɫɹ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɟɪɜɵɣ ɫɬɨɥɛɟɰ D – ɧɭ-

ɥɟɜɨɣ. Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɜɬɨɪɨɝɨ ɫɬɨɥɛɰɚ ɩɨɞɫɬɚɜɢɦ ɜ ɜɵ-

ɦɦ

ɪɚɠɟɧɢɟ (9.4c) ɜɟɤɬɨɪɵ AAc ɢ OA:

y

0,1

B B'

ª0.1ɦɦº

ª 0

º

A

A'

«

0

»= >D@

«

»,

5ɫɦ

¬

¼

¬5ɫɦ¼

ɢɥɢ

Ɉ

0.1ɦɦ= Dxx 0+Dxy 5ɫɦ, 0= Dyx 0+Dyy 5ɫɦ.

6ɫɦ

ɋ x

Ɉɬɫɸɞɚ Dxy=2 10

– 3

, ɨɫɬɚɥɶɧɵɟ Dij ɪɚɜɧɵ ɧɭɥɸ.

Ɋɢɫ. 9.11

Ɂɚɦɟɬɢɦ, ɱɬɨ ɜɬɨɪɨɣ ɫɬɨɥɛɟɰ ɦɚɬɪɢɰɵ D ɩɪɟɞɫɬɚɜɥɹɟɬ ɢɡɦɟɧɟɧɢɟ ɟɞɢɧɢɱɧɨɝɨ ɜɨɥɨɤɧɚ, ɨɪɢɟɧɬɢɪɨɜɚɧɧɨɝɨ ɜɞɨɥɶ ɨɫɢ y, ɢ ɞɥɹ ɟɝɨ ɨɩɪɟɞɟɥɟɧɢɹ ɞɨɫɬɚɬɨɱɧɨ ɪɚɡɞɟɥɢɬɶ ɜɟɤɬɨɪ AAc (ɢɡɦɟɧɟɧɢɟ ɜɨɥɨɤɧɚ OA ɞɥɢɧɨɣ 5 ɫɦ) ɧɚ 5 ɫɦ.

ɉɪɢɦɟɪ 7. Ⱦɥɹ ɩɪɢɜɟɞɟɧɧɨɝɨ ɩɪɢɦɟɪɚ ɧɚɣɬɢ ɜɟɤɬɨɪ BBc (ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ

B ɢɡɜɟɫɬɧɵ).

Ɋɟɲɟɧɢɟ. Ⱦɢɫɬɨɪɫɢɹ ɧɚɣɞɟɧɚ ɪɚɧɶɲɟ, ɡɧɚɱɢɬ, ɦɨɠɧɨ ɩɪɹɦɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɮɨɪɦɭɥɨɣ (9.4c):

>BBc @= >D@ (>6 ɫɦ; 5 ɫɦ@)T= >Dxy 5 ɫɦ; 0@T=>0,1 ɦɦ; 0@T.

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜɟɤɬɨɪ BBc ɪɚɜɟɧ AAc (ɢ ɪɚɡɦɟɪ 6ɫɦ ɧɢ ɧɚ ɱɬɨ ɧɟ ɩɨɜɥɢɹɥ). ɗɬɨ ɟɫɬɟɫɬɜɟɧɧɨ, ɩɨɫɤɨɥɶɤɭ ɜɨɥɨɤɧɨ AB ɩɚɪɚɥɥɟɥɶɧɨ ɨɫɢ x; ɨɧɨ ɧɟ ɞɟɮɨɪɦɢɪɭɟɬɫɹ ɢ ɧɟ ɩɨɜɨɪɚɱɢɜɚɟɬɫɹ.

ɉɪɢɦɟɪ 8. ɉɪɹɦɨɭɝɨɥɶɧɢɤ ɈȺȼɋ ɢɡ ɩɪɟɞɵɞɭɳɟɣ ɡɚɞɚɱɢ ɧɚ ɷɬɨɬ ɪɚɡ ɩɨɜɟɪɧɭɥɫɹ ɧɚ ɭɝɨɥ Z=10–3 ɛɟɡ ɞɟɮɨɪɦɚɰɢɢ (ɪɢɫ. 9.12). ɇɚɣɬɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɭɸ ɞɢɫ-

y			B'	ɬɨɪɫɢɸ.
A'		A	B	Ɋɟɲɟɧɢɟ. ɉɪɢ ɠɟɫɬɤɨɦ ɩɨɜɨɪɨɬɟ ɬɟɥɚ ɜɫɟ ɟɝɨ ɜɨɥɨɤɧɚ
				ɢɦɟɸɬ ɨɞɧɭ ɢ ɬɭ ɠɟ ɞɟɮɨɪɦɚɰɢɸ (ɧɨɥɶ) ɢ ɨɞɢɧ ɢ ɬɨɬ ɠɟ
			ɋ'
Ɉ			ɋ	x
			Ɋɢɫ. 9.12