Моделирование предохранительных конструкций. Учебное пособие
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ǵȘȔȕȉȢ ȗȇȘȞȌșȇ ȖȗȌȋȕȜȗȇȔȏșȌȒȣȔȢȜ ȑȕȔȘșȗȚȑȝȏȐ
3.2.1. ǴȌȗȇȎȗȚȟȇȌȓȢȌ ȖȗȌȋȕȜȗȇȔȏșȌȒȣȔȢȌ ȑȕȔȘșȗȚȑȝȏȏ ȖȗȕȏȎȉȕȋȘșȉȌȔȔȢȜ ȖȕȓȌȠȌȔȏȐ
ɉɪɢ ɧɟɩɪɟɞɧɚɦɟɪɟɧɧɵɯ ɩɪɨɦɵɲɥɟɧɧɵɯ ɜɡɪɵɜɚɯ ɜ ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɵɯ ɩɨɦɟɳɟɧɢɹɯ ɫɭɳɟɫɬɜɟɧɧɨ ɫɧɢɡɢɬɶ ɞɚɜɥɟɧɢɟ ɜ ɩɨɦɟɳɟɧɢɢ ɦɨɠɧɨ ɡɚ ɫɱɟɬ ɭɜɟ ɥɢɱɟɧɢɹ ɨɛɳɟɣ ɩɥɨɳɚɞɢ ɩɪɨɟɦɨɜ ɜ ɨɝɪɚɠɞɚɸɳɢɯ ɤɨɧɫɬɪɭɤɰɢɹɯ ɩɨɫɤɨɥɶɤɭ ɷɬɨ ɩɪɢɜɨɞɢɬ ɤ ɭɜɟɥɢɱɟɧɢɸ ɢɫɬɟɱɟɧɢɹ ɩɪɨɞɭɤɬɨɜ ɫɝɨɪɚɧɢɹ
Ⱦɥɹ ɨɬɜɨɞɚ ɷɧɟɪɝɢɢ ɜ ɩɪɨɰɟɫɫɟ ɝɨɪɟɧɢɹ ɜ ɨɝɪɚɠɞɟɧɢɹɯ ɩɨɦɟɳɟɧɢɹ ɩɪɟɞɭɫɦɚɬɪɢɜɚɟɬɫɹ ɧɟɨɛɯɨɞɢɦɨɟ ɤɨɥɢɱɟɫɬɜɨ ɨɬɜɟɪɫɬɢɣ ɤɨɬɨɪɵɟ ɞɨɥɠɧɵ ɨɛɟɫɩɟɱɢɬɶ ɩɪɨɩɭɫɤ ɬɪɟɛɭɟɦɨɝɨ ɤɨɥɢɱɟɫɬɜɚ ɤɚɤ ɫɝɨɪɟɜɲɟɝɨ ɬɚɤ ɢ ɯɨɥɨɞɧɨɝɨ ɝɚɡɚ ɗɬɢ ɨɬɜɟɪɫɬɢɹ ɧɚɡɵɜɚɸɬɫɹ ɫɛɪɨɫɧɵɦɢ ɫɛɪɨɫɨɜɵɦɢ ɚ ɭɫɬɪɨɣɫɬɜɚ ɢɯ ɡɚ ɤɪɵɜɚɸɳɢɟ ɩɪɟɞɨɯɪɚɧɢɬɟɥɶɧɵɦɢ ɤɨɧɫɬɪɭɤɰɢɹɦɢ ɉɄ
ɉɪɟɞɨɯɪɚɧɢɬɟɥɶɧɵɟ ɤɨɧɫɬɪɭɤɰɢɢ ɜɫɤɪɵɜɚɸɬɫɹ ɩɪɢ ɫɪɚɜɧɢɬɟɥɶɧɨ ɧɟ ɛɨɥɶɲɨɦ ɢɡɛɵɬɨɱɧɨɦ ɞɚɜɥɟɧɢɢ 'Pɜɫɤ ɢ ɬɟɦ ɫɚɦɵɦ ɨɛɟɫɩɟɱɢɜɚɸɬ ɜɨɡɦɨɠɧɨɫɬɶ ɢɧɬɟɧɫɢɜɧɨɝɨ ɢɫɬɟɱɟɧɢɹ ɝɚɡɚ ɩɪɨɞɭɤɬɨɜ ɝɨɪɟɧɢɹ ɢ ɧɟɩɪɨɪɟɚɝɢɪɨɜɚɜɲɟɣ ɱɚɫɬɢ ɝɚɡɨɜɨɣ ɫɦɟɫɢ Ƚɋ ɱɟɪɟɡ ɨɛɪɚɡɨɜɚɜɲɢɟɫɹ ɩɪɨɟɦɵ ɢɡ ɩɨɦɟɳɟɧɢɹ ɜ ɧɚɪɭɠɧɭɸ ɚɬɦɨɫɮɟɪɭ ɂɫɬɟɱɟɧɢɟ ɝɚɡɚ ɜ ɚɬɦɨɫɮɟɪɭ ɩɪɢɜɨɞɢɬ ɤ ɫɧɢɠɟɧɢɸ ɞɚɜɥɟɧɢɹ ɜ ɩɨ ɦɟɳɟɧɢɢ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɝɨɪɟɧɢɟɦ Ƚɋ ɜ ɡɚɦɤɧɭɬɨɦ ɨɛɴɟɦɟ ɫɦ ɪɢɫ
ɋɬɟɩɟɧɶ ɫɧɢɠɟɧɢɹ ɞɚɜɥɟɧɢɹ ɡɚɜɢɫɢɬ ɨɬ ɩɥɨɳɚɞɢ ɉɄ ɡɚɤɨɧɨɦɟɪɧɨɫɬɟɣ ɢɯɜɫɤɪɵɬɢɹ ɜɢɞɚȽɋ ɯɚɪɚɤɬɟɪɚɡɚɝɚɡɨɜɚɧɧɨɫɬɢ ɩɨɦɟɳɟɧɢɹ ɟɝɨɨɛɴɟɦɧɨ ɩɥɚ ɧɢɪɨɜɨɱɧɨɝɨ ɪɟɲɟɧɢɹ ɢ ɞɪɭɝɢɯ ɮɚɤɬɨɪɨɜ
ɉɨ ɤɨɧɫɬɪɭɤɬɢɜɧɵɦ ɪɟɲɟɧɢɹɦ ɉɄ ɞɟɥɹɬɫɹ ɧɚ ɪɚɡɪɭɲɚɟɦɵɟ ɢ ɧɟɪɚɡɪɭ ɲɚɟɦɵɟ
ɇɟɪɚɡɪɭɲɚɟɦɵɟ ɉɄ ɞɨɥɠɧɵ ɩɨɥɧɨɫɬɶɸ ɫɨɯɪɚɧɹɬɶɫɹ ɩɨɫɥɟ ɜɫɤɪɵɬɢɹ ɤɨɬɨɪɨɟ ɩɪɨɢɫɯɨɞɢɬ ɜ ɪɟɡɭɥɶɬɚɬɟ ɫɪɚɛɚɬɵɜɚɧɢɹ ɫɩɟɰɢɚɥɶɧɵɯ ɤɪɟɩɟɠɧɵɯ ɭɫɬɪɨɣɫɬɜ ɇɟɪɚɡɪɭɲɚɟɦɵɟ ɉɄ ɜ ɫɜɨɸ ɨɱɟɪɟɞɶ ɦɨɝɭɬ ɛɵɬɶ ɪɚɡɞɟɥɟɧɵ ɧɚ ɜɪɚɳɚɸɳɢɟɫɹ ɟɫɥɢ ɨɛɪɚɡɨɜɚɧɢɟ ɩɪɨɟɦɨɜ ɜ ɧɚɪɭɠɧɨɦ ɨɝɪɚɠɞɟɧɢɢ ɩɪɨɢɫɯɨ ɞɢɬ ɜ ɪɟɡɭɥɶɬɚɬɟ ɩɨɜɨɪɨɬɚ ɩɪɟɞɨɯɪɚɧɢɬɟɥɶɧɨɣ ɤɨɧɫɬɪɭɤɰɢɢ ɢ ɫɦɟɳɚɸɳɢ ɟɫɹ7 ɟɫɥɢ ɨɛɪɚɡɨɜɚɧɢɟ ɩɪɨɟɦɚ ɜ ɧɚɪɭɠɧɨɦ ɨɝɪɚɠɞɟɧɢɢ ɩɪɨɢɫɯɨɞɢɬ ɜɫɥɟɞ ɫɬɜɢɟ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɩɪɟɞɨɯɪɚɧɢɬɟɥɶɧɨɣ ɤɨɧɫɬɪɭɤɰɢɢ
ɇɚɱɚɥɨ ɞɜɢɠɟɧɢɹ ɧɟɪɚɡɪɭɲɚɸɳɢɯɫɹ ɉɄ ɨɛɭɫɥɚɜɥɢɜɚɟɬɫɹ ɫɪɚɛɚɬɵɜɚ ɧɢɟɦ ɤɨɬɨɪɨɟ ɧɟ ɢɫɤɥɸɱɚɟɬ ɢ ɪɚɡɪɭɲɟɧɢɹ ɤɪɟɩɟɠɧɵɯ ɭɫɬɪɨɣɫɬɜ ɛɥɚɝɨɞɚɪɹ ɤɨɬɨɪɵɦ ɩɪɟɞɨɯɪɚɧɢɬɟɥɶɧɚɹ ɤɨɧɫɬɪɭɤɰɢɹ ɧɚɯɨɞɢɬɫɹ ɜ ɡɚɤɪɵɬɨɦ ɩɨɥɨɠɟɧɢɢ Ɂɚɤɨɧɨɦɟɪɧɨɫɬɢ ɜɫɤɪɵɬɢɹ ɧɟɪɚɡɪɭɲɚɟɦɵɯ ɉɄ ɨɩɪɟɞɟɥɹɸɬɫɹ ɯɚɪɚɤɬɟ ɪɨɦ ɢɯ ɞɜɢɠɟɧɢɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɧɚɝɪɭɡɤɢ ɜɨɡɧɢɤɚɸɳɟɣ ɩɪɢ ɜɡɪɵɜɧɨɦ ɝɨɪɟ
ɧɢɢ ɜ ɩɨɦɟɳɟɧɢɢ ȼ ɜɢɞɟ ɧɟɪɚɡɪɭɲɚɟɦɵɯ ɉɄ ɦɨɝɭɬ ɛɵɬɶ ɜɵɩɨɥɧɟɧɵ ɨɤɨɧɧɵɟ ɩɟɪɟɩɥɟɬɵ ɫ
ɨɬɤɪɵɜɚɸɳɢɦɢɫɹ ɫɬɜɨɪɤɚɦɢ ɋɬɜɨɪɤɢ ɤɪɟɩɹɬɫɹ ɤ ɨɤɨɧɧɵɦ ɪɚɦɚɦ ɜɟɪɬɢɤɚɥɶ
7 Ɋɟɚɥɢɡɭɟɬɫɹ ɩɪɹɦɨɥɢɧɟɣɧɨɟ ɩɨɫɬɭɩɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɉɄ ɫ ɨɛɪɚɡɨɜɚɧɢɟɦ ɩɪɢ ɭɞɚɥɟɧɢɢ ɉɄ ɨɬ ɨɝɪɚɠɞɚɸɳɢɯ ɤɨɧɫɬɪɭɤɰɢɣ ɪɚɫɱɟɬɧɨɝɨ ɨɬɜɟɪɫɬɢɹ ɜ ɧɚɪɭɠɧɨɦ ɨɝɪɚɠɞɟɧɢɢ ɪɚɜɧɨɝɨ ɩɥɨɳɚɞɢ ɩɪɨɟɦɚ
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ɧɵɦ ɢɥɢ ɝɨɪɢɡɨɧɬɚɥɶɧɵɦ ɜɟɪɯɧɢɦ ɢɥɢ ɧɢɠɧɢɦ ɲɚɪɧɢɪɨɦ ɤ ɫɩɟɰɢɚɥɶɧɵɦ ɡɚɩɨɪɧɵɦ ɭɫɬɪɨɣɫɬɜɚɦ ɭɞɟɪɠɢɜɚɸɳɢɦ ɫɬɜɨɪɤɢ ɜ ɡɚɤɪɵɬɨɦ ɫɨɫɬɨɹɧɢɢ ɋɪɚ ɛɚɬɵɜɚɧɢɟ ɡɚɩɨɪɧɵɯ ɭɫɬɪɨɣɫɬɜ ɢ ɨɬɤɪɵɜɚɧɢɟ ɫɬɜɨɪɨɤ ɞɨɥɠɧɨ ɩɪɨɢɫɯɨɞɢɬɶ ɩɪɢ ɢɡɛɵɬɨɱɧɨɦ ɞɚɜɥɟɧɢɢ ɡɧɚɱɢɬɟɥɶɧɨ ɦɟɧɶɲɟ ɬɨɝɨ ɤɨɬɨɪɨɟ ɦɨɠɟɬ ɜɵɡɜɚɬɶ ɪɚɡɪɭɲɟɧɢɟ ɫɬɟɤɨɥ ɭɫɬɚɧɚɜɥɢɜɚɟɦɵɯ ɜ ɩɟɪɟɩɥɟɬɚɯ
W ɜɪɟɦɹ ɮɚɡɵ ɫɠɚɬɢɹ W | 5t ɝɪɚɮɢɤ ɢɡɦɟɧɟɧɢɹ ɞɚɜɥɟɧɢɹ ɞɥɹ ɩɨɦɟɳɟɧɢɣ ɛɟɡ ɩɪɨɟɦɨɜ ɝɪɚɮɢɤ ɢɡɦɟɧɟɧɢɹ ɞɚɜɥɟɧɢɹ ɞɥɹ ɩɨɦɟɳɟɧɢɣ F ɩɪɨɟɦɚɦɢ ɝɪɚɮɢɤ ɢɡɦɟɧɟɧɢɹ ɞɚɜɥɟɧɢɹ ɞɥɹ ɩɨɦɟɳɟɧɢɣ ɫ ɉɄ
'Ɋɞɨɩ ɛɟɡɨɩɚɫɧɵɣ ɭɪɨɜɟɧɶ ɞɚɜɥɟɧɢɹ ɞɥɹ ɥɸɞɟɣ ɢ ɤɨɧɫɬɪɭɤɰɢɣ ɜ ɩɨɦɟɳɟɧɢɢ 'Pɜɫɤ ɞɚɜɥɟɧɢɟ ɩɪɢ ɤɨɬɨɪɨɦ ɨɬɤɪɵɜɚɸɬɫɹ ɩɟɪɟɦɟɳɚɸɬɫɹ ɢɥɢ ɪɚɡɪɭɲɚɸɬɫɹ ɉɄ
ǷȏȘȚȔȕȑ 3 ǼȇȗȇȑșȌȗ ȏȎȓȌȔȌȔȏȦ ȋȇȉȒȌȔȏȦ ȉȔȚșȗȏ ȖȕȓȌȠȌȔȏȦ ȉ ȎȇȉȏȘȏȓȕȘșȏ ȕș ȉȗȌȓȌȔȏ ȊȕȗȌȔȏȦ ǪǸ
Ⱦɜɟɪɢ ɢ ɜɨɪɨɬɚ ɭɫɬɪɚɢɜɚɟɦɵɟ ɜ ɧɚɪɭɠɧɨɦ ɨɝɪɚɠɞɟɧɢɢ ɩɨɦɟɳɟɧɢɣ ɦɨ ɝɭɬ ɢɫɩɨɥɶɡɨɜɚɬɶɫɹ ɜ ɤɚɱɟɫɬɜɟ ɉɄ ɪɚɛɨɬɚɸɳɢɯ ɩɨɬɨɦɭ ɠɟ ɩɪɢɧɰɢɩɭ ɱɬɨ ɢ ɨɬɤɪɵɜɚɸɳɢɟɫɹ ɫɬɜɨɪɤɢ ɧɟɪɚɡɪɭɲɚɟɦɵɯ ɉɄ
ɂɫɯɨɞɹ ɢɡ ɡɧɚɱɟɧɢɹ ɦɚɫɫɵ ɜ ɡɚɤɨɧɨɦɟɪɧɨɫɬɹɯ ɜɫɤɪɵɬɢɹ ɉɄ ɦɨɝɭɬ ɛɵɬɶ ɭɫɥɨɜɧɨ ɪɚɡɞɟɥɟɧɵ ɧɚ ɢɧɟɪɰɢɨɧɧɵɟ ɢ ɛɟɡɵɧɟɪɰɢɨɧɧɵɟ Ʉ ɢɧɟɪɰɢɨɧɧɵɦ ɨɬɧɨ ɫɹɬɫɹɬɚɤɢɟɉɄ ɦɚɫɫɚɤɨɬɨɪɵɯɨɤɚɡɵɜɚɟɬɩɪɚɤɬɢɱɟɫɤɨɟɜɥɢɹɧɢɹɧɚɡɚɤɨɧɨɦɟɪ ɧɨɫɬɢ ɢɯ ɜɫɤɪɵɬɢɹ ɢ ɩɨɬɨɦɭ ɞɨɥɠɧɚ ɭɱɢɬɵɜɚɬɶɫɹ ɩɪɢ ɪɚɫɱɟɬɚɯ ȿɫɥɢ ɜɥɢɹɧɢɟ ɦɚɫɫɵ ɧɚ ɡɚɤɨɧɨɦɟɪɧɨɫɬɢ ɜɫɤɪɵɬɢɹ ɉɄ ɧɚɫɬɨɥɶɤɨ ɧɟɡɧɚɱɢɬɟɥɶɧɨ ɱɬɨ ɟɸ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɬɨ ɉɄ ɫɱɢɬɚɸɬɫɹ ɛɟɡɵɧɟɪɰɢɨɧɧɵɦɢ
ɇɚɡɚɤɨɧɨɦɟɪɧɨɫɬɢ ɜɫɤɪɵɬɢɹ ɢɧɟɪɰɢɨɧɧɵɯ ɉɄ ɦɨɠɟɬ ɫɭɳɟɫɬɜɟɧɧɨ ɜɥɢ ɹɬɶ ɢɯ ɩɨɥɨɠɟɧɢɟ ɜ ɨɝɪɚɠɞɚɸɳɢɯ ɤɨɧɫɬɪɭɤɰɢɹɯɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɩɨɜɟɪɯɧɨɫɬɢ ɡɟɦɥɢ ȼ ɫɜɹɡɢ ɫ ɷɬɢɦ ɫɥɟɞɭɟɬ ɪɚɡɥɢɱɚɬɶ ɜɟɪɬɢɤɚɥɶɧɵɟ ɧɚɤɥɨɧɧɵɟ ɢ ɝɨɪɢɡɨɧ ɬɚɥɶɧɵɟ ɉɄ
Ⱦɥɹ ɢɡɝɨɬɨɜɥɟɧɢɹ ɉɄ ɩɪɢɦɟɧɹɟɬɫɹ ɪɚɡɥɢɱɧɵɣ ɫɬɪɨɢɬɟɥɶɧɵɣ ɦɚɬɟɪɢɚɥ ȼ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɤɨɧɫɬɪɭɤɬɢɜɧɨɝɨ ɪɟɲɟɧɢɹ ɲɢɪɨɤɨ ɢɫɩɨɥɶɡɭɸɬ ɫɬɟɤɥɨ ɩɚ ɧɟɥɢ ɬɢɩɚ ©ɋɗɇȾȼɂɑª ɜɵɩɨɥɧɟɧɧɵɟ ɜ ɜɢɞɟ ɫɥɨɟɜ ɢɡ ɨɛɥɟɝɱɟɧɧɨɝɨ ɦɟɬɚɥɥɚ ɫ ɩɟɧɨɩɥɚɫɬɨɦ
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ȼɵɩɨɥɧɟɧɢɟ ɢɧɟɪɰɢɨɧɧɵɯ ɉɄ ɜ ɜɢɞɟ ɜɪɚɳɚɸɳɢɯɫɹ ɩɥɢɬ ɹɜɥɹɟɬɫɹ ɬɟɯ ɧɢɱɟɫɤɢ ɫɥɨɠɧɨɣ ɡɚɞɚɱɟɣ ɬɚɤ ɤɚɤ ɤɚɠɞɚɹ ɩɚɧɟɥɶ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɦɟɯɚɧɢɡɦ ɱɬɨ ɡɧɚɱɢɬɟɥɶɧɨ ɭɞɨɪɨɠɚɟɬ ɫɬɪɨɢɬɟɥɶɫɬɜɨ
ɒɚɪɧɢɪɵ ɤɨɬɨɪɵɟ ɨɛɟɫɩɟɱɢɜɚɸɬ ɩɨɜɨɪɨɬ ɜɪɚɳɚɸɳɢɯɫɹ ɉɄ ɜ ɨɬɥɢɱɢɟ ɨɬ ɞɪɭɝɢɯ ɤɪɟɩɟɠɧɵɯ ɭɫɬɪɨɣɫɬɜ ɪɚɫɫɱɢɬɵɜɚɸɬɫɹ ɧɚ ɩɪɨɱɧɨɫɬɶ Ɉɧɢ ɧɟ ɞɨɥɠɧɵ ɪɚɡɪɭɲɚɬɶɫɹ ɩɪɢ ɜɫɤɪɵɬɢɢ ɉɄ. ɋɯɟɦɵ ɜɪɚɳɚɸɳɢɯɫɹ ɉɄ ɪɚɡɥɢɱɚɸɬ ɩɨ ɩɨɥɨɠɟɧɢɸ ɨɫɢ ɩɨɜɨɪɨɬɚ ɤɨɧɫɬɪɭɤɰɢɢ ɝɨɪɢɡɨɧɬɚɥɶɧɚɹ ɨɫɶ ɩɨɜɨɪɨɬɚ ɜɟɪ ɬɢɤɚɥɶɧɚɹ ɨɫɶ ɩɨɜɨɪɨɬɚ .
3.2.2. dzȌșȕȋȏȑȇ ȗȇȘȞȌșȇ ȖȕȉȕȗȕșȔȕȐ ȔȌȗȇȎȗȚȟȇȌȓȕȐ ȖȗȌȋȕȜȗȇȔȏșȌȒȣȔȕȐ ȑȕȔȘșȗȚȑȝȏȏ
Ɋɚɫɱɟɬ ɉɄ ɜɤɥɸɱɚɟɬ ɜ ɫɟɛɹ ɨɩɪɟɞɟɥɟɧɢɟ ɪɚɡɦɟɪɨɜ ɢ ɤɨɥɢɱɟɫɬɜɚ ɉɄ
Ɉɩɪɟɞɟɥɟɧɢɟ ɩɥɨɳɚɞɢ ɜɫɤɪɵɬɢɹ
Ɍɪɟɛɭɟɦɚɹ ɩɥɨɳɚɞɶ ɩɪɨɟɦɨɜ ɩɪɢ ɭɫɬɪɨɣɫɬɜɟ ɤɨɬɨɪɵɯ ɞɚɜɥɟɧɢɟ ɩɪɢ ɞɟ ɮɥɚɝɪɚɰɢɨɧɧɨɦ ɜɡɪɵɜɟ ɜ ɩɨɦɟɳɟɧɢɢ ɛɭɞɟɬ ɧɟ ɛɨɥɟɟ ɞɨɩɭɫɬɢɦɨɝɨ ɡɧɚɱɟɧɢɹ 'Pɞɨɩ ɞɥɹ ɩɪɚɤɬɢɱɟɫɤɢɯ ɪɚɫɱɟɬɨɜ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ ɤɨɬɨɪɚɹ ɫ ɞɨɫɬɚ ɬɨɱɧɨɣ ɬɨɱɧɨɫɬɶɸ ɨɩɪɟɞɟɥɹɟɬ ɩɥɨɳɚɞɶ ɜɫɤɪɵɬɢɹ ɩɪɢ ɩɨɥɧɨɣ ɡɚɝɚɡɨɜɚɧɧɨɫɬɢ ɩɨɦɟɳɟɧɢɹ
ɉɪɢɛɥɢɠɟɧɧɨɟɭɪɚɜɧɟɧɢɟɞɥɹɨɩɪɟɞɟɥɟɧɢɹɩɥɨɳɚɞɢɜɫɤɪɵɬɢɹɢɦɟɟɬɜɢɞ
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4 Vɪɚɫɱ aUɧ H 1 Uɫɬɯ |
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ɝɞɟ Vɪɚɫɱ ®ɞɥɹVɉɪȼɋ |
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ɉɥȼɋ |
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D ɤɨɷɮɮɢɰɢɟɧɬ ɢɧɬɟɧɫɢɮɢɤɚɰɢɢ ɝɨɪɟɧɢɹ
Uɧ ɫɤɨɪɨɫɬɶɝɨɪɟɧɢɹɩɪɨɞɭɤɬɨɜɫɝɨɪɚɧɢɹɫɬɟɯɢɨɦɟɬɪɢɱɟɫɤɨɣ ɫɦɟɫɢ ɦ ɫ; H ɫɬɟɩɟɧɶ ɪɚɫɲɢɪɟɧɢɹ ɩɪɨɞɭɤɬɨɜ ɫɝɨɪɚɧɢɹ;
'Pɞɨɩ ɞɨɩɭɫɬɢɦɨɟ ɞɚɜɥɟɧɢɟ ɜ ɩɨɦɟɳɟɧɢɢ
Uɫɬɯ ɩɥɨɬɧɨɫɬɶ ɝɚɡɨɜ ɢɫɬɟɤɚɸɳɢɯ ɢɡ ɩɪɨɟɦɨɜ ɤɝ ɦ .
Ɉɬɧɨɫɢɬɟɥɶɧɨ ɩɥɨɬɧɨɫɬɢ ɢɫɬɟɤɚɸɳɢɯ ɢɡ ɩɪɨɟɦɨɜ ɝɚɡɨɜ
xɞɥɹ ɩɨɦɟɳɟɧɢɣ ɥɢɧɟɣɧɵɟ ɪɚɡɦɟɪɵ ɤɨɬɨɪɵɯ H B L ɛɥɢɡɤɢ ɞɪɭɝ ɤ ɞɪɭɝɭ Uɫɬɯ ɩɪɢɧɢɦɚɸɬ ɪɚɜɧɨɣ ɩɥɨɬɧɨɫɬɢ ɞɥɹ ɯɨɥɨɞɧɨɣ ɫɦɟɫɢ Uɫɦ
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xɟɫɥɢ ɫɛɪɨɫɧɵɟ ɨɬɜɟɪɫɬɢɹ ɫɨɫɪɟɞɨɬɨɱɟɧɵ ɜ ɨɞɧɨɦ ɤɨɧɰɟ ɜɵɬɹɧɭɬɨɝɨ ɩɨɦɟɳɟɧɢɹ ɬɨ ɩɪɢ ɡɚɠɢɝɚɧɢɢ Ƚȼɋ ɜɛɥɢɡɢ ɨɬɜɟɪɫɬɢɣ
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Uɫɬɯ Uɫɦ H |
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ɩɪɢ ɡɚɠɢɝɚɧɢɢ ɭ ɝɥɭɯɨɝɨ ɤɨɧɰɚ
Uɫɬɯ Uɫɦ ;
xɜ ɫɥɭɱɚɟ ɜɵɬɹɧɭɬɵɯ ɩɨɦɟɳɟɧɢɣ ɢ ɪɚɜɧɨɦɟɪɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɩɥɨ ɳɚɞɢ ɩɪɨɟɦɨɜ ɧɚ ɧɚɪɭɠɧɨɦ ɨɝɪɚɠɞɟɧɢɢ ɱɟɪɟɡ ɧɟɤɨɬɨɪɵɟ ɨɬɜɟɪɫɬɢɹ ɛɭɞɭɬ ɢɫɬɟɤɚɬɶ ɩɪɨɞɭɤɬɵ ɫɝɨɪɚɧɢɹ ɫ ɩɥɨɬɧɨɫɬɶɸ ȡɫɦ İ ɚ ɱɟɪɟɡ ɞɪɭ
ɝɢɟ ɯɨɥɨɞɧɚɹ ɫɦɟɫɶ ɫ ɩɥɨɬɧɨɫɬɶɸ ȡɫɦ ɉɪɢ ɷɬɨɦ ɪɚɫɱɟɬɧɚɹ ɷɮɮɟɤ ɬɢɜɧɚɹ ɩɥɨɬɧɨɫɬɶ ɝɚɡɨɜ ȡ ɤɝ ɦ3 ɦɨɠɟɬ ɛɵɬɶ ɨɰɟɧɟɧɚ ɡɚɜɢɫɢɦɨɫɬɶɸ
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ɫɦ |
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ȼ ɪɚɦɤɚɯ ɪɟɲɚɟɦɨɣ ɡɚɞɚɱɢ ɩɪɢɧɢɦɚɟɦ Uɫɬɯ = Uɫɦ.
ɉɪɢ ɛɨɥɶɲɨɣ ɜɵɬɹɧɭɬɨɫɬɢ ɩɨɦɟɳɟɧɢɹ ɩɪɨɰɟɫɫ ɢɡ ɞɟɮɥɚɝɪɚɰɢɨɧɧɨɝɨ ɩɟɪɟɪɚɫɬɚɟɬ ɜ ɞɟɬɨɧɚɰɢɨɧɧɵɣ ɤɨɬɨɪɵɣ ɨɩɢɫɵɜɚɟɬɫɹ ɭɠɟ ɞɪɭɝɢɦɢ ɭɪɚɜɧɟɧɢ ɹɦɢ
Ɉɩɪɟɞɟɥɟɧɢɟ ɪɚɡɦɟɪɨɜ ɉɄ
ɂɧɟɪɰɢɨɧɧɵɟ ɉɄ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɩɥɢɬɵ ɪɚɡɦɟɪɨɦ O ǜ b ɜɵɩɨɥ ɧɟɧɧɵɟ ɢɡ ɫɪɚɜɧɢɬɟɥɶɧɨ ɥɟɝɤɨɝɨ ɦɚɬɟɪɢɚɥɚ ɦɚɫɫɨɣ ɧɚ ɟɞɢɧɢɰɭ ɩɥɨɳɚɞɢ P … ɤɝ ɦ Ɉɧɢ ɩɟɪɟɤɪɵɜɚɸɬ ɩɪɨɟɦɵ ɧɚ ɜɟɪɬɢɤɚɥɶɧɵɯ ɧɚɪɭɠɧɵɟ ɫɬɟɧɵ ɢɥɢ ɝɨɪɢɡɨɧɬɚɥɶɧɵɯ ɨɝɪɚɠɞɟɧɢɹɯ ɩɨɤɪɵɬɢɹ ɡɞɚɧɢɣ ɉɪɢ ɷɬɨɦ ɨɧɢ ɢɦɟɸɬ ɤɪɟɩɥɟɧɢɟ ɤɨɬɨɪɨɟ ɞɨɥɠɧɨ ɪɚɡɪɭɲɚɬɶɫɹ ɬɨ ɟɫɬɶ ɉɄ ɞɨɥɠɧɵ ɧɚɱɚɬɶ ɜɫɤɪɵɜɚɬɶɫɹ ɩɪɢ ɞɚɜɥɟɧɢɢ
'Pɜɫɤ < 'Pɞɨɩ.
Ɉɛɵɱɧɨ 'Pɜɫɤ … ɤɉɚ ɚ 'Pɞɨɩ …5 ɤɉɚ Ⱦɨ ɦɨɦɟɧɬɚ ɜɫɤɪɵɬɢɹ ɞɚɜɥɟɧɢɟɜɩɨɦɟɳɟɧɢɢɢɡɦɟɧɹɟɬɫɹɤɚɤɜɡɚɦɤɧɭɬɨɦɨɛɴɟɦɟ ɉɨɫɥɟɪɚɡɪɭɲɟɧɢɹ ɤɪɟɩɥɟɧɢɹ ɩɪɢ 'Pɜɫɤ ɧɟɨɛɯɨɞɢɦɨ ɱɬɨ ɛɵ ɬɟɤɭɳɟɟ ɢɡɛɵɬɨɱɧɨɟ ɞɚɜɥɟɧɢɟ 'P t ɧɟɩɪɟɜɵɲɚɥɨ'Pɞɨɩ ɫɦ ɪɢɫ ɗɬɨɞɨɫɬɢɠɢɦɨ ɟɫɥɢɤɦɨɦɟɧɬɭɨɛɪɚɡɨɜɚɧɢɹ ɫɩɥɨɲɧɨɝɨ ɩɪɨɟɦɚ tɫɩ ɩɪɢɪɨɫɬ ɨɛɴɟɦɚ ɝɨɪɸɱɟɝɨ ɝɚɡɚ ɧɟ ɛɭɞɟɬ ɩɪɟɜɵɲɚɬɶ ɨɛɴ ɟɦɚ ɝɚɡɚ ɯɨɥɨɞɧɨɝɨ ɢɥɢ ɝɨɪɹɱɟɝɨ ɜɵɬɟɤɚɸɳɟɝɨ ɢɡ ɜɫɤɪɵɜɚɸɳɢɯɫɹ ɨɬɜɟɪ ɫɬɢɣ ɋɥɟɞɨɜɚɬɟɥɶɧɨ ɞɥɹ ɷɮɮɟɤɬɢɜɧɨɫɬɢ ɪɚɛɨɬɵ ɉɄ ɜɪɟɦɹ ɨɛɪɚɡɨɜɚɧɢɹ ɫɩɥɨɲɧɨɝɨ ɩɪɨɟɦɚ tɫɩ ɧɟ ɞɨɥɠɧɨ ɩɪɟɜɵɲɚɬɶ ɜɪɟɦɹ ɝɨɪɟɧɢɹ t .
ȾɨɜɫɤɪɵɬɢɹɉɄɧɚɝɪɭɡɤɚ ɞɟɣɫɬɜɭɸɳɚɹɧɚɧɢɯ ɪɚɜɧɚɢɡɛɵɬɨɱɧɨɦɭɞɚɜ ɥɟɧɢɸ ɜ ɩɨɦɟɳɟɧɢɢ ɨɬ ɜɡɪɵɜɧɨɝɨ ɝɨɪɟɧɢɹ Ƚɋ ɋɪɚɡɭ ɠɟ ɩɨɫɥɟ ɪɚɡɪɭɲɟɧɢɹ ɤɪɟɩɟɠɧɵɯ ɭɫɬɪɨɣɫɬɜ ɫ ɧɚɱɚɥɨɦ ɞɜɢɠɟɧɢɹ ɉɄ ɦɟɠɞɭ ɧɢɦɢ ɢ ɨɝɪɚɠɞɟɧɢɟɦ ɩɨɦɟɳɟɧɢɹ ɨɛɪɚɡɭɸɬɫɹ ɳɟɥɢ ɱɟɪɟɡ ɤɨɬɨɪɵɟ ɩɪɨɢɫɯɨɞɢɬ ɢɧɬɟɧɫɢɜɧɨɟ ɢɫɬɟɱɟ ɧɢɟ ɝɚɡɚ ɢɡ ɩɨɦɟɳɟɧɢɹ ɜ ɚɬɦɨɫɮɟɪɭ
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɨɰɟɫɫ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɧɚɝɪɭɡɤɢ ɫ ɉɄ ɜɪɚɳɚɸɳɟɣɫɹ ɜɨ ɤɪɭɝ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ ɪɢɫ
ɇɚ ɷɥɟɦɟɧɬ ɉɄ ɜɪɚɳɚɸɳɢɣɫɹ ɜɨɤɪɭɝ ɨɫɢ Oy ɞɟɣɫɬɜɭɟɬ ɜɡɪɵɜɧɚɹ ɧɚɝɪɭɡɤɚ ɢ ɢɧɟɪɰɢɨɧɧɵɟ ɫɢɥɵ ɋɨɫɬɚɜɢɦ ɭɪɚɜɧɟɧɢɟ ɫɢɥ ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɷɥɟ ɦɟɧɬ ɉɄ ɦɨɦɟɧɬ ɫɢɥɵ ɩɪɨɢɡɜɟɞɟɧɢɟ ɫɢɥɵ ɧɚ ɩɥɟɱɨ ɪɚɜɟɧ ɩɪɨɢɡɜɟɞɟɧɢɸ ɦɨ
64
ɦɟɧɬɚ ɢɧɟɪɰɢɢ ɦɚɫɫɚ ɧɚ ɤɜɚɞɪɚɬ ɪɚɫɫɬɨɹɧɢɹ ɨɬ ɨɫɢ ɜɪɚɳɟɧɢɹ ɧɚ ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ
ǷȏȘȚȔȕȑ 3 ǸȜȌȓȇ ȑ ȖȕȦȘȔȌȔȏȥ ȗȇȘȞȌșȕȉ ȎȇȑȕȔȕȓȌȗȔȕȘșȌȐ ȉȘȑȗȢșȏȦ
Ⱦɥɹ ɷɥɟɦɟɧɬɚɪɧɨɝɨ ɨɛɴɟɦɚ 'V ɭɪɚɜɧɟɧɢɟ ɦɨɦɟɧɬɨɜ ɡɚɩɢɲɟɬɫɹ ɫɥɟɞɭɸ ɳɢɦ ɨɛɪɚɡɨɦ
'P t 'X 'Y x J 'V x |
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'P t 'X 'Y x J 'X 'Y 'Z x |
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ɝɞɟ J ɩɥɨɬɧɨɫɬɶ ɦɚɬɟɪɢɚɥɚ ɉɄ ɤɝ ɦ ;
d M GW ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɫ ɪɚɞɢɚɧ ɜ ɫɟɤɭɧɞɭ ɡɚ ɫɟɤɭɧɞɭ 'X 'Y 'Z ɥɢɧɟɣɧɵɟ ɪɚɡɦɟɪɵ ɷɥɟɦɟɧɬɚɪɧɨɝɨ ɨɛɴɟɦɚ 'V8.
ɉɟɪɟɯɨɞɹ ɤ ɩɪɟɞɟɥɭ ɢ ɢɧɬɟɝɪɢɪɭɹ ɩɨ ɩɥɨɳɚɞɢ ɢ ɨɛɴɟɦɭ ɉɄ ɩɨɥɭɱɢɦ
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³³'P t xdxdy ³³³JMx dxdydz |
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ɚ ɩɨɫɥɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ |
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ɉɪɢɧɢɦɚɹ ɱɬɨ ɧɚɝɪɭɡɤɚ 'Ɋ t ɞɥɹ ɢɧɬɟɪɜɚɥɚ ɜɪɟɦɟɧɢ tɜɫɤ ÷ t ɢɡɦɟɧɹɟɬɫɹ |
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ɩɨ ɥɢɧɟɣɧɨɦɭ ɡɚɤɨɧɭ ɫɦ ɪɢɫ |
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'P t |
'Pɞɨɩ í 'Pɜɫɤ t í tɜɫɤ t í tɜɫɤ 'Pɜɫɤ § 'Pɞɨɩ í 'Pɜɫɤ ǜt t + 'Pɜɫɤ. |
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8 ɍɪɚɜɧɟɧɢɟ ɪɚɡɦɟɪɧɨɫɬɟɣ ɦɨɦɟɧɬ ɫɢɥɵ x F = 'P S x ɇ ɦ2 ɦ ɦ =ɇ ɦ = ɤɝ ɦ ɫ ; ɫɢɥɚ ɢɧɟɪɰɢɢ m x d M dt ɤɝ ɦ ɫ ɤɝ ɦ ɫ .
65
ɍɱɢɬɵɜɚɹ ɱɬɨ tɜɫɤ W 'Pɜɫɤ '3 W ɩɨɥɚɝɚɟɦ tɜɫɤ ɢ ɞɜɚɠɞɵ ɢɧɬɟ ɝɪɢɪɭɹ ɜɵɪɚɠɟɧɢɟ ɞɥɹ M ɩɨɥɭɱɢɦ:
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t J l G ǜ 'Pɞɨɩ 'Pɜɫɤ . |
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ɍɱɢɬɵɜɚɹ ɱɬɨɩɪɢɜɫɤɪɵɬɨɦɩɪɨɟɦɟM S § ɢW Wɫɩ |
W ɩɨɥɭɱɢɦ |
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t 'Pɞɨɩ 'Pɜɫɤ J G M . |
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ɂɡ ɩɨɫɥɟɞɧɟɝɨ ɭɪɚɜɧɟɧɢɹ ɫɞɟɥɚɟɦ ɫɥɟɞɭɸɳɢɟ ɜɵɜɨɞɵ.
1.ɇɚ ɡɚɤɨɧɨɦɟɪɧɨɫɬɢ ɜɫɤɪɵɬɢɹ ɉɄ ɜɪɚɳɚɸɳɟɣɫɹ ɜɨɤɪɭɝ ɜɟɪɬɢɤɚɥɶ ɧɨɣɨɫɢ ɧɟɜɥɢɹɟɬɜɵɫɨɬɚɩɪɟɞɨɯɪɚɧɢɬɟɥɶɧɨɣɤɨɧɫɬɪɭɤɰɢɢ b Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɜɟɥɢɱɢɧɭ b ɦɨɠɧɨ ɜɵɛɢɪɚɬɶ ɪɭɤɨɜɨɞɫɬɜɭɹɫɶ ɱɢɫɬɨ ɬɟɯɧɨ ɥɨɝɢɱɟɫɤɢɦɢ ɫɨɨɛɪɚɠɟɧɢɹɦɢ ɜɵɫɨɬɨɣ ɩɨɦɟɳɟɧɢɹ ɭɞɨɛɫɬɜɨɦ ɬɪɚɧɫɩɨɪɬɢɪɨɜɤɢ ɢ ɦɨɧɬɚɠɚ ɢ ɬ ɩ
Ⱦɪɭɝɢɟ ɪɚɡɦɟɪɵ l G ɩɪɟɞɨɯɪɚɧɢɬɟɥɶɧɨɣ ɤɨɧɫɬɪɭɤɰɢɢ ɨɩɪɟɞɟɥɹ ɸɬɫɹ ɦɚɬɟɪɢɚɥɨɦ ɉɄ J ɜɢɞɨɦ ɝɨɪɸɱɟɝɨ ɩɪɢ ɜɡɪɵɜɟ t ɢ ɫɜɹɡɚɧɵ ɦɟɠɞɭ ɫɨɛɨɣ Ɂɚɞɚɜɚɹɫɶ ɨɞɧɢɦ ɪɚɡɦɟɪɨɦ ɢɡ ɩɨɫɥɟɞɧɟɝɨ ɜɵɪɚɠɟɧɢɹ ɩɨɥɭɱɢɦ ɞɪɭɝɨɣ
ȼɚɪɢɚɧɬ ɤɨɧɫɬɪɭɤɰɢɢ ɉɄ
Ɉɱɟɧɶ ɱɚɫɬɨ ɜ ɤɚɱɟɫɬɜɟ ɤɨɧɫɬɪɭɤɰɢɢ ɞɥɹ ɉɄ ɢɫɩɨɥɶɡɭɸɬ ɫɥɨɢɫɬɵɟ ɫɬɪɭɤ ɬɭɪɵ ɦɟɬɚɥɥ ɩɟɧɨɩɥɚɫɬ ɦɟɬɚɥɥ ɞɟɪɟɜɨ ɢ ɞɪɭɝɢɟ ɫɨɱɟɬɚɧɢɹ ɉɨɩɟɪɟɱɧɵɣ ɪɚɡɪɟɡ ɬɚɤɨɣ ɉɄ ɩɪɢɜɟɞɟɧ ɧɚ ɪɢɫɭɧɤɟ
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ǷȏȘȚȔȕȑ ǷȇȎȗȌȎ ǶDZ ȘȒȕȏȘșȕȐ ȘșȗȚȑșȚȗȢ
ȼ ɫɥɭɱɚɟ ɟɫɥɢ ɭɞɟɥɶɧɚɹ ɦɚɫɫɚ ɨɫɧɨɜɧɨɝɨ ɦɚɬɟɪɢɚɥɚ ɉɄ ɧɚɩɪɢɦɟɪ ɦɟ ɬɚɥɥɚ ɡɧɚɱɢɬɟɥɶɧɨ ɜɵɲɟ ɭɞɟɥɶɧɨɣ ɦɚɫɫɵ ɡɚɩɨɥɧɢɬɟɥɹ ɩɟɧɨɩɥɚɫɬ ɬɨ ɦɚɫɫɨɣ ɡɚɩɨɥɧɢɬɟɥɹ ɩɪɟɧɟɛɪɟɝɚɸɬ ȿɫɥɢ ɠɟ ɷɬɢ ɭɞɟɥɶɧɵɟ ɦɚɫɫɵ ɫɨɢɡɦɟɪɢɦɵ ɜ ɪɚɫ ɱɟɬɚɯ ɢɫɩɨɥɶɡɭɸɬ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɨɩɪɟɞɟɥɹɟɦɨɟ ɩɨ ɮɨɪɦɭɥɟ
66
n
Jɫɪ ^J G G G GQ J G G G GQ ` ¦GL
i 1
ɝɞɟ J ɩɥɨɬɧɨɫɬɶ ɨɫɧɨɜɧɨɝɨ ɦɚɬɟɪɢɚɥɚ ɉɄ ɤɝ ɦ ; J ɩɥɨɬɧɨɫɬɶ ɡɚɩɨɥɧɢɬɟɥɹ ɤɝ ɦ ; G G … Gn ɬɨɥɳɢɧɵ ɫɥɨɟɜ ɨɫɧɨɜɧɨɝɨ ɦɚɬɟɪɢɚɥɚ ɤɝ ɦ ; G G … Gní1 ɬɨɥɳɢɧɵ ɫɥɨɟɜ ɡɚɩɨɥɧɢɬɟɥɹ ɤɝ ɦ ; Q ɨɛɳɟɟ ɤɨɥɢɱɟɫɬɜɨ ɨɫɧɨɜɧɵɯ ɫɥɨɟɜ ɜ ɉɄ
ǮȇȋȇȔȏȌ Ȕȇ ȗȇȘȞȌș ȖȗȌȋȕȜȗȇȔȏșȌȒȣȔȢȜ ȑȕȔȘșȗȚȑȝȏȐ
Ɍɪɟɛɭɟɬɫɹ ɪɚɫɫɱɢɬɚɬɶ ɤɨɥɢɱɟɫɬɜɨ ɢ ɪɚɡɦɟɪɵ ɩɨɜɨɪɨɬɧɵɯ ɩɪɟɞɨɯɪɚɧɢ ɬɟɥɶɧɵɯɤɨɧɫɬɪɭɤɰɢɣɫɜɟɪɬɢɤɚɥɶɧɵɦɲɚɪɧɢɪɨɦ ɫɧɢɠɚɸɳɢɯɩɨɫɥɟɞɫɬɜɢɹɞɟ ɮɥɚɝɪɚɰɢɨɧɧɨɝɨ ɜɡɪɵɜɚ ɞɨ ɞɨɩɭɫɬɢɦɵɯ
ɂɫɯɨɞɧɵɦɢ ɞɚɧɧɵɦɢ ɞɥɹ ɪɚɫɱɟɬɨɜ ɹɜɥɹɸɬɫɹ
xɪɟɡɭɥɶɬɚɬɵɜɵɩɨɥɧɟɧɢɹɪɚɫɱɟɬɧɨɝɨɡɚɞɚɧɢɹɜɬɨɪɨɣɝɥɚɜɵ H t 'P
xɢɫɯɨɞɧɵɟ ɞɚɧɧɵɟ ɞɥɹ ɡɚɞɚɧɧɨɝɨ ɩɪɟɩɨɞɚɜɚɬɟɥɟɦ ɜɚɪɢɚɧɬɚ ɚɜɚɪɢɢV ɚ Uɧ 'Pɞɨɥ 'Pɜɫɤ m J1 J G1 G b
Ɉɩɪɟɞɟɥɟɧɢɟ ɧɟɨɛɯɨɞɢɦɨɣ ɩɥɨɳɚɞɢ ɜɫɤɪɵɬɢɹ
Ɋɚɫɱɟɬɵ ɩɪɨɜɨɞɹɬɫɹ ɩɪɢ ɫɥɟɞɭɸɳɢɯ ɞɨɩɭɳɟɧɢɹɯ
xɦɟɫɬɨ ɨɛɪɚɡɨɜɚɧɢɹ ɜɡɪɵɜɨɨɩɚɫɧɨɝɨ ɨɛɥɚɤɚ ɜ ɩɨɦɟɳɟɧɢɢ ɡɚɪɚɧɟɟ ɧɟ ɢɡɜɟɫɬɧɨ
xɢɫɩɨɥɶɡɭɟɬɫɹ ɷɦɩɢɪɢɱɟɫɤɚɹ ɮɨɪɦɭɥɚ ɩɨɥɭɱɟɧɧɚɹ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɞɢɚɝɪɚɦɦɵ ɪɚɡɪɭɲɟɧɢɣ ɬɢɩɨɜɨɝɨ ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɨɝɨ ɡɞɚɧɢɹ
ɉɥɨɳɚɞɶ ɜɫɤɪɵɬɢɹ ɩɨɦɟɳɟɧɢɹ ɩɪɢ ɞɟɮɥɚɝɪɚɰɢɨɧɧɨɦ ɜɡɪɵɜɟ
4 V aU H 1 U
F ɪɚɫɱ ɧ ɫɬɯ ɦ . 
'Ɋɞɨɩ
ɉɥɨɳɚɞɶ ɉɄ ɧɟɞɨɫɬɚɸɳɚɹ ɩɥɨɳɚɞɶ ɜɫɤɪɵɬɢɹ
Fɩɤ = F í Fɩɪ.
Ɉɩɪɟɞɟɥɟɧɢɟ ɩɚɪɚɦɟɬɪɨɜ ɉɄ ɢ ɢɯ ɤɨɥɢɱɟɫɬɜɚ
Ɋɚɫɱɟɬɵ ɩɪɨɜɨɞɹɬɫɹ ɩɪɢ ɫɥɟɞɭɸɳɢɯ ɞɨɩɭɳɟɧɢɹɯ
xɬɪɟɧɢɟɦ ɜ ɲɚɪɧɢɪɟ ɢ ɪɚɦɟ ɩɪɟɧɟɛɪɟɝɚɟɦ
xɜɪɟɦɹ ɨɛɪɚɡɨɜɚɧɢɹ ɫɩɥɨɲɧɨɝɨ ɩɪɨɟɦɚ ɦɟɧɶɲɟ ɜɪɟɦɟɧɢ ɜɡɪɵɜɧɨɝɨ ɝɨɪɟɧɢɹ
xɪɚɫɱɟɬ ɜɟɞɟɦ ɬɨɥɶɤɨ ɩɨ ɞɟɣɫɬɜɢɸ ɢɡɛɵɬɨɱɧɨɝɨ ɞɚɜɥɟɧɢɹ ɫɠɚɬɢɹ
67
Ɋɚɫɱɟɬɧɚɹ ɲɢɪɢɧɚ ɉɄ
l t 'Pɞɨɩ 'Pɜɫɤ JɩɤGɩɤM
ɝɞɟ Jɩɤ =^m J G m J G ` Gɩɤ ɫɪɟɞɧɢɣ ɭɞɟɥɶɧɵɣ ɜɟɫ ɉɄ
Gɩɤ = m G m G ɬɨɥɳɢɧɚ ɉɄ ɬɢɩɚ ©ɋɷɧɞɜɢɱª.
Ɂɧɚɱɟɧɢɟ ɪɚɫɱɟɬɧɨɣɲɢɪɢɧɵ ɉɄ Oɢɡɩɪɚɤɬɢɱɟɫɤɢɯɫɨɨɛɪɚɠɟɧɢɣɨɤɪɭɝ ɥɹɟɬɫɹ ɜ ɦɟɧɶɲɭɸ ɫɬɨɪɨɧɭ ɫ ɲɚɝɨɦ ɜ ɫɦ ɢ ɜ ɩɪɚɤɬɢɱɟɫɤɨɣ ɪɟɚɥɢɡɚɰɢɢ ɉɄ ɢɦɟɟɬ ɜ ɲɢɪɢɧɭ
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l = ɨɤɪɭɝɥ ɜɧɢɡ l*; . |
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Ʉɨɥɢɱɟɫɬɜɨ ɢ ɦɚɫɫɚ ɉɄ |
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ɉɪɢ ɡɚɞɚɧɧɨɣ ɜɵɫɨɬɟ ɉɄ Eɢ ɲɢɪɢɧɟ l ɩɥɨɳɚɞɶ ɉɄ |
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F1ɩɤ = b ǜ l. |
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Ɇɚɫɫɚ ɨɞɧɨɣ ɉɄ |
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mɩɤ blGJɩɤ |
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Ʉɨɥɢɱɟɫɬɜɨ ɉɄ |
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Nɩɤ ɰɟɥɨɟ ^Fɩɤ F1ɩɤ} + 1. |
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Ɋɟɡɭɥɶɬɚɬɨɦ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ ɩɨ ɪɚɫɱɟɬɭ ɩɪɟɞɨɯɪɚɧɢɬɟɥɶɧɵɯ ɤɨɧ ɫɬɪɭɤɰɢɣ ɹɜɥɹɸɬɫɹ ɪɚɡɦɟɪɵ b l ɢ ɤɨɥɢɱɟɫɬɜɨ ɉɄ Nɩɤ ɤɨɬɨɪɵɦɢ ɞɨɩɨɥɧɢ ɬɟɥɶɧɨ ɧɟɨɛɯɨɞɢɦɨ ɨɛɨɪɭɞɨɜɚɬɶ ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɨɟ ɩɨɦɟɳɟɧɢɟ
ȼɵɜɨɞ
ɉɪɢ ɨɛɨɪɭɞɨɜɚɧɢɢ ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɨɝɨ ɩɨɦɟɳɟɧɢɹ Nɩɤ ɫɛɪɨɫɨɜɵɦɢ ɨɬ ɜɟɪɫɬɢɹɦɢ ɪɚɡɦɟɪɚɦɢ E ǜl ɢɡɛɵɬɨɱɧɨɟ ɞɚɜɥɟɧɢɟ ɫɠɚɬɢɹ ɩɪɢ ɞɟɮɥɚɝɪɚɰɢɨɧ ɧɨɦ ɜɡɪɵɜɟ ɧɟ ɩɪɟɜɵɫɢɬ ɞɨɩɭɫɬɢɦɨɟ ɡɧɚɱɟɧɢɟ 'Pɞɨɩ ɩɟɪɫɨɧɚɥ ɧɟ ɩɨɥɭɱɢɬ ɫɭ ɳɟɫɬɜɟɧɧɵɯ ɩɨɜɪɟɠɞɟɧɢɣ ɚ ɨɛɨɪɭɞɨɜɚɧɢɟ ɢ ɩɨɦɟɳɟɧɢɟ ɪɚɡɪɭɲɟɧɢɣ ɢ ɫɥɟ ɞɨɜɚɬɟɥɶɧɨ ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɵɣɩɪɨɰɟɫɫɩɨɫɥɟɡɚɦɟɧɵɢɥɢɪɟɦɨɧɬɚɚɜɚɪɢɣɧɨɝɨ ɨɛɨɪɭɞɨɜɚɧɢɹ ɦɨɠɟɬ ɛɵɬɶ ɩɪɨɞɨɥɠɟɧ
Ɉɰɟɧɤɚ ɞɚɧɚ ɩɨ ɩɨɪɚɠɚɸɳɟɦɭ ɞɟɣɫɬɜɢɸ ɢɡɛɵɬɨɱɧɨɝɨ ɞɚɜɥɟɧɢɹ ɫɠɚɬɢɹ ɛɟɡ ɭɱɟɬɚ ɞɟɣɫɬɜɢɹ ɞɪɭɝɢɯ ɩɨɪɚɠɚɸɳɢɯ ɮɚɤɬɨɪɨɜ ɜ ɬɨɦ ɱɢɫɥɟ ɤɨɫɜɟɧɧɵɯ
68
ǶǷǯdzǬǷ ǶǷǵǩǬǫǬǴǯȆ ǷǧǸǾǬǹǵǩ
ǯȘȜȕȋȔȢȌ ȋȇȔȔȢȌ ȋȒȦ ȗȇȘȞȌșȇ
ȼ ɡɚɝɨɬɨɜɢɬɟɥɶɧɨɦ ɰɟɯɭ ɩɨ ɨɬɪɟɡɤɟ ɡɚɝɨɬɨɜɨɤ ɢɡ ɚɥɸɦɢɧɢɹ ɧɚ ɥɟɧɬɨɱɧɨ ɩɢɥɶɧɵɯ ɫɬɚɧɤɚɯ ɬɟɯɧɨɥɨɝɢɟɣ ɩɪɟɞɭɫɦɨɬɪɟɧɨ ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɜ ɤɚɱɟɫɬɜɟ ɫɦɚ ɡɨɱɧɨ ɨɯɥɚɠɞɚɸɳɢɯ ɠɢɞɤɨɫɬɟɣ ɋɈɀ ɬɟɯɧɢɱɟɫɤɨɣ ɫɦɟɫɢ ɧɚ ɨɫɧɨɜɟ ɤɟɪɨ ɫɢɧɚ ȼ ɤɚɱɟɫɬɜɟ ɜɡɪɵɜɨɨɩɚɫɧɨɝɨ ɨɛɴɟɤɬɚ ɜɨɡɶɦɟɦ ɟɦɤɨɫɬɶ ɫ ɤɟɪɨɫɢɧɨɦ ɨɛɴɟ ɦɨɦ V ɉɪɨɢɡɜɟɞɟɦ ɨɰɟɧɤɭ ɜɨɡɦɨɠɧɵɯ ɩɨɫɥɟɞɫɬɜɢɣ ɜɡɪɵɜɚ ɜɨɡɞɭɲɧɨɣ ɫɦɟɫɢ ɩɚɪɨɜ ɤɟɪɨɫɢɧɚ ɢ ɜɨɡɞɭɯɚ ɜ ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɨɦ ɩɨɦɟɳɟɧɢɢ ɩɪɢ ɚɜɚɪɢɣɧɨɦ ɪɚɡ ɥɢɜɟ ɢ ɢɫɩɚɪɟɧɢɢ ɟɦɤɨɫɬɢ ɫ ɤɟɪɨɫɢɧɨɦ
ɐɟɯɡɚɝɨɬɨɜɢɬɟɥɶɧɵɯɨɩɟɪɚɰɢɣ– ɨɞɧɨɷɬɚɠɧɨɟɩɪɨɢɡɜɨɞɫɬɜɟɧɧɨɟɡɞɚɧɢɟ ɫ ɪɚɡɦɟɪɚɦɢ ɜ ɩɥɚɧɟ LǜB ɢ ɜɵɫɨɬɨɣ H ɉɥɨɳɚɞɶ ɨɤɨɧɧɵɯ ɩɪɨɟɦɨɜ ɪɚɜɧɚ Fɨɛɳ. ɋɪɟɞɧɹɹɬɟɦɩɟɪɚɬɭɪɚɜɰɟɯɟtɩ ɐɟɯɨɛɨɪɭɞɨɜɚɧɚɜɚɪɢɣɧɨɣɜɟɧɬɢɥɹɰɢɟɣɫɤɪɚɬ ɧɨɫɬɶɸ ɜɨɡɞɭɯɨɨɛɦɟɧɚ Ⱥɜ Ɍɟɯɧɢɱɟɫɤɚɹ ɫɦɟɫɶ ɤɟɪɨɫɢɧɚ ɡɚɞɚɧɚ ɛɪɭɬɬɨ ɮɨɪɦɭ
ɥɨɣ &86 %H14 %.
ɂɫɯɨɞɧɵɟ ɞɚɧɧɵɟ |
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Ɂɧɚɱɟɧɢɟ |
|
|
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Ɇɨɥɹɪɧɚɹ ɦɚɫɫɚ |
M |
155 ɝ ɦɨɥɶ |
ɉɥɨɬɧɨɫɬɶ |
ȡ ɤɝ ɦ |
|
Ɇɚɫɫɚ ɠɢɞɤɨɫɬɢ |
mɠ = ɤɝ |
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Ɍɟɦɩɟɪɚɬɭɪɚ ɜ ɰɟɯɟ |
Tɩ Ʉ |
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Ɋɚɡɦɟɪɵ ɰɟɯɚ |
LǜBǜ+ ǜ ǜ4 ɦ |
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Ʉɨɷɮɮɢɰɢɟɧɬ ɪɚɫɬɟɤɚɟɦɨɫɬɢ ɠɢɞɤɨɫɬɢ |
I |
|
Ʉɨɷɮɮɢɰɢɟɧɬɵ ɞɥɹ ɪɚɫɱɟɬɚ ɞɚɜɥɟɧɢɹ |
Ⱥ ; ȼ ; |
|
ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ |
ɋ |
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Ʉɪɚɬɧɨɫɬɶ ɜɟɧɬɢɥɹɰɢɢ |
Ⱥɜ=5 |
|
ɉɥɨɳɚɞɶ ɨɤɨɧɧɵɯ ɩɪɨɟɦɨɜ |
Fɨɛɳ ɦ |
|
Ɋɚɫɫɬɨɹɧɢɟ ɞɨ ɫɦɟɠɧɨɝɨ ɨɛɴɟɤɬɚ |
Rɨɛ ɦ |
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Ʉɨɷɮɮɢɰɢɟɧɬ ɢɧɬɟɧɫɢɮɢɤɚɰɢɢ ɝɨɪɟɧɢɹ |
a |
4 |
ɉɪɨɰɟɧɬ ɪɚɫɬɜɨɪɢɬɟɥɹ ɠɢɞɤɨɫɬɢ |
w |
% |
ɇɨɪɦɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɝɨɪɟɧɢɹ |
Uɧ ɦ ɫ |
|
Ⱦɨɩɭɫɬɢɦɨɟ ɢɡɛɵɬɨɱɧɨɟ ɞɚɜɥɟɧɢɟ |
ǻPɞɨɩ 5 ɤɉɚ |
|
ɂɡɛɵɬɨɱɧɨɟ ɞɚɜɥɟɧɢɟ ɜɫɤɪɵɬɢɹ |
ǻPɜɫɤ ɤɉɚ |
|
ȼɵɫɨɬɚ ɩɪɟɞɨɯɪɚɧɢɬɟɥɶɧɵɯ ɤɨɧɫɬɪɭɤɰɢɣ |
E ɦ |
|
Ʉɨɥɢɱɟɫɬɜɨ ɧɟɫɭɳɢɯ ɫɥɨɟɜ |
N 4 |
|
Ɍɨɥɳɢɧɚ ɧɟɫɭɳɟɝɨ ɫɥɨɹ |
į1 ɦ |
|
Ɍɨɥɳɢɧɚ ɫɥɨɹ ɡɚɩɨɥɧɟɧɢɹ |
į |
ɦ |
ɉɥɨɬɧɨɫɬɶ ɧɟɫɭɳɟɝɨ ɫɥɨɹ |
Ȗ1 ɤɝ ɦ |
|
ɉɥɨɬɧɨɫɬɶ ɫɥɨɹ ɡɚɩɨɥɧɟɧɢɹ |
Ȗ |
ɤɝ ɦ |
69
ǵȝȌȔȑȇ ȉȕȎȓȕȍȔȢȜ ȖȕȘȒȌȋȘșȉȏȐ ȉȎȗȢȉȇ ȖȇȗȕȉȕȎȋȚȟȔȕȐ ȘȓȌȘȏ ȉ ȖȗȕȏȎȉȕȋȘșȉȌȔȔȕȓ ȖȕȓȌȠȌȔȏȏ Ȗȗȏ ȇȉȇȗȏȐȔȕȓ ȗȇȎȒȏȉȌ ȌȓȑȕȘșȏ
4.2 ǴȇȜȕȍȋȌȔȏȌ ȓȇȑȘȏȓȇȒȣȔȕȐ șȌȕȗȌșȏȞȌȘȑȕȐ șȌȓȖȌȗȇșȚȗȢ ȊȕȗȌȔȏȦ
Ȼɭɞɟɦ ɫɱɢɬɚɬɶ ɱɬɨ
xɤɨɧɰɟɧɬɪɚɰɢɹ ɫɦɟɫɢ ɝɨɪɸɱɟɝɨ ɫ ɜɨɡɞɭɯɨɦ – ɫɬɟɯɢɨɦɟɬɪɢɱɟɫɤɚɹ ɬɨ
ɟɫɬɶ ɨɛɟɫɩɟɱɢɜɚɟɬɫɹ ɩɨɥɧɨɟ ɫɝɨɪɚɧɢɟ ɛɟɡ ɢɡɛɵɬɤɚ ɢɥɢ ɧɟɞɨɫɬɚɬɤɚ ɤɢɫɥɨɪɨɞɚ;
xɜɵɞɟɥɹɟɦɚɹɩɪɢɝɨɪɟɧɢɢɷɧɟɪɝɢɹɢɞɟɬɬɨɥɶɤɨɧɚɩɨɞɨɝɪɟɜɩɪɨɞɭɤɬɨɜ ɫɝɨɪɚɧɢɹ ɬɨ ɟɫɬɶ ɩɪɨɰɟɫɫ ɝɨɪɟɧɢɹ ɚɞɢɚɛɚɬɢɱɟɫɤɢɣ
ɉɫɟɜɞɨɯɢɦɢɱɟɫɤɭɸ ɮɨɪɦɭɥɭ ɝɨɪɸɱɟɝɨ – ɋwɇm ɩɨɥɭɱɢɦ ɧɚ ɨɫɧɨɜɚɧɢɢ
ɛɪɭɬɬɨ ɮɨɪɦɭɥɵ &86 %H14 % ɝɞɟ w = |
M 86 |
P |
M 14 |
|
M& |
|
M H |
ɬɨɝɞɚ ɮɨɪɦɭɥɚ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ ɋ ɇ . Ʉɨɥɢɱɟɫɬɜɨ ɦɨɥɟɤɭɥ ɤɢɫɥɨɪɨɞɚ ɧɟɨɛɯɨɞɢɦɨɟ ɞɥɹ ɩɨɥɧɨɝɨ ɫɝɨɪɚɧɢɹ ɦɨ
ɥɟɤɭɥɵ ɤɟɪɨɫɢɧɚ ȕ Z m4 =
ɍɪɚɜɧɟɧɢɟ ɪɟɚɤɰɢɢ ɝɨɪɟɧɢɹ
ɋ ɇ Ɉ N ɋɈ ɇ Ɉ Â N .
Ɋɚɫɱɟɬ ɬɟɩɥɨɬɵ ɫɝɨɪɚɧɢɹ ɤɟɪɨɫɢɧɚ ɩɪɨɢɡɜɟɞɟɧ ɩɨ ɮɨɪɦɭɥɟ Ɇɟɧɞɟɥɟɟɜɚ
Qɧ ·[ɋ] Â[ɇ] ɤȾɠ
ɝɞɟ [ɋ] ɢ [H] – ɩɪɨɰɟɧɬɧɵɟ ɞɨɥɢ ɭɝɥɟɪɨɞɚ ɢ ɜɨɞɨɪɨɞɚ ɜ ɛɪɭɬɬɨ ɮɨɪɦɭɥɟ
Ʉɨɷɮɮɢɰɢɟɧɬ ɢɡɦɟɧɟɧɢɟ ɱɢɫɥɚ ɦɨɥɟɣ ɜ ɯɨɞɟ ɪɟɚɤɰɢɢ n= 6Qi ɤ =
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6Qi ɧ |
ɝɞɟ ȈȞi ɧ = Q& + |
Q2 Q1 ȕ Âȕ ɦɨɥɶ |
|||
ȈȞi ɤ = Q&2 QH 2 |
QN |
= w + |
m |
Âȕ ɦɨɥɶ. |
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||||
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Ɍɟɦɩɟɪɚɬɭɪɚ ɪɚɡɜɢɜɚɟɦɚɹ ɜ ɩɨɦɟɳɟɧɢɢ ɩɪɢ ɢɡɨɛɚɪɧɨ ɚɞɢɚɛɚɬɢɱɟɫɤɨɦ ɩɪɨɰɟɫɫɟ ɝɨɪɟɧɢɹ ɪɚɫɫɱɢɬɵɜɚɟɬɫɹ ɧɚ ɨɫɧɨɜɟ ɡɚɤɨɧɚ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ
Tp Ɍɩ |
P Qɧ |
. |
¦Qi C pi ɤ |
Ɍɟɦɩɟɪɚɬɭɪɚ ɪɚɡɜɢɜɚɟɦɚɹ ɜ ɩɨɦɟɳɟɧɢɢ ɩɪɢ ɢɡɨɯɨɪɧɨ ɚɞɢɚɛɚɬɢɱɟɫɤɨɦ ɩɪɨɰɟɫɫɟ ɝɨɪɟɧɢɹ
70