Сопротивление материалов / Nesmeyanov - Soprotivleniye materialov. Nestandartniye zadachi i podkhodi k ikh resheniyu 2005

(ɬ.ɟ. ɨɰɟɧɤɭ "ɫɧɢɡɭ" ɞɥɹ ɩɨɫɥɟɞɧɟɝɨ). Ɉɲɢɛɤɚ ɜ ɬɚɤɨɦ ɩɪɢɛɥɢɠɟɧɧɨɦ ɦɟɬɨɞɟ ɢɞɟɬ ɜ ɡɚɩɚɫ ɩɪɨɱɧɨɫɬɢ.

ɉɪɢɦɟɪ 1 – ɮɟɪɦɚ ɧɚ ɪɢɫ.7.1. Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɩɪɟɞɟɥɶɧɭɸ ɧɚɝɪɭɡɤɭ. Ɋɟɲɟɧɢɟ ɫɬɚɬɢɱɟɫɤɢɦ ɦɟɬɨɞɨɦ. ɂɳɟɦ ɦɚɤɫɢɦɭɦ

Pmax=?

(7.1)

ɩɪɢ ɭɫɥɨɜɢɹɯ ɢ ɨɝɪɚɧɢɱɟɧɢɹɯ:

			P						N1+N2+N3=P,		(7.2)
S	2S 2S								2N1+N2+P= 0,		(7.3)
									«N1 /S «d VT, «N2 /2S «dVT, «N3 /2S « dVT .		(7.4)
a	a	a	Ɂɞɟɫɶ (7.2)							ɢ (7.3) – ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ; (7.4) – ɭɫɥɨɜɢɹ
	Ɋɢɫ.7.1		ɨɝɪɚɧɢɱɟɧɧɨɫɬɢ ɧɚɩɪɹɠɟɧɢɣ; ɫɬɟɪɠɧɢ ɩɪɨɧɭɦɟɪɨɜɚɧɵ ɫɥɟ-
			ɜɚ ɧɚɩɪɚɜɨ.
Ɋɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ ɪɚɜɧɨɜɟɫɢɹ ɢɦɟɟɬ ɜɢɞ:
		N1=X1,		N2= –P – 2X1, N3=2P+ X1.
ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɨɝɪɚɧɢɱɟɧɢɹ (7.4) ɩɪɢɧɢɦɚɸɬ ɮɨɪɦɭ:
–VTSd X1 d VTS, –2VTSd–P–2X1d2VTS, –2VTSd P + 2X1 d 2VTS.											(7.5)
ɇɚ ɩɥɨɫɤɨɫɬɢ ^X1,P` ɢɦ ɨɬɜɟɱɚɟɬ ɨɛɥɚɫɬɶ ɞɨɩɭɫɬɢɦɵɯ ɡɧɚɱɟɧɢɣ, ɩɨɤɚɡɚɧɧɚɹ ɧɚ
	P .P0									ɪɢɫ.7.2. ɇɟɬɪɭɞɧɨ ɧɚɣɬɢ ɫɚɦɨɟ ɛɨɥɶɲɨɟ ɡɧɚ-
										ɱɟɧɢɟ P (ɷɬɨ ɢ ɟɫɬɶ ɩɪɟɞɟɥɶɧɚɹ ɧɚɝɪɭɡɤɚ) ɢ ɫɨ-
S										ɨɬɜɟɬɫɬɜɭɸɳɟɟ ɡɧɚɱɟɧɢɟ N1: P0=1,5VT S	ɩɪɢ
			N3							X1= – VT S = N1, N2=VTS/2, N3 =2VT S.
Ɍ	S			=
1.5V
	Ɍ			2V						Ʉɢɧɟɦɚɬɢɱɟɫɤɢɣ ɦɟɬɨɞ ɛɨɥɟɟ ɧɚɝɥɹɞɟɧ ɢ
	V						S
		0			Ɍ					ɱɚɳɟ ɢɫɩɨɥɶɡɭɟɬɫɹ ɜ ɢɧɠɟɧɟɪɧɨɣ ɩɪɚɤɬɢɤɟ.
	-VɌ S	VɌS		N				X1		ɍɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ ɢ ɞɨɩɭɫɬɢɦɨɫɬɶ ɧɚɩɪɹɠɟ-
				2						ɧɢɣ ɩɪɢ ɷɬɨɦ ɧɟ ɪɚɫɫɦɚɬɪɢɜɚɸɬ. Ⱦɥɹ ɨɩɪɟɞɟ-
S					=
						2				ɥɟɧɢɹ ɧɚɝɪɭɡɤɢ P0 ɢɫɩɨɥɶɡɭɸɬ ɩɪɢɧɰɢɩ ɜɨɡ-
Ɍ							V
-V
	N						Ɍ
							S			ɦɨɠɧɵɯ ɩɟɪɟɦɟɳɟɧɢɣ: ɜ ɩɪɟɞɟɥɶɧɨɦ ɫɨɫɬɨɹ-
=		-P0		N
	2
1
N	=					=-
	-				3
	2.									ɧɢɢ ɪɚɛɨɬɚ ɜɧɟɲɧɢɯ ɫɢɥ 'W ɧɚ ɷɥɟɦɟɧɬɚɪɧɨɦ
	V						2
	Ɍ							V	S
			S					Ɍ		ɞɜɢɠɟɧɢɢ ɩɥɚɫɬɢɱɟɫɤɨɝɨ ɦɟɯɚɧɢɡɦɚ ɪɚɜɧɚ ɪɚ-
		S
			Ɍ
	Ɋɢɫ.7.2		1=V							ɛɨɬɟ ɜɧɭɬɪɟɧɧɢɯ ɫɢɥ – ɚ ɢɦɟɧɧɨ, ɪɚɫɫɟɹɧɢɸ
			N							ɷɧɟɪɝɢɢ 'D ɧɚ ɩɥɚɫɬɢɱɟɫɤɨɟ ɞɟɮɨɪɦɢɪɨɜɚɧɢɟ

'Hp. ɉɨɫɥɟɞɧɹɹ ɜ ɭɫɥɨɜɢɹɯ ɪɚɫɬɹɠɟɧɢɹ-ɫɠɚɬɢɹ ɩɪɟɞɫɬɚɜɥɹɟɬ ɢɧɬɟɝɪɚɥ

'D = ³V'HpdV.

V

ȼ ɮɟɪɦɟ ɷɬɨɬ ɢɧɬɟɝɪɚɥ ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɜɢɞɟ ɫɭɦɦɵ

'D =¦VT Si «'lpi «,

ɝɞɟ i – ɧɨɦɟɪ ɫɬɟɪɠɧɹ, S – ɩɥɨɳɚɞɶ ɟɝɨ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ, 'lpi=Hpili, li – ɞɥɢɧɚ ɫɬɟɪɠɧɹ. Ʉɨɝɞɚ ɩɥɚɫɬɢɱɟɫɤɢɣ ɦɟɯɚɧɢɡɦ ɨɱɟɜɢɞɟɧ, ɧɚɯɨɠɞɟɧɢɟ ɩɪɟɞɟɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ ɧɚɝɪɭɡɤɢ ɨɤɚɡɵɜɚɟɬɫɹ ɧɟɫɥɨɠɧɵɦ. ɇɨ ɟɫɥɢ ɜɨɡɦɨɠɧɵ ɧɟɫɤɨɥɶɤɨ ɦɟɯɚɧɢɡ-

Ɋɢɫ.7.3

ɦɨɜ ɪɚɡɪɭɲɟɧɢɹ, ɬɨ ɩɪɢɯɨɞɢɬɫɹ ɧɚɯɨɞɢɬɶ ɩɪɟɞɟɥɶɧɭɸ ɧɚɝɪɭɡɤɭ ɞɥɹ ɤɚɠɞɨɝɨ ɢɡ
ɧɢɯ. Ɋɟɡɭɥɶɬɚɬ ɞɥɹ ɥɸɛɨɝɨ ɦɟɯɚɧɢɡɦɚ, ɤɪɨɦɟ ɞɟɣɫɬɜɢɬɟɥɶɧɨɝɨ, ɩɨɥɭɱɚɟɬɫɹ ɫ ɡɚ-
ɜɵɲɟɧɢɟɦ.
	ȼ ɩɪɢɛɥɢɠɟɧɧɨɦ ɤɢɧɟɦɚɬɢɱɟɫɤɨɦ ɦɟɬɨɞɟ ɪɚɫɫɦɚɬɪɢɜɚɸɬ ɧɟ ɜɫɟ, ɚ ɬɨɥɶɤɨ ɧɟ-
ɤɨɬɨɪɵɣ ɧɚɛɨɪ ɦɟɯɚɧɢɡɦɨɜ. ɇɚɢɦɟɧɶɲɚɹ ɢɡ ɧɚɣɞɟɧɧɵɯ ɧɚɝɪɭɡɨɤ ɞɚɟɬ ɨɰɟɧɤɭ ɩɪɟ-
ɞɟɥɶɧɨɣ ɧɚɝɪɭɡɤɢ "ɫɜɟɪɯɭ".
	ɂɧɨɝɞɚ ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɫɨɱɟɬɚɧɢɟ ɩɪɢɛɥɢɠɟɧɧɵɯ ɫɬɚɬɢɱɟɫɤɨɝɨ ɢ ɤɢɧɟɦɚɬɢɱɟ-
ɫɤɨɝɨ ɦɟɬɨɞɨɜ. Ɉɧɨ ɞɚɟɬ "ɜɢɥɤɭ", ɜɧɭɬɪɢ ɤɨɬɨɪɨɣ ɧɚɯɨɞɢɬɫɹ ɬɨɱɧɨɟ ɪɟɲɟɧɢɟ. ȿɫ-
ɥɢ "ɜɢɥɤɚ" ɧɟ ɫɥɢɲɤɨɦ ɲɢɪɨɤɚ ɫ ɩɪɢɤɥɚɞɧɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ, ɬɨ ɩɨɢɫɤ ɛɨɥɟɟ ɬɨɱ-
ɧɵɯ ɪɟɲɟɧɢɣ ɦɨɠɧɨ ɩɪɟɤɪɚɬɢɬɶ.
	ɉɪɢɦɟɪ 2 (ɪɢɫ.7.3ɚ). ɋɬɟɩɟɧɶ ɫɬɚɬɢɱɟɫɤɨɣ ɧɟɨɩɪɟɞɟɥɢɦɨɫɬɢ k ɜ ɷɬɨɣ ɡɚɞɚɱɟ
ɪɚɜɧɚ ɟɞɢɧɢɰɟ ɢ ɞɥɹ ɨɛɪɚɡɨɜɚɧɢɹ ɦɟɯɚɧɢɡɦɚ ɞɨɫɬɚɬɨɱɧɨ ɩɥɚɫɬɢɱɟɫɤɨɝɨ ɬɟɱɟɧɢɹ ɜ
ɞɜɭɯ ɫɬɟɪɠɧɹɯ (k+1). ɉɨɫɤɨɥɶɤɭ ɫɬɟɪɠɧɟɣ ɬɪɢ, ɬɨ ɫɭɳɟɫɬɜɭɸɬ ɬɪɢ ɜɨɡɦɨɠɧɵɯ
ɩɥɚɫɬɢɱɟɫɤɢɯ ɦɟɯɚɧɢɡɦɚ.
	ɉɭɫɬɶ "ɬɟɤɭɬ" ɞɜɚ ɤɪɚɣɧɢɯ ɫɬɟɪɠɧɹ, ɫɪɟɞɧɢɣ ɨɫɬɚɟɬɫɹ ɭɩɪɭɝɢɦ (ɢɥɢ, ɱɬɨ ɬɨ
	a	a	a	2a	ɠɟ, ɠɟɫɬɤɢɦ) – ɪɢɫ.7.3ɛ. Ɉɛɨɡɧɚɱɢɦ:				' – ɫɦɟɳɟ-
					ɧɢɟ ɤɨɧɰɚ ɧɢɠɧɟɝɨ ɠɟɫɬɤɨɝɨ ɫɬɟɪɠɧɹ. Ɉɧ ɩɨ-
a)	P	S	3S	2S	ɜɟɪɧɟɬɫɹ ɜɨɤɪɭɝ ɨɩɨɪɵ O2 ɧɚ ɭɝɨɥ 'M ='/(5a),
					ɩɪɢ ɷɬɨɦ ɫɪɟɞɧɢɣ, ɧɟɞɟɮɨɪɦɢɪɭɟɦɵɣ, ɫɬɟɪɠɟɧɶ
					ɩɟɪɟɦɟɫɬɢɬɫɹ ɜɧɢɡ ɧɚ ɜɟɥɢɱɢɧɭ 'M 3a=3'/5.
					ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɜɟɪɯɧɢɣ ɠɟɫɬɤɢɣ ɫɬɟɪɠɟɧɶ
	Ɉ1				ɞɨɥɠɟɧ	ɩɨɜɟɪɧɭɬɶɫɹ	ɧɚ	ɭɝɨɥ	3'/(5 2a)=
ɛ)					=3'/(10a). Ɂɧɚɹ ɭɝɥɵ ɩɨɜɨɪɨɬɚ ɠɟɫɬɤɢɯ ɫɬɟɪɠ-
					ɧɟɣ, ɧɟɬɪɭɞɧɨ ɜɵɱɢɫɥɢɬɶ ɭɞɥɢɧɟɧɢɟ ɥɟɜɨɝɨ ɢ
					Ɉ2 ɭɤɨɪɨɱɟɧɢɟ ɩɪɚɜɨɝɨ «ɬɟɤɭɳɢɯ» ɫɬɟɪɠɧɟɣ:

Ɉ1				'lp1= ' /(5a) 4a – 3'/(10a) a=' /2,
VɌ S		VɌ	2S	'lp3=3' /(10a) 3a – ' /(5a) 2a=' /2.
				ɍɪɚɜɧɟɧɢɟ ɛɚɥɚɧɫɚ ɪɚɛɨɬ ɞɚɟɬ
ɜ)	N2 N2
VɌ S		VɌ	2S	P0'=VTS'/2+VT 2S' /2,
P0
			Ɉ2	ɢɥɢ P0=3VTS/2.	(7.6)

Ɉɞɧɚɤɨ ɷɬɨ ɟɳɟ ɧɟ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ: ɪɚɫɫɦɨɬɪɟɧ ɥɢɲɶ ɨɞɢɧ ɜɨɡɦɨɠɧɵɣ ɦɟɯɚɧɢɡɦ. Ɍɚɤɢɟ ɠɟ ɪɚɫɱɟɬɵ ɫɥɟɞɭɟɬ ɩɪɨɢɡɜɟɫɬɢ ɢ ɞɥɹ ɞɜɭɯ ɞɪɭɝɢɯ, ɢ ɢɡ ɬɪɟɯ ɡɧɚɱɟɧɢɣ ɩɪɟɞɟɥɶɧɨɣ ɧɚɝɪɭɡɤɢ ɜɵɛɪɚɬɶ ɦɢɧɢɦɚɥɶɧɭɸ.

ɂɧɨɝɞɚ ɦɟɯɚɧɢɡɦ ɩɨɱɬɢ ɨɱɟɜɢɞɟɧ, ɢ ɩɟɪɟɛɨɪ ɜɫɟɯ ɞɪɭɝɢɯ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɩɨɱɬɢ ɧɚɩɪɚɫɧɨɣ ɬɪɚɬɨɣ ɜɪɟɦɟɧɢ. Ɍɨɝɞɚ ɜɨɡɦɨɠɟɧ ɢɧɨɣ ɩɭɬɶ: ɧɚɣɞɹ ɞɥɹ ɨɱɟɜɢɞɧɨɝɨ ɦɟɯɚɧɢɡɦɚ ɩɪɟɞɟɥɶɧɭɸ ɧɚɝɪɭɡɤɭ, ɩɪɨɜɟɪɢɬɶ, ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɥɢ ɩɨɥɭɱɟɧɧɨɟ ɪɟɲɟɧɢɟ ɫɬɚɬɢɱɟɫɤɢɦ ɨɝɪɚɧɢɱɟɧɢɹɦ. ȿɫɥɢ ɞɚ, ɬɨ ɪɟɲɟɧɢɟ ɧɚɣɞɟɧɨ, ɢɧɚɱɟ – ɧɢɱɟɝɨ ɧɟ ɩɨɬɟɪɹɧɨ, ɧɭɠɧɨ ɪɚɫɫɦɨɬɪɟɬɶ ɞɪɭɝɨɣ ɦɟɯɚɧɢɡɦ (ɞɪɭɝɢɟ ɦɟɯɚɧɢɡɦɵ).

ɇɚɩɪɢɦɟɪ, ɜ ɪɚɫɫɦɨɬɪɟɧɧɨɦ ɩɪɢɦɟɪɟ ɧɚɦ ɩɨɤɚɡɚɥɨɫɶ, ɱɬɨ ɡɧɚɱɟɧɢɟ (7.6) ɞɨɥɠɧɨ ɛɵɬɶ ɜɟɪɧɨ. ɉɪɢ ɞɚɧɧɨɦ ɡɧɚɱɟɧɢɢ P =1,5VTS ɢɡ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ ɜɟɪɯɧɟɝɨ ɫɬɟɪɠɧɹ ɧɚɣɞɟɦ ɭɫɢɥɢɟ ɜ ɫɪɟɞɧɟɦ:

	VTSa+N22a – 2VTS3a=0,
ɬɨ ɟɫɬɶ N2=2.5VTS. ɇɟɬɪɭɞɧɨ ɩɪɨɜɟɪɢɬɶ, ɱɬɨ ɢ ɭɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ ɧɢɠɧɟɝɨ
ɫɬɟɪɠɧɹ ɭɞɨɜɥɟɬɜɨɪɹɟɬɫɹ ɩɪɢ ɬɨɦ ɠɟ ɡɧɚɱɟɧɢɢ N2. ɗɬɨ ɭɫɢɥɢɟ ɦɟɧɶɲɟ ɫɨɨɬɜɟɬɫɬ-
ɜɭɸɳɟɝɨ ɭɫɢɥɢɹ ɬɟɤɭɱɟɫɬɢ (N02=3VTS). Ɂɧɚɱɢɬ, ɜɵɩɨɥɧɹɸɬɫɹ ɧɟ ɬɨɥɶɤɨ ɤɢɧɟɦɚ-
ɬɢɱɟɫɤɢɟ, ɧɨ ɢ ɫɬɚɬɢɱɟɫɤɢɟ ɭɫɥɨɜɢɹ (ɪɚɜɧɨɜɟɫɢɟ) ɢ ɨɝɪɚɧɢɱɟɧɢɹ (_V_ dVT). Ɋɟɲɟ-
ɧɢɟ, ɩɨɥɭɱɟɧɧɨɟ ɧɚ ɨɫɧɨɜɟ ɞɚɧɧɨɝɨ ɦɟɯɚɧɢɡɦɚ, ɨɤɚɡɚɥɨɫɶ ɜɟɪɧɵɦ; ɞɪɭɝɢɟ ɦɟɯɚ-
ɧɢɡɦɵ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɧɟ ɧɭɠɧɨ.
ɉɪɢɦɟɪ 3 (ɪɢɫ.7.4ɚ) – ɫɥɭɱɚɣ, ɤɨɝɞɚ ɩɪɨɫɬɨɣ ɩɟɪɟɛɨɪ ɦɟɯɚɧɢɡɦɨɜ ɧɟɜɨɡɦɨ-
ɠɟɧ. Ɂɚɞɚɱɚ ɨɞɧɚɠɞɵ ɫɬɚɬɢɱɟɫɤɢ ɧɟɨɩɪɟɞɟɥɢɦɚ ɢ ɩɨɬɨɦɭ ɞɥɹ ɨɛɪɚɡɨɜɚɧɢɹ ɦɟɯɚ-
ɧɢɡɦɚ ɬɪɟɛɭɸɬɫɹ ɞɜɚ ɩɥɚɫɬɢɱɟɫɤɢɯ ɲɚɪɧɢɪɚ. Ɉɞɢɧ ɢɡ ɧɢɯ – ɜ ɡɚɳɟɦɥɟɧɢɢ, ɧɨ ɤɨ-
ɨɪɞɢɧɚɬɭ x ɜɬɨɪɨɝɨ, ɤɨɬɨɪɵɣ ɨɛɪɚɡɭɟɬɫɹ ɜ ɩɪɨɥɟɬɟ (ɪɢɫ.7.4ɛ), ɫɪɚɡɭ ɭɤɚɡɚɬɶ ɧɟɥɶ-
a)		q	ɡɹ.
	l		Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɛɚɥɚɧɫɚ ɪɚɛɨɬ. ȼ
z		x	ɥɟɜɨɦ ɩɥɚɫɬɢɱɟɫɤɨɦ		ɲɚɪɧɢɪɟ ɩɪɨɢɫɯɨɞɢɬ
	dz		ɩɨɜɨɪɨɬ ɧɚ ɭɝɨɥ 'M x/(l – x); ɜ ɫɪɟɞɧɟɦ
ɛ)		q0	ɲɚɪɧɢɪɟ	ɩɨɜɨɪɨɬ	ɪɚɜɟɧ	ɫɭɦɦɟ	'M+
x			+'M x/(l–x). Ɂɚɦɟɬɢɦ, ɱɬɨ ɜ ɩɪɚɜɨɦ ɲɚɪɧɢ-
			ɪɟ, ɝɞɟ ɩɪɨɢɫɯɨɞɢɬ		ɩɨɜɨɪɨɬ	ɧɚ ɭɝɨɥ	'M,
l-x	x
			ɞɢɫɫɢɩɚɰɢɢ ɧɟɬ, ɩɨɫɤɨɥɶɤɭ ɬɚɦ ɨɛɵɱɧɵɣ, ɚ
			ɧɟ ɩɥɚɫɬɢɱɟɫɤɢɣ ɲɚɪɧɢɪ. ɋɢɥɚ q0dz, ɩɪɢ-
ɜ)	Ɇ0	ɗɆ(q0 )	ɯɨɞɹɳɚɹɫɹ ɧɚ ɷɥɟɦɟɧɬ ɞɥɢɧɵ ɛɚɥɤɢ dz (ɤɨ-
Ɇ0			ɨɪɞɢɧɚɬɚ ɷɥɟɦɟɧɬɚ – z, q0 – ɩɪɟɞɟɥɶɧɨɟ ɡɧɚ-
ɝ)		ɗQ(q0 )	ɱɟɧɢɟ ɩɚɪɚɦɟɬɪɚ ɧɚɝɪɭɡɤɢ), ɩɪɢ ɩɟɪɟɦɟɳɟ-
q0 l-R 1		R 1	ɧɢɢ ɷɥɟɦɟɧɬɚ ɜɧɢɡ ɧɚ ɜɟɥɢɱɢɧɭ 'X(z) ɩɪɨ-
	Ɋɢɫ. 7.4		ɢɡɜɨɞɢɬ ɪɚɛɨɬɭ q0dz'X(z).
			Ɋɚɛɨɬɚ ɜɫɟɣ ɧɚɝɪɭɡɤɢ

³q0dz'X(z)=q0 ³'X(z)dz.

l

l

ɂɧɬɟɝɪɚɥ ³'X(z)dz ɟɫɬɶ ɩɥɨɳɚɞɶ, ɨɝɪɚɧɢɱɟɧɧɚɹ ɧɚɱɚɥɶɧɵɦ ɢ ɧɨɜɵɦ ɩɨɥɨɠɟɧɢɹ-

l

ɦɢ ɨɫɟɜɨɣ ɥɢɧɢɢ ɫɬɟɪɠɧɹ (ɪɢɫ.7.4ɛ). ɍɪɚɜɧɟɧɢɟ ɛɚɥɚɧɫɚ ɪɚɛɨɬ

0.5q0lx'M =M0 x'M /(l – x)+M0('M+'M x /(l – x))

ɩɨɫɥɟ ɫɨɤɪɚɳɟɧɢɹ ɧɚ 'M ɞɚɟɬ ɡɧɚɱɟɧɢɟ q0 ɤɚɤ ɮɭɧɤɰɢɸ ɤɨɨɪɞɢɧɚɬɵ x:

q0=2M0(x –1+l –1)/(l – x).

(7.7)

ɂɫɬɢɧɧɵɣ ɦɟɯɚɧɢɡɦ ɪɚɡɪɭɲɟɧɢɹ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɧɚɢɦɟɧɶɲɟɦɭ ɡɧɚɱɟɧɢɸ ɩɪɟɞɟɥɶɧɨɣ ɧɚɝɪɭɡɤɢ; ɨɬɫɸɞɚ ɫɥɟɞɭɟɬ ɩɭɬɶ ɤ ɨɩɪɟɞɟɥɟɧɢɸ ɜɟɥɢɱɢɧɵ x:

dq0 /dx=2M0(– (l – x)x–2+(x–1+l–1))/(l – x)2= 0, x2+2lx – l2= 0, x = – l± l 2 .

Ɉɞɢɧ ɢɡ ɤɨɪɧɟɣ ɹɜɧɨ ɩɪɨɬɢɜɨɪɟɱɢɬ ɮɢɡɢɱɟɫɤɨɦɭ ɫɦɵɫɥɭ ɡɚɞɚɱɢ (x 0). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, x=l( 2 – 1) ɢ ɢɡ (7.7) ɩɨɥɭɱɢɦ:

q0=2M0(3 – 2 2 )–1/l2.

ɉɪɢɦɟɪ 4. ɉɪɢɜɟɞɟɦ ɪɟɲɟɧɢɟ ɬɨɣ ɠɟ ɡɚɞɚɱɢ ɫɬɚɬɢɱɟɫɤɢɦ ɦɟɬɨɞɨɦ. ɂɡ ɭɫɥɨɜɢɣ ɪɚɜɧɨɜɟɫɢɹ ɫɥɟɞɭɟɬ, ɱɬɨ ɷɩɸɪɚ ɢɡɝɢɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ ɞɨɥɠɧɚ ɢɦɟɬɶ ɜɢɞ, ɩɨɤɚɡɚɧɧɵɣ ɧɚ ɪɢɫ.7.4ɜ; ɦɟɯɚɧɢɡɦ, ɢɡɨɛɪɚɠɟɧɧɵɣ ɧɚ ɪɢɫ.7.4ɛ, ɩɨɞɫɤɚɡɵɜɚɟɬ, ɱɬɨ ɜ ɡɚɞɟɥɤɟ ɦɨɦɟɧɬ ɪɚɜɟɧ ɩɪɟɞɟɥɶɧɨɦɭ, ɢ ɧɢɠɧɢɟ ɫɥɨɢ ɛɚɥɤɢ ɫɠɚɬɵ; ɜ ɩɪɚɜɨɦ ɩɥɚɫɬɢɱɟɫɤɨɦ ɲɚɪɧɢɪɟ ɦɨɦɟɧɬ ɬɚɤɠɟ ɪɚɜɟɧ ɩɪɟɞɟɥɶɧɨɦɭ, ɧɨ ɫɠɚɬɵ ɜɟɪɯɧɢɟ ɫɥɨɢ. ȼ ɬɨɱɤɟ ɷɤɫɬɪɟɦɭɦɚ ɦɨɦɟɧɬɚ ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ, ɩɨ ɭɫɥɨɜɢɹɦ ɪɚɜɧɨɜɟɫɢɹ, ɪɚɜɧɚ ɧɭɥɸ. ɗɩɸɪɚ ɩɨɩɟɪɟɱɧɵɯ ɫɢɥ ɩɨɤɚɡɚɧɚ ɧɚ ɪɢɫ.7.4ɝ; ɨɩɨɪɧɚɹ ɪɟɚɤɰɢɹ R1 ɫɜɹɡɚɧɚ ɫ ɪɚɫɫɬɨɹɧɢɟɦ x ɭɪɚɜɧɟɧɢɟɦ ɪɚɜɧɨɜɟɫɢɹ

dQ/dz= – R1 /x= – q0, R1=q0 x.

ɍɱɢɬɵɜɚɹ, ɱɬɨ ɦɨɦɟɧɬ ɜ ɩɪɚɜɨɦ ɩɥɚɫɬɢɱɟɫɤɨɦ ɲɚɪɧɢɪɟ ɪɚɜɟɧ ɩɥɨɳɚɞɢ ɱɚɫɬɢ ɷɩɸɪɵ Q(z), ɪɚɫɩɨɥɨɠɟɧɧɨɣ ɫɩɪɚɜɚ ɨɬ ɲɚɪɧɢɪɚ, ɩɨɥɭɱɢɦ

R1x/2=M0, x= 2M 0 / q0 .

ɉɥɨɳɚɞɶ ɷɩɸɪɵ Q(z) ɫɥɟɜɚ ɨɬ ɩɪɚɜɨɝɨ ɲɚɪɧɢɪɚ ɪɚɜɧɚ ɢɡɦɟɧɟɧɢɸ ɦɨɦɟɧɬɚ ɧɚ ɷɬɨɦ ɭɱɚɫɬɤɟ (2M0):

(q0l – R1)(l – x)/2=2M0 .

ɉɨɞɫɬɚɧɨɜɤɚ ɜ ɷɬɨ ɜɵɪɚɠɟɧɢɟ ɜɟɥɢɱɢɧ R1=q0x ɢ q0=2M0 /x2 ɞɚɟɬ ɭɠɟ ɡɧɚɤɨɦɵɟ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɤɨɨɪɞɢɧɚɬɵ x ɢ ɩɪɟɞɟɥɶɧɨɣ ɧɚɝɪɭɡɤɢ q0.

ɉɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱ ɩɨɥɟɡɧɨ ɭɱɢɬɵɜɚɬɶ, ɱɬɨ ɜɩɥɨɬɶ ɞɨ ɞɨɫɬɢɠɟɧɢɹ ɩɪɟɞɟɥɶɧɨɣ ɧɚɝɪɭɡɤɢ ɤɨɧɫɬɪɭɤɰɢɹ ɭɫɬɨɣɱɢɜɚ, ɢ ɩɨɬɨɦɭ ɨɫɬɚɟɬɫɹ ɫɩɪɚɜɟɞɥɢɜɵɦ ɡɚɤɨɧ ɫɢɦɦɟɬɪɢɢ: ɩɪɟɞɟɥɶɧɨɟ ɫɨɫɬɨɹɧɢɟ ɫɢɦɦɟɬɪɢɱɧɨɣ ɤɨɧɫɬɪɭɤɰɢɢ ɩɪɢ ɫɢɦɦɟɬɪɢɱɧɨɦ ɜɨɡɞɟɣɫɬɜɢɢ ɫɢɦɦɟɬɪɢɱɧɨ. ȿɫɥɢ ɭɱɟɫɬɶ ɞɨɩɨɥɧɢɬɟɥɶɧɨ, ɱɬɨ ɩɪɟɞɟɥɵ ɬɟɤɭɱɟɫɬɢ ɩɪɢ ɪɚɫɬɹɠɟɧɢɢ ɢ ɫɠɚɬɢɢ ɩɪɢɧɢɦɚɸɬɫɹ ɨɛɵɱɧɨ ɨɞɢɧɚɤɨɜɵɦɢ ɢ ɞɢɚɝɪɚɦɦɚ ɞɟɮɨɪɦɢɪɨɜɚɧɢɹ ɨɤɚɡɵɜɚɟɬɫɹ ɧɟɱɟɬɧɨɣ, ɬɨ ɭɜɢɞɢɦ, ɱɬɨ ɫɩɪɚɜɟɞɥɢɜ ɢ ɡɚɤɨɧ ɤɨɫɨɣ ɫɢɦɦɟɬɪɢɢ: ɩɪɢ ɤɨɫɨɫɢɦɦɟɬɪɢɱɧɨɦ ɜɨɡɞɟɣɫɬɜɢɢ ɧɚ ɫɢɦɦɟɬɪɢɱɧɭɸ ɤɨɧɫɬɪɭɤɰɢɸ ɩɪɟɞɟɥɶɧɨɟ ɫɨɫɬɨɹɧɢɟ (ɤɚɤ ɢ ɥɸɛɨɟ ɩɪɨɦɟɠɭɬɨɱɧɨɟ) ɤɨɫɨɫɢɦɦɟɬɪɢɱɧɨ. ɋɥɟɞɭɟɬ, ɨɞɧɚɤɨ, ɢɦɟɬɶ ɜ ɜɢɞɭ, ɱɬɨ ɩɪɢ ɧɟɫɢɦɦɟɬɪɢɱɧɨɦ ɧɚɝɪɭɠɟɧɢɢ ɪɚɡɥɨɠɟɧɢɟ ɧɚɝɪɭɡɤɢ ɧɚ ɫɢɦɦɟɬɪɢɱɧɭɸ ɢ ɤɨɫɨɫɢɦɦɟɬɪɢɱɧɭɸ ɜɨɡɦɨɠɧɨ, ɧɨ ɛɟɫɩɨɥɟɡɧɨ: ɩɪɢɧɰɢɩ ɫɭɩɟɪɩɨɡɢɰɢɢ ɜ ɧɟɥɢɧɟɣɧɵɯ ɡɚɞɚɱɚɯ ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ.

ɋɩɟɰɢɮɢɱɟɫɤɢɣ ɤɥɚɫɫ ɡɚɞɚɱ ɩɪɟɞɟɥɶɧɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɩɪɟɞɫɬɚɜɥɹɟɬ ɪɚɫɱɟɬ ɧɟɫɭɳɟɣ ɫɩɨɫɨɛɧɨɫɬɢ ɡɚɤɥɟɩɨɤ. Ɂɚɦɟɬɢɦ, ɱɬɨ ɭɩɪɭɝɢɟ ɪɚɫɱɟɬɵ ɡɚɤɥɟɩɨɤ ɢ ɭɫɥɨɜɧɵɟ ɫɯɟɦɵ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɜ ɧɢɯ ɭɫɢɥɢɣ, ɜɫɬɪɟɱɚɸɳɢɟɫɹ ɜ ɧɟɤɨɬɨɪɵɯ ɭɱɟɛɧɢɤɚɯ ɢ ɩɨɫɨɛɢɹɯ ɩɨ ɫɨɩɪɨɬɢɜɥɟɧɢɸ ɦɚɬɟɪɢɚɥɨɜ, ɮɢɡɢɱɟɫɤɢ ɦɚɥɨ ɨɩɪɚɜɞɚɧɵ. Ɋɚɫɱɟɬɵ ɩɨ ɩɪɟɞɟɥɶɧɨɦɭ ɫɨɫɬɨɹɧɢɸ ɩɪɟɞɫɬɚɜɥɹɸɬɫɹ ɧɚɦ ɛɨɥɟɟ ɤɨɪɪɟɤɬɧɵɦɢ.

ɉɪɢɦɟɪ 5 (ɪɢɫ.7.5): ɬɪɟɛɭɟɬɫɹ ɨɰɟɧɢɬɶ ɧɟɫɭɳɭɸ ɫɩɨɫɨɛɧɨɫɬɶ ɤɨɧɫɬɪɭɤɰɢɢ ɢɡ ɞɜɭɯ ɩɨɥɨɫ, ɫɤɪɟɩɥɟɧɧɵɯ ɱɟɬɵɪɶɦɹ ɡɚɤɥɟɩɤɚɦɢ. Ɏɨɪɦɚ ɩɨɥɨɫɵ, ɩɨɥɚɝɚɟɦɨɣ ɜ ɩɪɟɞɟɥɶɧɨɦ ɫɨɫɬɨɹɧɢɢ ɭɩɪɭɝɨɣ (ɢɥɢ, ɱɬɨ ɬɨ ɠɟ, ɚɛɫɨɥɸɬɧɨ ɠɟɫɬɤɨɣ), ɧɟ ɢɦɟɟɬ ɡɧɚɱɟɧɢɹ, ɢ ɡɚɞɚɱɚ ɷɤɜɢɜɚɥɟɧɬɧɚ ɩɨɤɚɡɚɧɧɨɣ ɧɚ ɪɢɫ.7.6, ɹɜɥɹɸɳɟɣɫɹ ɤɨɫɨɫɢɦɦɟɬ-

a 2

b

P

P

Ɋɢɫ. 7.5

a a
		P
a	a	b
		Ɋɢɫ. 7.6

ɪɢɱɧɨɣ. ɉɨɫɥɟɞɧɟɟ ɨɛɫɬɨɹɬɟɥɶɫɬɜɨ ɩɨɡɜɨɥɹɟɬ ɩɪɢ ɚɧɚɥɢɡɟ ɨɝɪɚɧɢɱɢɬɶɫɹ ɤɨɫɨɫɢɦɦɟɬɪɢɱɧɵɦɢ ɦɟɯɚɧɢɡɦɚɦɢ ɩɪɟɞɟɥɶɧɨɝɨ ɪɚɜɧɨɜɟɫɢɹ – ɷɬɨ ɩɨɜɨɪɨɬ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɤɨɬɨɪɨɣ ɬɨɱɤɢ C, ɥɟɠɚɳɟɣ ɧɚ ɨɫɢ ɫɢɦɦɟɬɪɢɢ ɤɨɧɫɬɪɭɤɰɢɢ. ɐɟɧɬɪ ɩɨɜɨɪɨɬɚ ɢɡ ɨɱɟɜɢɞɧɵɯ ɫɨɨɛɪɚɠɟɧɢɣ ɪɚɜɧɨɜɟɫɢɹ ɧɟ ɦɨɠɟɬ ɛɵɬɶ ɧɢ ɥɟɜɟɟ, ɧɢ ɩɪɚɜɟɟ ɨɬɪɟɡɤɚ ɞɥɢɧɨɣ a , ɩɨɤɚɡɚɧɧɨɝɨ ɧɚ ɪɢɫ.7.7 (ɞɨɫɬɚɬɨɱɧɨ ɪɚɫɫɦɨɬɪɟɬɶ ɧɚɩɪɚɜɥɟɧɢɹ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɩɨɥɨɫɭ ɜ ɷɬɢɯ ɞɜɭɯ ɫɥɭɱɚɹɯ; ɡɧɚɱɟɧɢɹ ɫɢɥ ɩɪɢ ɷɬɨɦ ɧɟ ɢɝɪɚɸɬ ɪɨɥɢ). ɉɨɜɨɪɨɬ ɜɨɡɦɨɠɟɧ ɥɢɛɨ ɜɨɤɪɭɝ ɥɟɜɨɣ ɡɚɤɥɟɩɤɢ (ɫɪɟɡɚɸɬɫɹ ɬɪɢ ɡɚɤɥɟɩɤɢ), ɥɢɛɨ ɜɨɤɪɭɝ ɬɨɱɤɢ ɜɧɭɬɪɢ ɤɜɚɞɪɚɬɚ, ɨɛɪɚɡɨɜɚɧɧɨɝɨ ɡɚɤɥɟɩɤɚɦɢ (ɫɪɟɡɚɸɬɫɹ ɱɟɬɵɪɟ ɡɚɤɥɟɩɤɢ).

ɋ		a a	b				ɉɟɪɜɚɹ ɫɢɬɭɚɰɢɹ ɞɨɫ-
				Q	Q		ɬɚɬɨɱɧɨ ɩɪɨɫɬɚ ɞɥɹ ɪɚɫɱɟɬɚ;
			Q
	Q						ɧɚ ɪɢɫ.7.8 ɩɨɤɚɡɚɧɵ ɫɢɥɵ,
				ɋ
							ɞɟɣɫɬɜɭɸɳɢɟ	ɧɚ	ɩɨɥɨɫɭ,
		ɋ	Q	R			ɫɢɥɚ ɫɪɟɡɚ ɡɚɤɥɟɩɤɢ ɨɛɨɡɧɚ-
							ɱɟɧɚ Q. ɉɪɢɧɰɢɩ ɜɨɡɦɨɠ-
				Q
	Q		P0			P0	ɧɵɯ ɩɟɪɟɦɟɳɟɧɢɣ ɡɞɟɫɶ ɷɤ-
		e					ɜɢɜɚɥɟɧɬɟɧ ɭɫɥɨɜɢɸ ɪɚɜɧɨ-
			Ɋɢɫ.7.7		Ɋɢɫ.7.8		ɜɟɫɢɹ ɞɥɹ ɦɨɦɟɧɬɚ ɫɢɥ ɨɬ-
			P0 (b + 2a) = Q2a + 2Qa			2 ,	ɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ ɋ:
	ɢɥɢ
			P0 = QD E /(1+E), D = 1+			2 , E = 2a/b.
	ɂɡ ɫɭɦɦɵ ɩɪɨɟɤɰɢɣ ɫɢɥ ɧɚ ɜɟɪɬɢɤɚɥɶɧɭɸ ɨɫɶ ɧɚɣɞɟɦ ɭɫɢɥɢɟ ɜ ɱɟɬɜɟɪɬɨɣ ɡɚɤɥɟɩɤɟ

R=Q + Q 2 – P0 =D/(E + 1)Q.

ɗɬɨ ɭɫɢɥɢɟ ɧɟ ɩɪɟɜɵɲɚɟɬ Q, ɟɫɥɢ

1+E t D, at b/ 2 ;

ɜ ɷɬɢɯ ɫɥɭɱɚɹɯ ɩɨɥɭɱɟɧɧɨɟ ɪɟɲɟɧɢɟ ɹɜɥɹɟɬɫɹ ɢɫɬɢɧɧɵɦ.

Ɉɞɧɚɤɨ ɩɪɢ ɛɨɥɶɲɟɦ ɭɞɚɥɟɧɢɢ ɡɚɤɥɟɩɨɤ ɨɬ ɬɨɱɤɢ ɩɪɢɥɨɠɟɧɢɹ ɫɢɥɵ (b>a 2 ) ɩɪɟɞɩɨɥɨɠɟɧɢɟ ɨ ɬɨɦ, ɱɬɨ ɥɟɜɚɹ ɡɚɤɥɟɩɤɚ ɧɟ ɪɚɡɪɭɲɚɟɬɫɹ, ɧɟ ɨɬɜɟɱɚɟɬ ɪɚɫɱɟɬɭ ɢ ɪɚɫɫɦɨɬɪɟɧɧɵɣ ɦɟɯɚɧɢɡɦ ɨɤɚɡɵɜɚɟɬɫɹ ɧɟɞɟɣɫɬɜɢɬɟɥɶɧɵɦ. Ʉɚɤ ɧɢ ɤɚɠɟɬɫɹ ɩɚɪɚɞɨɤɫɚɥɶɧɵɦ, ɩɪɢ ɷɬɨɦ ɫɪɟɡɚɸɬɫɹ ɜɫɟ ɡɚɤɥɟɩɤɢ ɨɞɧɨɜɪɟɦɟɧɧɨ (ɪɢɫ.7.7).

ɉɪɢ ɪɚɫɱɟɬɟ ɷɬɨɝɨ ɩɥɚɫɬɢɱɟɫɤɨɝɨ ɦɟɯɚɧɢɡɦɚ ɩɨɹɜɥɹɟɬɫɹ ɧɨɜɚɹ ɧɟɢɡɜɟɫɬɧɚɹ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɚɹ ɩɨɥɨɠɟɧɢɟ ɰɟɧɬɪɚ ɜɪɚɳɟɧɢɹ (ɪɚɡɦɟɪ e), ɡɚɬɨ ɭɫɢɥɢɹ ɜɨ ɜɫɟɯ ɡɚɤɥɟɩɤɚɯ ɢɡɜɟɫɬɧɵ. ɍɪɚɜɧɟɧɢɟ ɩɪɢɧɰɢɩɚ ɜɨɡɦɨɠɧɵɯ ɩɟɪɟɦɟɳɟɧɢɣ ɫɜɨɞɢɬɫɹ ɤ ɪɚɜɟɧɫɬɜɭ ɧɭɥɸ ɫɭɦɦɵ ɦɨɦɟɧɬɨɜ ɫɢɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ ɋ

P0(b+a+c)=2aQ+2QU, U = ɟ2 ɚ2 .

Ʉɚɠɞɨɦɭ ɢɡ ɬɚɤɢɯ ɦɟɯɚɧɢɡɦɨɜ, ɨɩɪɟɞɟɥɹɟɦɵɯ ɡɚɞɚɧɢɟɦ ɩɨɥɨɠɟɧɢɹ ɰɟɧɬɪɚ ɜɪɚɳɟɧɢɹ e, ɨɬɜɟɱɚɟɬ ɫɜɨɟ ɡɧɚɱɟɧɢɟ ɩɪɟɞɟɥɶɧɨɣ ɧɚɝɪɭɡɤɢ P0 (ɫɢɥɚ Q ɩɨɥɚɝɚɟɬɫɹ ɢɡɜɟɫɬɧɨɣ – ɷɬɨ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɡɚɤɥɟɩɤɢ); ɢɫɬɢɧɧɨɦɭ ɦɟɯɚɧɢɡɦɭ ɨɬɜɟɱɚɟɬ ɦɢɧɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɫɢɥɵ P0.

ȼɵɪɚɠɟɧɢɟ ɞɥɹ ɩɪɨɢɡɜɨɞɧɨɣ dP0 /de ɜ ɷɬɨɣ ɡɚɞɚɱɟ ɝɪɨɦɨɡɞɤɨ; ɩɪɨɳɟ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɜɬɨɪɵɦ ɭɫɥɨɜɢɟɦ ɪɚɜɧɨɜɟɫɢɹ (ɩɨɫɤɨɥɶɤɭ ɞɥɹ ɢɫɬɢɧɧɨɝɨ ɦɟɯɚɧɢɡɦɚ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ ɫɩɪɚɜɟɞɥɢɜɵ) – ɩɨ ɫɢɥɚɦ:

P0=2QsinD, sinD=e/U.

(7.8)

ɋɨɜɦɟɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɫɥɨɜɢɣ ɪɚɜɧɨɜɟɫɢɹ ɨɩɪɟɞɟɥɹɟɬ e=2a2(a+b)/(2ab+b2).

ɇɚɣɞɹ e, ɧɟɬɪɭɞɧɨ ɨɩɪɟɞɟɥɢɬɶ P0 ɢɡ ɜɵɪɚɠɟɧɢɹ (7.8).

ɉɪɨɜɟɪɤɚ: ɪɚɫɱɟɬ ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɩɪɢ ɜɫɟɯ ɡɧɚɱɟɧɢɹɯ ɩɚɪɚɦɟɬɪɚ e ɜ ɭɩɨɦɹɧɭɬɵɯ ɩɪɟɞɟɥɚɯ 0d ed a ɡɧɚɱɟɧɢɟ P0 ɞɥɹ ɜɬɨɪɨɝɨ ɦɟɯɚɧɢɡɦɚ (7.8) ɧɢɠɟ, ɱɟɦ ɞɥɹ ɩɟɪɜɨɝɨ.

ɋɚɦɨɫɬɨɹɬɟɥɶɧɵɣ ɤɪɭɝ ɡɚɞɚɱ ɩɪɟɞɫɬɚɜɥɹɸɬ ɪɚɫɱɟɬɵ ɩɪɟɞɟɥɶɧɨɝɨ ɢɡɝɢɛɚɸɳɟɝɨ ɦɨɦɟɧɬɚ ɩɪɢ ɤɨɫɨɦ ɢɡɝɢɛɟ. ɗɬɚ ɡɚɞɚɱɚ ɬɪɢɜɢɚɥɶɧɚ, ɟɫɥɢ ɡɚɞɚɧɚ ɧɟɣɬɪɚɥɶɧɚɹ ɥɢɧɢɹ (ɧɚɩɪɹɠɟɧɢɹ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɢɡɜɟɫɬɧɵ ɢ ɧɚɣɬɢ ɢɯ ɦɨɦɟɧɬ ɞɨɫɬɚɬɨɱɧɨ ɩɪɨɫɬɨ). ȿɫɥɢ, ɨɞɧɚɤɨ, ɡɚɞɚɧɵ ɢɡɝɢɛɚɸɳɢɟ ɦɨɦɟɧɬɵ, ɬɨ ɨɩɪɟɞɟɥɟɧɢɟ ɩɨɥɨɠɟɧɢɹ ɧɟɣɬɪɚɥɶɧɨɣ ɥɢɧɢɢ ɫɢɥɶɧɨ ɭɫɥɨɠɧɹɟɬɫɹ.

ɉɪɢɦɟɪ 6 – ɢɡɹɳɧɨɟ ɢɫɤɥɸɱɟɧɢɟ ɢɡ ɷɬɨɝɨ ɩɪɚɜɢɥɚ. ɉɪɹɦɨɭɝɨɥɶɧɵɣ ɫɬɟɪɠɟɧɶ ɢɡ ɢɞɟɚɥɶɧɨ ɩɥɚɫɬɢɱɟɫɤɨɝɨ ɦɚɬɟɪɢɚɥɚ ɢɡɝɢɛɚɟɬɫɹ ɬɚɤ, ɱɬɨ ɩɥɨɫɤɨɫɬɶ ɫɭɦɦɚɪɧɨɝɨ ɦɨɦɟɧɬɚ ɩɪɨɯɨɞɢɬ ɩɨ ɞɢɚɝɨɧɚɥɢ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ (ɪɢɫ.7.9). ɇɚɣɬɢ ɩɪɟɞɟɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɦɨɦɟɧɬɚ.

ɋɬɚɧɞɚɪɬɧɨɟ ɪɟɲɟɧɢɟ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨɛɵ ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɩɨɥɨɠɟɧɢɹ ɧɟɣɬɪɚɥɶɧɨɣ ɥɢɧɢɢ ɧɚɣɬɢ ɢɡɝɢɛɚɸɳɢɟ ɦɨɦɟɧɬɵ, ɩɥɨɫɤɨɫɬɶ ɢɯ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɟɣ ɢ ɩɨ ɡɚɞɚɧɧɨɦɭ ɩɨɥɨɠɟɧɢɸ ɷɬɨɣ ɩɥɨɫɤɨɫɬɢ ɧɚɣɬɢ ɩɨɥɨɠɟɧɢɟ ɷɬɨɣ ɥɢɧɢɢ. Ⱦɚɥɶɲɟ ɡɚɞɚɱɚ ɬɪɢɜɢɚɥɶɧɚ.

ɗɬɚ ɡɚɞɚɱɚ, ɨɞɧɚɤɨ, ɨɛɥɚɞɚɟɬ ɫɥɟɞɭɸɳɟɣ ɨɫɨɛɟɧɧɨɫɬɶɸ: ɩɪɢ ɡɚɞɚɧɧɨɦ ɢɡɝɢɛɟ ɭɩɪɭɝɨɣ ɛɚɥɤɢ ɧɟɣɬɪɚɥɶɧɚɹ ɥɢɧɢɹ ɩɪɨɯɨɞɢɬ ɩɨ ɞɪɭɝɨɣ ɞɢɚɝɨɧɚɥɢ ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ. ɗɬɨ ɦɨɠɟɬ (ɧɨ, ɟɫɬɟɫɬɜɟɧɧɨ, ɧɟ ɨɛɹɡɚɬɟɥɶɧɨ ɞɨɥɠɧɨ) ɨɤɚɡɚɬɶɫɹ ɫɩɪɚɜɟɞɥɢɜɵɦ ɢ ɞɥɹ ɫɥɭɱɚɹ ɩɪɟɞɟɥɶɧɨɝɨ ɫɨɫɬɨɹɧɢɹ.

ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɷɬɨ ɫɩɪɚɜɟɞɥɢɜɨ. Ɍɨɝɞɚ ɜ ɩɪɟɞɟɥɶɧɨɦ ɫɨɫɬɨɹɧɢɢ ɪɚɫɬɹɝɢɜɚɸɳɢɟ ɧɚɩɪɹɠɟɧɢɹ, ɪɚɜɧɵɟ VT, ɨɯɜɚɬɵɜɚɸɬ ɬɪɟɭɝɨɥɶɧɭɸ ɨɛɥɚɫɬɶ (ɪɢɫ.7.10); ɢɯ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ P1 ɪɚɫɩɨɥɚɝɚɟɬɫɹ ɜ ɰɟɧɬɪɟ ɬɹɠɟɫɬɢ ɬɪɟɭɝɨɥɶɧɢɤɚ – ɬɨ ɟɫɬɶ ɧɚ ɦɟɞɢɚɧɟ. Ɍɨ ɠɟ ɨɬɧɨɫɢɬɫɹ ɢ ɤ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɟɣ P2 ɫɠɢɦɚɸɳɢɯ ɧɚɩɪɹɠɟɧɢɣ. Ʉɚɤ ɜɢɞɢɦ, ɩɥɨɫɤɨɫɬɶ ɢɡɝɢɛɚɸɳɟɝɨ ɦɨɦɟɧɬɚ ɞɟɣɫɬɜɢɬɟɥɶɧɨ ɨɬɜɟɱɚɟɬ ɞɢɚɝɨɧɚɥɢ.

Ɉɫɬɚɟɬɫɹ ɥɢɲɶ ɧɚɣɬɢ ɦɨɦɟɧɬ ɩɚɪɵ

b/3	ɫɢɥ P1, P2: ɷɬɨ ɢ ɟɫɬɶ ɢɫɤɨɦɨɟ ɡɧɚ-

	P1
b	ɗV
h	P2
Ɇ	VT
Ɋɢɫ.7.9	Ɋɢɫ.7.10

ɱɟɧɢɟ M0. ȼɟɥɢɱɢɧɚ ɫɢɥɵ P1 ɪɚɜɧɚ h/3 ɩɪɨɢɡɜɟɞɟɧɢɸ VT bh/2; ɩɨɥɨɠɟɧɢɟ ɰɟɧɬɪɨɜ ɬɹɠɟɫɬɢ ɩɪɹɦɨɭɝɨɥɶɧɵɯ ɬɪɟɭɝɨɥɶɧɢɤɨɜ ɢɡɜɟɫɬɧɨ (ɫɦ. ɪɢɫ.7.10). Ɉɬɫɸɞɚ ɧɚɯɨɞɢɦ

M0=VTbh b2 h2 /6.

8. ɍɋɌɈɃɑɂȼɈɋɌɖ ɍɉɊɍȽɂɏ ɋɂɋɌȿɆ

8.1. ɑɬɨ ɬɚɤɨɟ ɤɪɢɬɢɱɟɫɤɚɹ ɫɢɥɚ?

Ɋɚɫɫɦɚɬɪɢɜɚɹ ɷɬɨɬ ɪɚɡɞɟɥ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɦɚɬɟɪɢɚɥɨɜ, ɫɥɟɞɭɟɬ ɨɛɪɚɬɢɬɶ ɜɧɢɦɚɧɢɟ ɧɚ ɫɥɟɞɭɸɳɢɟ ɟɝɨ ɨɫɨɛɟɧɧɨɫɬɢ. ȼɨ-ɩɟɪɜɵɯ, ɟɫɥɢ ɪɚɧɶɲɟ ɧɚɫ ɢɧɬɟɪɟɫɨɜɚɥɨ ɥɢɲɶ, ɩɪɢ ɤɚɤɢɯ ɭɫɥɨɜɢɹɯ ɤɨɧɫɬɪɭɤɰɢɹ ɧɚɯɨɞɢɬɫɹ ɜ ɪɚɜɧɨɜɟɫɢɢ, ɬɨ ɫɟɣɱɚɫ ɦɵ ɚɧɚɥɢɡɢɪɭɟɦ ɤɚɱɟɫɬɜɨ ɪɚɜɧɨɜɟɫɢɹ: ɨɤɚɡɵɜɚɟɬɫɹ, ɞɥɹ ɩɪɨɱɧɨɫɬɢ ɤɨɧɫɬɪɭɤɰɢɢ ɷɬɨ ɛɵɜɚɟɬ ɤɪɚɣɧɟ ɜɚɠɧɨ. ȿɫɥɢ ɪɚɜɧɨɜɟɫɢɟ ɧɟɭɫɬɨɣɱɢɜɨ, ɬɨ ɩɪɢ ɥɸɛɨɦ ɦɚɥɨɦ ɜɨɡɦɭɳɟɧɢɢ (ɚ ɢɡɛɟɠɚɬɶ ɟɝɨ ɩɪɚɤɬɢɱɟɫɤɢ ɧɟɜɨɡɦɨɠɧɨ) ɫɢɫɬɟɦɚ ɩɟɪɟɯɨɞɢɬ ɜ ɞɪɭɝɨɟ ɩɨɥɨɠɟɧɢɟ, ɧɟ ɩɪɟɞɭɫɦɨɬɪɟɧɧɨɟ ɭɫɥɨɜɢɹɦɢ ɟɟ ɪɚɛɨɬɵ; ɩɪɢ ɷɬɨɦ ɩɪɨɢɫɯɨɞɢɬ ɧɟɠɟɥɚɬɟɥɶɧɨɟ ɩɟɪɟɪɚɫɩɪɟɞɟɥɟɧɢɟ ɭɫɢɥɢɣ ɜ ɟɟ ɷɥɟɦɟɧɬɚɯ.

ȼɨ-ɜɬɨɪɵɯ, ɩɪɢɯɨɞɢɬɫɹ ɨɬɤɚɡɚɬɶɫɹ ɨɬ ɨɞɧɨɣ ɢɡ ɨɫɧɨɜɧɵɯ ɝɢɩɨɬɟɡ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɦɚɬɟɪɢɚɥɨɜ – ɝɢɩɨɬɟɡɵ ɦɚɥɨɫɬɢ ɩɟɪɟɦɟɳɟɧɢɣ (ɧɚɡɵɜɚɟɦɨɣ ɩɪɢɧɰɢɩɨɦ ɧɚɱɚɥɶɧɵɯ ɪɚɡɦɟɪɨɜ), ɬɚɤ ɤɚɤ ɜ ɪɚɦɤɚɯ ɷɬɨɣ ɝɢɩɨɬɟɡɵ ɪɚɜɧɨɜɟɫɢɟ ɜɫɟɝɞɚ ɭɫɬɨɣɱɢɜɨ, ɟɫɥɢ ɭɫɬɨɣɱɢɜ ɦɚɬɟɪɢɚɥ ɤɨɧɫɬɪɭɤɰɢɢ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɡɭɱɚɹ ɩɨ-ɩɪɟɠɧɟɦɭ ɨɬɧɨɫɢɬɟɥɶɧɨ ɦɚɥɵɟ ɫɦɟɳɟɧɢɹ ɬɨɱɟɤ ɬɟɥɚ ɩɪɢ ɟɝɨ ɧɚɝɪɭɠɟɧɢɢ, ɦɵ ɭɠɟ ɧɟ ɪɚɫɫɦɚɬɪɢɜɚɟɦ ɢɯ ɤɚɤ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɟ. C ɷɬɢɦ ɫɜɹɡɚɧɚ ɬɪɟɬɶɹ ɨɫɨɛɟɧɧɨɫɬɶ: ɞɚɠɟ ɜ ɩɪɟɞɟɥɚɯ ɡɚɤɨɧɚ Ƚɭɤɚ ɡɚɞɚɱɚ ɩɟɪɟɫɬɚɟɬ ɛɵɬɶ ɥɢɧɟɣɧɨɣ; ɫɬɚɧɨɜɢɬɫɹ ɧɟɫɩɪɚɜɟɞɥɢɜɵɦ ɢ ɩɪɢɧɰɢɩ ɫɭɩɟɪɩɨɡɢɰɢɢ.

Ɍɢɩɵ ɪɚɜɧɨɜɟɫɢɹ ɩɪɨɳɟ ɜɫɟɝɨ ɭɜɢɞɟɬɶ ɧɚ ɩɪɢɦɟɪɟ ɜɟɫɨɦɨɝɨ ɲɚɪɢɤɚ, ɥɟɠɚɳɟɝɨ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ, ɪɟɥɶɟɮ ɤɨɬɨɪɨɣ ɩɨɤɚɡɚɧ ɧɚ ɪɢɫ.8.1. Ɂɞɟɫɶ ɉ – ɜɵɫɨɬɚ ɪɟɥɶɟɮɚ ɤɚɤ ɮɭɧɤɰɢɹ ɤɨɨɪɞɢɧɚɬɵ ɯ; ɨɞɧɨɜɪɟɦɟɧɧɨ, ɷɬɨ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɲɚɪɢɤɚ, ɞɟɥɟɧɧɚɹ ɧɚ ɩɨɫɬɨɹɧɧɭɸ ɜɟɥɢɱɢɧɭ – ɜɟɫ ɲɚɪɢɤɚ.

ɇɟɬɪɭɞɧɨ ɜɢɞɟɬɶ, ɱɬɨ ɪɚɜɧɨɜɟɫɢɟ ɧɚɛɥɸɞɚɟɬɫɹ ɥɢɲɶ ɜ ɱɟɬɵɪɟɯ ɩɨɥɨɠɟɧɢɹɯ ɢɡ ɩɨɤɚɡɚɧɧɵɯ ɩɹɬɢ.

ɉɨɥɨɠɟɧɢɟ 3 ɧɚɢɛɨɥɟɟ ɬɢɩɢɱɧɨ ɞɥɹ ɦɟɯɚɧɢɱɟɫɤɢɯ ɫɢɫɬɟɦ (ɩɪɨɫɬɟɣɲɭɸ ɢɡ

ɤɨɬɨɪɵɯ ɦɵ ɪɚɫɫɦɚɬɪɢɜɚɟɦ): ɷɬɨ ɭɫɬɨɣɱɢɜɨɟ ɪɚɜɧɨɜɟɫɢɟ. Ɇɚɥɵɟ ɜɨɡɦɭɳɟɧɢɹ,
ɉ		ɫɦɟɳɚɸɳɢɟ ɲɚɪɢɤ ɜɥɟɜɨ ɢɥɢ ɜɩɪɚɜɨ, ɜɵɜɨɞɹɬ
		ɟɝɨ ɢɡ ɧɚɱɚɥɶɧɨɝɨ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɢ ɩɟɪɟ-
1 2	4	ɜɨɞɹɬ ɜ ɫɦɟɠɧɨɟ ɩɨɥɨɠɟɧɢɟ – ɛɟɫɤɨɧɟɱɧɨ ɛɥɢɡ-
		ɤɨɟ ɤ ɢɫɯɨɞɧɨɦɭ, ɧɨ ɩɨɫɥɟ ɢɯ ɫɧɹɬɢɹ ɲɚɪɢɤ ɜɨɡ-

35 ɜɪɚɳɚɟɬɫɹ ɜ ɢɫɯɨɞɧɨɟ ɩɨɥɨɠɟɧɢɟ.

Ɋɢɫ.8.1	ɯ	ɉɨɥɨɠɟɧɢɟ 4	ɜ ɷɬɨɦ ɨɬɧɨɲɟɧɢɢ ɧɚɢɛɨɥɟɟ
		ɧɟɩɪɢɹɬɧɨ: ɥɸɛɨɟ ɦɚɥɨɟ ɜɨɡɦɭɳɟɧɢɟ ɜɵɜɨɞɢɬ

ɲɚɪɢɤ ɢɡ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ, ɢ ɨɧ ɫɤɚɬɵɜɚɟɬɫɹ ɞɨ ɧɚɯɨɠɞɟɧɢɹ ɞɪɭɝɨɝɨ. Ɍɚɤɨɟ ɫɨɫɬɨɹɧɢɟ ɫɢɫɬɟɦɵ ɧɚɡɵɜɚɟɬɫɹ ɧɟɭɫɬɨɣɱɢɜɵɦ ɪɚɜɧɨɜɟɫɢɟɦ.

Ɉɫɨɛɨɟ ɦɟɫɬɨ ɡɚɧɢɦɚɟɬ ɫɨɫɬɨɹɧɢɟ ɛɟɡɪɚɡɥɢɱɧɨɝɨ ɪɚɜɧɨɜɟɫɢɹ 5. ɗɬɨ – ɟɞɢɧɫɬɜɟɧɧɚɹ ɫɢɬɭɚɰɢɹ, ɤɨɝɞɚ ɫɦɟɧɚ ɩɨɥɨɠɟɧɢɹ ɧɟ ɧɚɪɭɲɚɟɬ ɪɚɜɧɨɜɟɫɢɹ. ɉɟɪɟɯɨɞ ɤɨɧɫɬɪɭɤɰɢɢ ɢɡ ɭɫɬɨɣɱɢɜɨɝɨ ɫɨɫɬɨɹɧɢɹ ɜ ɧɟɭɫɬɨɣɱɢɜɨɟ (ɩɪɢ ɪɨɫɬɟ ɧɚɝɪɭɡɤɢ) ɩɪɨɢɫɯɨɞɢɬ ɱɟɪɟɡ ɷɬɭ ɮɚɡɭ, ɧɚɡɵɜɚɟɦɭɸ ɤɪɢɬɢɱɟɫɤɨɣ; ɨɩɪɟɞɟɥɟɧɢɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ (ɤɪɢɬɢɱɟɫɤɨɝɨ) ɡɧɚɱɟɧɢɹ ɧɚɝɪɭɡɤɢ ɢ ɩɪɟɞɫɬɚɜɥɹɟɬ ɨɛɵɱɧɨ ɡɚɞɚɱɭ ɪɚɫɱɟɬɱɢɤɚ.

ɇɚɤɨɧɟɰ, ɜɫɬɪɟɱɚɟɬɫɹ "ɭɫɬɨɣɱɢɜɨɟ ɜ ɦɚɥɨɦ" ɫɨɫɬɨɹɧɢɟ 1; ɨɧɨ ɜ ɦɟɯɚɧɢɤɟ ɩɨɱɬɢ ɧɟ ɢɡɭɱɚɟɬɫɹ. Ɂɞɟɫɶ ɩɪɢ ɦɚɥɵɯ ɜɨɡɦɭɳɟɧɢɹɯ ɲɚɪɢɤ ɭɫɬɨɣɱɢɜ, ɨɞɧɚɤɨ, ɟɫɥɢ

Ɋɢɫ.8.2

ɜɨɡɦɭɳɟɧɢɹ ɛɨɥɶɲɟ ɧɟɤɨɬɨɪɵɯ ɧɟɛɨɥɶɲɢɯ, ɧɨ ɤɨɧɟɱɧɵɯ ɜɟɥɢɱɢɧ, ɬɨ ɩɪɨɢɫɯɨɞɢɬ ɩɨɬɟɪɹ ɞɚɧɧɨɣ ɮɨɪɦɵ ɪɚɜɧɨɜɟɫɢɹ.

ɇɟɬɪɭɞɧɨ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɜ ɭɫɬɨɣɱɢɜɨɦ ɩɨɥɨɠɟɧɢɢ ɲɚɪɢɤ ɨɛɥɚɞɚɟɬ ɦɢɧɢɦɚɥɶɧɨɣ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɟɣ. ȼ ɛɟɡɪɚɡɥɢɱɧɨɦ ɫɨɫɬɨɹɧɢɢ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɧɟ ɢɡɦɟɧɹɟɬɫɹ ɩɪɢ ɫɦɟɧɟ ɩɨɥɨɠɟɧɢɹ.

Ⱦɥɹ ɞɟɮɨɪɦɢɪɭɟɦɵɯ ɤɨɧɫɬɪɭɤɰɢɣ ɫɢɬɭɚɰɢɹ ɚɧɚɥɨɝɢɱɧɚ, ɟɫɥɢ ɫɱɢɬɚɬɶ, ɱɬɨ ɜɟɥɢɱɢɧɚ ɉ ɫɨɫɬɨɢɬ ɢɡ «ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɫɢɥɵ» (ɫɜɹɡɚɧɧɨɣ ɫ ɬɟɤɭɳɢɦ ɩɨɥɨɠɟɧɢɟɦ ɬɨɱɤɢ ɩɪɢɥɨɠɟɧɢɹ ɫɢɥɵ) ɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɞɟɮɨɪɦɚɰɢɢ. ɗɬɭ ɫɭɦɦɭ ɧɚɡɵɜɚɸɬ ɩɨɥɧɨɣ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɟɣ ɫɢɫɬɟɦɵ. ȼɫɟ ɩɪɢɜɟɞɟɧɧɵɟ ɜɵɲɟ ɫɨɨɛɪɚɠɟɧɢɹ ɨ ɜɢɞɚɯ ɪɚɜɧɨɜɟɫɢɹ (ɢɥɢ ɧɟɪɚɜɧɨɜɟɫɢɹ) ɨɫɬɚɸɬɫɹ ɜ ɫɢɥɟ, ɧɨ ɪɟɥɶɟɮ ɩɨɜɟɪɯɧɨɫɬɢ ɉ ɡɚɜɢɫɢɬ ɨɬ ɡɧɚɱɟɧɢɣ ɞɟɣɫɬɜɭɸɳɢɯ ɫɢɥ. ɉɪɢ ɧɟɜɵɫɨɤɢɯ ɧɚɝɪɭɡɤɚɯ ɤɨɧɫɬɪɭɤɰɢɹ ɨɛɵɱɧɨ ɭɫɬɨɣɱɢɜɚ, ɧɨ ɩɪɢ ɜɨɡɪɚɫɬɚɧɢɢ ɧɚɝɪɭɡɤɢ (ɢɥɢ ɬɟɦɩɟɪɚɬɭɪɵ, ɢɥɢ ɩɚɪɚɦɟɬɪɚ ɤɢɧɟɦɚɬɢɱɟɫɤɨɝɨ ɜɨɡɞɟɣɫɬɜɢɹ) ɞɨ ɧɟɤɨɬɨɪɨɣ ɜɟɥɢɱɢɧɵ ɫɢɬɭɚɰɢɹ ɦɨɠɟɬ ɢɡɦɟɧɢɬɶɫɹ: ɩɨɹɜɥɹɟɬɫɹ ɧɨɜɨɟ, ɛɥɢɡɤɨɟ ɤ ɧɚɱɚɥɶɧɨɦɭ (ɫɦɟɠɧɨɟ), ɫɨɫɬɨɹɧɢɟ ɪɚɜɧɨɜɟɫɢɹ ɤɨɧɫɬɪɭɤɰɢɢ, ɩɪɢɱɟɦ ɩɪɟɠɧɟɟ ɨɫɬɚɟɬɫɹ ɪɚɜɧɨɜɟɫɧɵɦ (ɛɢɮɭɪ-

ɤɚɰɢɹ) ɢɥɢ ɨɤɚɡɵɜɚɟɬɫɹ ɧɟɪɚɜɧɨɜɟɫɧɵɦ, ɢ ɤɨɧɫɬɪɭɤɰɢɹ, ɦɟɧɹɹ	ɉ	ɚ)
ɫɜɨɸ ɮɨɪɦɭ, ɩɟɪɟɯɨɞɢɬ ɜ ɧɨɜɨɟ ɫɨɫɬɨɹɧɢɟ ɪɚɜɧɨɜɟɫɢɹ. ɗɬɨ		. Ⱥ
ɡɧɚɱɟɧɢɟ ɩɚɪɚɦɟɬɪɚ ɧɚɝɪɭɡɤɢ ɧɚɡɵɜɚɸɬ ɤɪɢɬɢɱɟɫɤɢɦ.

ȼ ɱɚɫɬɧɨɫɬɢ, ɩɪɢ ɫɠɚɬɢɢ ɩɪɹɦɨɝɨ ɜɟɪɬɢɤɚɥɶɧɨɝɨ ɫɬɟɪɠɧɹ (ɡɚɞɚɱɚ ɗɣɥɟɪɚ, ɪɢɫ.8.2) ɝɪɭɡɨɦ G ɫɨɫɬɨɹɧɢɟ ɫɢɫɬɟɦɵ «ɫɬɟɪɠɟɧɶ – ɝɪɭɡ» ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɫɬɪɟɥɨɣ ɩɪɨɝɢɛɚ f. ɉɪɢ ɧɟɜɵɫɨɤɨɣ ɧɚɝɪɭɡɤɟ ɭɫ- f ɬɨɣɱɢɜɨ ɫɨɫɬɨɹɧɢɟ f=0. ɗɬɨ ɦɨɠɧɨ ɭɜɢɞɟɬɶ ɢɡ ɷɧɟɪɝɟɬɢɱɟɫɤɨɝɨ ɪɟɥɶɟɮɚ (ɪɢɫ.8.3ɚ), ɝɞɟ ɉ – ɩɨ- ɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɝɪɭɡɚ ɜ ɫɭɦɦɟ ɫ ɷɧɟɪɝɢɟɣ ɫɠɚɬɢɹ ɫɬɨɣɤɢ (ɩɪɚɤɬɢɱɟɫɤɢ ɧɟ ɡɚɜɢɫɹɳɟɣ ɨɬ f) ɢ ɷɧɟɪɝɢɟɣ ɢɡɝɢɛɚ ɫɬɨɣɤɢ, ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɣ ɤɜɚɞɪɚɬɭ f . ɍɦɟɧɶɲɟɧɢɟ ɜɵɫɨɬɵ ɫɬɨɣɤɢ ɩɪɢ ɟɟ

ɢɡɝɢɛɟ (ɢ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɭɦɟɧɶɲɟɧɢɟ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɝɪɭɡɚ) ɬɚɤɠɟ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɤɜɚɞɪɚɬɭ f, ɩɨɷɬɨɦɭ ɪɟɥɶɟɮ ɨɩɢɫɵɜɚɟɬɫɹ

ɤɜɚɞɪɚɬɢɱɧɨɣ ɩɚɪɚɛɨɥɨɣ. ɉɨɤɚ ɝɪɭɡ ɧɟɜɟɥɢɤ, ɫ ɪɨɫɬɨɦ ɩɪɨɝɢɛɚ ɜɟɥɢɱɢɧɚ ɉ ɜɨɡɪɚɫɬɚɟɬ (ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɢɡɝɢɛɚ ɪɚɫɬɟɬ ɛɵɫɬɪɟɟ, ɱɟɦ ɭɛɵɜɚɟɬ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɝɪɭɡɚ, ɪɢɫ.8.3ɚ), ɧɨ ɩɪɢ ɛɨɥɶɲɨɦ ɜɟɫɟ ɝɪɭɡɚ ɫɢɬɭɚɰɢɹ ɨɛɪɚɬɧɚ (ɪɢɫ.8.3ɜ). Ʉɪɢɬɢɱɟɫɤɨɦɭ ɡɧɚɱɟɧɢɸ ɧɚɝɪɭɡɤɢ G = G* ɨɬɜɟɱɚɟɬ ɩɪɨɦɟɠɭɬɨɱɧɚɹ ɫɢɬɭɚɰɢɹ (ɪɢɫ.8.3ɛ).

8.2. Ʉɚɤ ɧɚɣɬɢ ɤɪɢɬɢɱɟɫɤɭɸ ɫɢɥɭ?

<	*	f
	0
	ɉ
	Ⱥ.	ɛ)
=	*	f
	0
	ɉ
	.Ⱥ	ɜ)
>	*	f
	0

Ɋɢɫ.8.3

Ɋɚɫɫɦɨɬɪɟɧɧɚɹ ɫɢɬɭɚɰɢɹ ɩɨɞɫɤɚɡɵɜɚɟɬ ɞɜɚ ɩɭɬɢ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ. Ɉɞɢɧ – ɷɧɟɪɝɟɬɢɱɟɫɤɢɣ, ɨɧ ɨɩɢɪɚɟɬɫɹ ɧɚ ɪɚɜɟɧɫɬɜɨ ɩɨɬɟɧɰɢɚɥɶɧɵɯ ɷɧɟɪɝɢɣ ɉ ɜ ɫɦɟɠɧɵɯ ɫɨɫɬɨɹɧɢɹɯ (ɪɢɫ.8.3ɛ). Ⱦɪɭɝɨɣ ɧɚɡɵɜɚɸɬ ɨɛɵɱɧɨ ɦɟɬɨɞɨɦ ɗɣɥɟɪɚ. Ɉɧ ɪɟɚɥɢɡɭɟɬ ɡɚɦɟɱɟɧɧɭɸ ɗɣɥɟɪɨɦ ɨɫɨɛɟɧɧɨɫɬɶ ɫɨɫɬɨɹɧɢɹ G = G*: ɬɨɥɶɤɨ ɩɪɢ ɬɚɤɨɣ ɧɚɝɪɭɡɤɟ

ɢɡɨɝɧɭɬɵɣ ɫɬɟɪɠɟɧɶ ɧɚɯɨɞɢɬɫɹ ɜ ɪɚɜɧɨɜɟɫɢɢ (ɫɪɚɜɧɢɬɟ ɫɨɫɬɨɹɧɢɹ fz0 – ɬɨɱɤɚ Ⱥ

– ɜ ɬɪɟɯ ɜɚɪɢɚɧɬɚɯ ɧɚɝɪɭɡɤɢ ɧɚ ɪɢɫ.8.3). Ⱦɨɫɬɚɬɨɱɧɨ ɡɚɩɢɫɚɬɶ ɭɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ ɞɥɹ ɫɦɟɠɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɫɢɫɬɟɦɵ, ɱɬɨɛɵ ɩɨɥɭɱɢɬɶ ɡɧɚɱɟɧɢɟ G*.

		ɉɪɢɦɟɪ 1 (ɪɢɫ.8.4) [4]. Ɍɪɟɛɭɟɬɫɹ ɢɫɫɥɟɞɨɜɚɬɶ ɭɫɬɨɣɱɢɜɨɫɬɶ
P	P	ɫɢɫɬɟɦɵ, ɫɨɫɬɨɹ-
		ɳɟɣ ɢɡ ɚɛɫɨɥɸɬɧɨ ɠɟɫɬɤɨɝɨ ɫɬɟɪɠɧɹ ɞɥɢɧɨɣ l ɢ ɫɜɹɡɚɧɧɨɣ ɫ
		ɧɢɦ ɩɪɭɠɢɧɵ ɠɟɫɬɤɨɫɬɢ ɫ. ɇɚ ɫɜɨɛɨɞɧɵɣ ɤɨɧɟɰ ɫɬɟɪɠɧɹ ɞɟɣɫɬɜɭɟɬ
l		ɫɢɥɚ Ɋ. ɉɪɢ ɩɟɪɟɯɨɞɟ ɫɢɫɬɟɦɵ ɜ ɫɦɟɠɧɨɟ ɩɨɥɨɠɟɧɢɟ ɫɢɥɚ ɧɟ ɦɟɧɹɟɬ
		ɫɜɨɟɝɨ ɧɚɩɪɚɜɥɟɧɢɹ.
	A	Ɋɟɲɟɧɢɟ. ɂɫɩɨɥɶɡɭɟɦ ɜɧɚɱɚɥɟ ɷɧɟɪɝɟɬɢɱɟɫɤɢɣ ɫɩɨɫɨɛ: ɧɚɣɞɟɦ
ɋ		ɡɚɜɢɫɢɦɨɫɬɶ ɩɨɥɧɨɣ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɨɬ ɭɝɥɚ ɩɨɜɨ-
Ɋɢɫ. 8.4		ɪɨɬɚ ɫɬɟɪɠɧɹ M :
		ɉ = ɫM 2/2+PlcosM .

Ɂɞɟɫɶ ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ – ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɞɟɮɨɪɦɚɰɢɢ, ɡɚɩɚɫɚɟɦɚɹ ɩɪɭɠɢɧɨɣ ɩɪɢ ɩɨɜɨɪɨɬɟ ɫɬɟɪɠɧɹ ɧɚ ɭɝɨɥ M, ɜɬɨɪɨɟ – ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ «ɝɪɭɡɚ» P, ɟɫɥɢ ɡɚ ɟɟ ɧɭɥɟɜɨɣ ɭɪɨɜɟɧɶ ɩɪɢɧɹɬɶ ɫɨɫɬɨɹɧɢɟ ɩɪɢ M = S/2. ɍɞɨɛɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɨɬɧɨɫɢɬɟɥɶɧɵɟ ɩɚɪɚɦɟɬɪɵ:

ɉc = M 2/2 + Pc cosM,

(8.1)

ɝɞɟ ɉc – ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɉ/ɫ, Pc – ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɧɚɝɪɭɡɤɚ Pl/c. Ƚɪɚɮɢɤ ɮɭɧɤɰɢɢ ɉc(M) (8.1) ɩɪɟɞɫɬɚɜɥɟɧ ɧɚ ɪɢɫ.8.5. Ʉɪɢɜɵɟ 1, 2 ɢ 3 ɨɬɜɟɱɚɸɬ ɡɧɚɱɟɧɢɹɦ Pc, ɪɚɜɧɵɦ 1, 1.09 ɢ 1.2. ɉɪɢ Pc =1 (ɤɪɢɜɚɹ 1) ɪɚɜɧɨɜɟɫɢɟ ɨɬɜɟɱɚɟɬ ɡɧɚɱɟɧɢɸ M = 0; ɷɬɨ ɩɨɥɨɠɟɧɢɟ ɪɚɜɧɨɜɟɫɢɹ ɭɫɬɨɣɱɢɜɨ. ɉɪɢ Pc >1 ɪɚɜɧɨɜɟɫɧɵɦɢ ɹɜɥɹɸɬɫɹ ɞɜɚ ɫɨɫɬɨɹɧɢɹ, ɩɪɢɱɟɦ ɫɨɫɬɨɹɧɢɟ M = 0 ɧɟɭɫɬɨɣɱɢɜɨ, ɚ ɨɬɤɥɨɧɟɧɧɨɟ (ɦɢɧɢɦɭɦ ɉc ɫɦɟɳɚɟɬɫɹ ɩɨ ɨɫɢ M ɧɚ ɪɢɫ.8.5 ɜɩɪɚɜɨ) – ɭɫɬɨɣɱɢɜɨ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ,

ɉ /					ɤɪɢɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ	Pc ɪɚɜɧɨ ɟɞɢɧɢɰɟ; ɫɨɨɬɜɟɬ-
1.24					ɫɬɜɟɧɧɨ, P*=c/l.
		3			Ɇɟɬɨɞɨɦ ɗɣɥɟɪɚ	ɷɬɚ ɡɚɞɚɱɚ ɪɟɲɚɟɬɫɹ ɱɭɬɶ
					ɩɪɨɳɟ. ɍɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ ɞɥɹ ɫɬɟɪɠɧɹ, ɩɨɜɟɪɧɭ-
1.12		2			ɬɨɝɨ ɧɚ ɭɝɨɥ M (ɫɭɦɦɚ ɦɨɦɟɧɬɨɜ ɫɢɥ, ɞɟɣɫɬɜɭɸ-
					ɳɢɯ ɧɚ ɠɟɫɬɤɢɣ ɫɬɟɪɠɟɧɶ, ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ A –
					ɪɢɫ.8.4 – ɞɨɥɠɧɚ ɛɵɬɶ ɪɚɜɧɚ ɧɭɥɸ) ɢɦɟɟɬ ɜɢɞ
1.0		1
					PlsinM = cM .
				,ɪɚɞ
			1.2
0		0.6			Ɉɬɫɸɞɚ, ɟɫɬɟɫɬɜɟɧɧɨ, ɩɨɥɭɱɚɟɦ ɬɨ ɠɟ ɡɧɚɱɟ-

Ɋɢɫ.8.5

ɧɢɟ ɤɪɢɬɢɱɟɫɤɨɣ ɫɢɥɵ P* ɩɪɢ ɭɫɥɨɜɢɢ ɦɚɥɨɫɬɢ ɭɝ-

ɥɚ M (sinM # M).

ɇɚɡɜɚɧɧɵɟ ɦɟɬɨɞɵ ɨɩɪɟɞɟɥɟɧɢɹ ɤɪɢɬɢɱɟɫɤɨɣ ɧɚɝɪɭɡɤɢ ɦɨɠɧɨ ɩɪɢɦɟɧɹɬɶ ɢ ɞɥɹ ɫɬɟɪɠɧɟɣ, ɧɟ ɩɨɞɜɟɪɠɟɧɧɵɯ ɫɠɚɬɢɸ.