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ȡǫȈȭȋǯǹؕᄽ

(ਰѣܖȷᚐௌщܖ)

ޛɦ ᘽ

ஜᜒ፯᝻૰

2020ᖺᗘ

URL:

http://stlab.ssi.ist.hokudai.ac.jp/yuhyama/lecture/mechatronics

2

ਰѣܖ

ᇹ1ᇘ

ਰѣܖ

= ཋƷਰѣǛৢƍLJƢ

` <sub>ᮦᩱຊᏛ䛿静的なኚᙧ䞉ෆຊ䜢ᢅ䛳䛶䛔䜎䛧䛯䚹</sub> ` _{䛣䛣䛷䛿䚸ື䛝䜢క䛖動的なኚᙧ䚸≉䛻振動䜢ᢅ䛔䜎䛩䚹} ` 䛺䛬䚸᣺ື䛻╔┠䛩䜛䛾䛛䠛(ᶵᲔⓗ䛺᣺ື䛾ሙྜ) ` ไ᣺䞉ච㟈(ᘓ≀䞉㌴୧䞉⢭ᐦᶵჾetc…) ` ᵓ㐀䞉ᶵᵓୖ䛾ၥ㢟䛷䚸䛒䜛䛔䛿䛭䜒䛭䜒䛾┠ⓗ䛸䛧䛶䚸䛂≀䜢᣺ື䛥䛫 䜛䛃䛣䛸䛜ᚲせ䛺ሙྜ䛜䛒䜛䚹 ` ඹ᣺䜢㑊䛡䜛 (ඹ᣺䛩䜛䛸䚸᭱ᝏ䚸◚ቯ䛜㉳䛝䜛䚹䛯䛸䛘䜀䚸㍈䛾ᅇ㌿ 䛾༴㝤㏿ᗘ)

ؕஜႎƳᢃѣ૾ᆉࡸ

` ᣺ື䛜㉳䛝䜛䛾䛿䚸(ᐇ㉁ⓗ䛻)2㝵௨ୖ䛾ᚤศ᪉⛬ᘧ ` Mass-Spring-Damper 䝅䝇䝔䝮 (୍␒༢⣧䛺䝅䝇䝔䝮) ᚤศ᪉⛬ᘧ: ຍ㏿ᗘ㡯 ⢓ᛶ 䜀䛽 እຊ ᦶ᧿ ຊ _n : ↓ῶ⾶⣔䛾ᅛ᭷ゅ࿘Ἴᩘ : ῶ⾶ẚ m k c 䜀䛽䛾⮬↛㛗 x 2020ᖺᗘ 5

2᨞ዴ࢟ࠝࣇЎ૾ᆉࡸƷᚐ (1)

` 䛧䜀䜙䛟䛿እຊ䝊䝻(f = 0)䛸䛧䛶⪃䛘䜛䚹Ѝ ⮬⏤᣺ື ` ≉ᛶ᪉⛬ᘧ: m 2_{+ c + k = 0} ` ┦␗䛺䜛ᐇᩘゎ <sub>1</sub> <sub>2</sub>䜢ᣢ䛴ሙྜ(c2 ± 4mk > 0 䛒䜛䛔䛿 > 1) x(t) = C<sub>1</sub>exp( <sub>1</sub>t) + C<sub>2</sub>exp( <sub>2</sub>t) ` 㔜ゎ ₁䜢ᣢ䛴ሙྜ(c2 _{± 4mk = 0 䛒䜛䛔䛿 = 1)} x(t) = C₁exp( ₁t) + C₂texp( ₁t) ` ⹫ᩘゎ r ± j 䜢ᣢ䛴ሙྜ(c2<sub>± 4mk < 0 䛒䜛䛔䛿 < 1)</sub> x(t) = C<sub>1</sub>ert<sub>sin(t) + C</sub> 2ertcos(t) ` ⹫ᩘゎ䜢ᣢ䛴ሙྜ䛾䜏䚸᣺ືⓗ䚹 ` ≉ᛶ᪉⛬ᘧ䛾ゎ䛾ᐇᩘ㒊䛜㈇䛷䛒䜛䛣䛸䛜䚸ゎ䛜䝊䝻䛻཰᮰䛩䜛䛯䜑䛾 ᚲせ༑ศ᮲௳䚹(₞㏆Ᏻᐃᛶ) 2020ᖺᗘ 6

2᨞ዴ࢟ࠝࣇЎ૾ᆉࡸƷᚐ (2)

` m > 0, c > 0, k > 0 䛺䜙䜀䚸ᚲ䛪≉ᛶ᪉⛬ᘧ䛾ゎ䛾ᐇ㒊䛿㈇䚸䛩䛺䜟 䛱₞㏆Ᏻᐃ䚹 ` m > 0, c = 0, k > 0 䛺䜙䜀䚸ᚲ䛪≉ᛶ᪉⛬ᘧ䛾ゎ䛿⣧⹫ᩘ䛷䚸Ᏻᐃ䛰 䛜₞㏆Ᏻᐃ䛷䛿䛺䛔䚹䛣䛾ሙྜ䚸ᚤศ᪉⛬ᘧ䛾ゎ䛿ᐃᖖⓗ䛻᣺ື䛩䜛 (ῶ⾶䛧䛺䛔)䚹 ` m > 0, c > 0, k > 0 䛾⠊ᅖ䛷䚸c 䛰䛡䜢ᑠ䛥䛟䛧䛶䛔䛟䛸≉ᛶ᪉⛬ᘧ䛾ゎ 䛿⹫ᩘ䛸䛺䜚䚸᣺ືⓗ䛻䛺䜛䚹 ` c 䛜ᑠ䛥䛔䛸䛝(䛜ᑠ䛥䛔䛸䛝)䚸≉ᛶ᪉⛬ᘧ䛾ゎ䛿䚸 ` c 䛜ᑠ䛥䛔䛸䛝䚸䛭䛾᣺ືゅ࿘Ἴᩘ䛿䚸䜋䛸䜣䛹 2஌䛾䜸䞊䝎 Ѝ ᚤᑠ

໯ถᘛƷئӳ

(ᛦԧਰѣ܇)

` ῶ⾶㡯䛜䛺䛔⮬⏤᣺ື(c = 0 䛒䜛䛔䛿 = 0) ` ᚤศ᪉⛬ᘧ䛾ゎ: э ↓ῶ⾶䛾ሙྜ䛿ᣢ⥆ⓗ䛺᣺ື ` ᣺ືゅ࿘Ἴᩘ䛿 <sub>n</sub>(↓ῶ⾶⣔䛾ᅛ᭷ゅ࿘Ἴᩘ) ` A䛿᣺ᖜ*䛿఩┦䛸࿧䜀䜜䜛䚹

ᛦԧਰѣ܇ƷႻ׋

(Phase portrait)

2020ᖺᗘ ` ┦ᅗ(ᶓ㍈ x, ⦪㍈ 䛷䝥䝻䝑䝖) ` 䜶䝛䝹䜼䞊ಖᏑ๎: э 䜶䝛䝹䜼䞊୍ᐃ䛾᭤⥺ୖ䜢ື䛟 x ‡ ྑᅇ䜚 ‡ ⦪/ᶓ = _n ‡ ࿘ᮇ䛿2$ / _n 9

ถᘛਰѣ

2020ᖺᗘ ` ῶ⾶᣺ື(0 < < 1)䛾ሙྜ: ` ࿘ᮇᚋ(t + T )䛻x(t)䛿䛹䜜䛰䛡ῶ⾶䛧䛶䛔䜛䛛? (࿘ᮇ: T = 2$ / _d) ` ᑐᩘῶ⾶⋡: э ᫬㛫㍈䛾ఙ⦰䜢チ䛧䛯ᛂ⟅Ἴᙧ䛾 䛂ぢ䛯┠䛃䛿 䛾䜏䛻䜘䛳䛶Ỵ䜎䜛䚹 Ѝ ῶ⾶᣺ື䛾ᅛ᭷ゅ࿘Ἴᩘ t t + T x _T 10

᣻щƕ΁Ƙئӳ

` 㔜ຊ(gravity)䛜ാ䛟Mass-Spring-Damper⣔ ` ᪂䛧䛔ᗙᶆ: z = x ± mg / k ` ᪂䛧䛔ᗙᶆ䛷⪃䛘䛯ᚤศ᪉⛬ᘧ э㔜ຊ䛺䛧䛾ሙྜ䛸ྠ䛨ᘧ ` 䛴䜎䜚䚸 㔜ຊ䛜ാ䛟ሙྜ䛿䚸఩⨨ኚᩘ䛾ཎⅬ䜢⮬↛㛗䛸ྲྀ䜛䛾䛷䛿䛺䛟䚸 㔜ຊ䛸䛴䜚䛒䛳䛶㟼Ṇ䛧䛶䛔䜛఩⨨䜢ཎⅬ䛸䛩䜜䜀䚸 㔜ຊ䛺䛧䛾ሙྜ䛸ྠ䛨ᚤศ᪉⛬ᘧ䛻䛺䜛䚹 m k c x 㔜ຊ㡯(gravity force)

ƞLJƟLJƳᲫᐯဌࡇਰѣኒ

(1)

` 䛽䛨䜚Წ䜀䛽(torsion bar) 䛻䜘䜛᣺ື ᙎᛶ㍈⮬య䛾័ᛶ䝰䞊䝯䞁䝖䛿↓ど J: 䛚䜒䜚䛾័ᛶ䝰䞊䝯䞁䝖 (Inertia moment) k: ᙎᛶ㍈䛾䛽䛨䜚䛣䜟䛥 (୸Წ䛾ሙྜ) G: ᶓᙎᛶಀᩘ d: 䛽䛨䜚Წ䜀䛽䛾┤ᚄ `: 䛽䛨䜚Წ䜀䛽䛾㛗䛥 J k 

ƞLJƟLJƳᲫᐯဌࡇਰѣኒ

(2)

2020ᖺᗘ ` 䛿䜚(ᱱ; beam)䛾᣺ື: k: 䛿䜚䛾䛣䜟䛥 ` 䛥䜎䛦䜎䛺䛿䜚䛻ᑐ䛩䜛 k ∦ᣢ䛱ᱱ ୧➃ᨭᣢ ୧➃ᅛᐃ ᕥᅛᐃ-ྑᨭᣢ x E: ᙎᛶಀᩘ I: ᩿㠃2ḟ䝰䞊䝯䞁䝖 `: ᱱ䛾㛗䛥 `₁: 㗽䛛䜙ᕥ➃䜎䛷䛾㛗䛥 `₂: 㗽䛛䜙ྑ➃䜎䛷䛾㛗䛥 m 13

ƞLJƟLJƳᲫᐯဌࡇਰѣኒ

(3)

2020ᖺᗘ ` ᾮᰕ䛾᣺ື: `: ᾮᰕ䛾㛗䛥 A: 䝟䜲䝥䛾᩿㠃✚ : ẚ㔜 g: 㔜ຊຍ㏿ᗘ x 14

2ឋໜኒ

` 2䛴䛾㗽䛜䝞䝛䛷⤖ྜ䛥䜜䛶䛔䜛䚹 ` 2䛴䛾㗽䜢ྜ䜟䛫䛯㔜ᚰ఩⨨䛾ື䛝䛻䛿⯆࿡䛜䛺䛔䚹┦ᑐ఩⨨䛾ື䛝䛰 䛡䛻⯆࿡䛜䛒䜛䚹Ѝ⮬⏤ᗘ⣔䛻䛺䜛 ` m<sub>1</sub>䛾఩⨨: x<sub>1</sub>, m<sub>2</sub>䛾఩⨨: x<sub>2</sub> ` x = x₁± x₂± d (d : 䝞䝛䛾⮬↛㛗) ` 㐠ື᪉⛬ᘧ: ` ┦ᑐ㐠ື䛰䛡䜢グ㏙䛩䜛㐠ື᪉⛬ᘧ: m₁ k c m₂ э ➼౯㉁㔞

LCRׅែNjӷơࡸ (1)

` ୪ิᅇ㊰(䜰䝗䝭䝑䝍䞁䝇䛾ぢ᪉) C R L e

LCRׅែNjӷơࡸ (2)

2020ᖺᗘ ` ┤ิᅇ㊰ (䜲䞁䝢䞊䝎䞁䝇䛾ぢ᪉) C R L i 17

ҥਰǓ܇ኒ

2020ᖺᗘ ` ᚤศ᪉⛬ᘧ(䜶䝛䝹䜼䞊ಖᏑ๎䛻䜘䜚࿘ᮇ᣺ື䛩䜛) 䛯䛰䛧䚸 э 䛜ᑠ䛥䛔䛸䛝( )䛾ゅ࿘Ἴᩘ ` ཝᐦゎ 䛯䛰䛧䚸sn(a, k) 䛿䝲䝁䝡䛾ᴃ෇㛵ᩘ, ₀䛿᣺䜚Ꮚ䛾᭱኱᣺ᖜ䚹 ` ࿘ᮇ T = 4K(k)/ (K(k)䛿➨䠍✀᏶඲ᴃ෇✚ศ) 9 ᣺ᖜ䛜ᑠ䛥䛡䜜䜀༢᣺ື䛾ሙྜT = 2$/ 䛸ྠ䛨 9 䛂᣺䜚Ꮚ䛾➼᫬ᛶ䛃䛿኱䛝䛔᣺ᖜ䛷䛿ᔂ䜜䜛 l  18

ࢍСਰѣƱƸ

` እຊ䜢᣺ື䛥䛫䛯ሙྜ䛾㐠ື᪉⛬ᘧ: ` a 䛜እຊ䛾᣺ᖜ䜢⾲䛩䚹 ` ୍⯡ゎ(ᐇᩘᅛ᭷್䜢䜒䛴ሙྜ): ` 䛂⮬⏤᣺ື䛾୍⯡ゎ䛃䛾㒊ศ䛾ᮍᐃಀᩘ䛿እຊ㡯䛾ᙳ㡪䜢ཷ䛡䜛䚹 ` 䛣䛾䜘䛖䛻䚸୍⯡ゎ䛜⮬⏤᣺ືゎ䛸ᐃᖖ᣺ືゎ䛾䛂࿴䛃䛻䛺䜛䛾䛿䚸”⥺ᙧ ᚤศ᪉⛬ᘧ”䛰䛛䜙䛷䛒䜛䚹༢᣺䜚Ꮚ䛾䜘䛖䛻㠀⥺ᙧ䛾ሙྜ䛿䚸ཝᐦ䛻䛿 ୍⯡ゎ䛜䛂࿴䛃䛻䛺䜙䛺䛔䚹 m k c x ⮬⏤᣺ື䛾୍⯡ゎ ᐃᖖ᣺ືゎ

ᐃᖖ᣺ື䛾᣺ᖜ(Amplitude) ఩┦㐜䜜(Phase delay; Phase lag) f = acos t

ࢍСਰѣƷᚐǛቇҥƴ൭NJǔ

` ᙉไ᣺ື䛾ゎ䜢ồ䜑䜛䛻䛿䝷䝥䝷䝇ኚ᥮(Laplace transform)䜢⏝䛔䜛䚹 䛯䛰䛧䚸L[x(t)] = X(s)䚹 ` බᘧ: ` X(s)䛻䛴䛔䛶ゎ䛟 ᐃᖖ᣺ື (Stationary vibration)

ܭࠝਰѣ᪮ƩƚƴბႸ

2020ᖺᗘ

` ఏ㐩㛵ᩘG(s)䛻 cos t 䜢ධຊ䛧䛯ሙྜ䛾ᐃᖖ᣺ື㡯䛾䝷䝥䝷䝇ኚ᥮䛿䚸

` ᫬㛫㡿ᇦ䛷᭩䛟䛸䚸

` a = 1䛸䛧䛯ሙྜ䛾๓䝨䞊䝆䛾⤖ᯝ䜒䛭䛖䛺䛳䛶䛔䜛䚹

䝀䜲䞁(Gain) ఩┦㐍䜏 䠄Phase lead; ㈇䛺䜙఩┦䛜㐜䜜䛶䛔䜛)

21

˷ᛩ

2020ᖺᗘ ` 䛔䛴䜒ఏ㐩㛵ᩘ䛻 s = j 䜢௦ධ䛧䛶⪃䛘䜜䜀䜘䛔䜟䛡䛷䛿䛺䛔䚹 ` ఏ㐩㛵ᩘ䛻 s = j 䜢௦ධ䛩䜛䜿䞊䝇䛿௨ୗ䛾2䛴䚹 ‡ ࿘Ἴᩘᛂ⟅(frequency response)䜢ㄪ䜉䜛ሙྜ 䠄௒ᅇ䛾ሙྜ䠅 9 ṇᘻἼ᣺ື䜢ຍ䛘䛯䛸䛝䛻ᐃᖖⓗ䛻ṧ䜛᣺ື䜢ㄪ䜉䜛䚹 9 s = j 䜢௦ධ䛩䜛䛣䛸䛷䚸㐣Ώᛂ⟅䜢↓ど䛧䛶䛔䜛䚹 9 ఏ㐩㛵ᩘ䛜Ᏻᐃ䛷䛒䜛䛣䛸䜢௬ᐃ䛧䛶䛔䜛 9 㟁Ẽᅇ㊰䛜஺ὶ㟁ᅽ※䛻䜘䜚ບ㉳䛥䜜䛶䛔䜛ሙྜ䛾ᐃᖖ≧ែ䜢ㄪ䜉 䜛䛾䜒䚸䛣䛾ሙྜ䛻┦ᙜ䛩䜛䚹㐣Ώᛂ⟅䜢ㄪ䜉䜛䛾䛻䛿౑䛘䛺䛔䚹 9 ఏ㐩㛵ᩘ䛿䜲䞁䝟䝹䝇ᛂ⟅䛾཰᮰ᇦ䛰䛡䜢ᐃ⩏ᇦ䛸䛩䜛䚹

‡ 䝘䜲䜻䝇䝖䛾Ᏻᐃุูἲ(Nyquist stability criterion)䜢౑䛖ሙྜ

9 䝘䜲䜻䝇䝖⥺ᅗ(Nyquist plot)䜢ᥥ䛟䛯䜑䛻 s = j 䜢௦ධ䛩䜛䚹 9 ఏ㐩㛵ᩘ䛿ゎᯒ᥋⥆䛥䜜䛶䛔䛶ᴟ䜢㝖䛟」⣲ᖹ㠃ୖ䛷ᐃ⩏䛥䜜䜛䚹 ` Bode⥺ᅗ䛿ୖグ䛾2䛴䛾୧᪉䛾┠ⓗ䛷౑䜟䜜䜛䚹 22

DzǤȳዴ׋

` ᶓ㍈(ᑐᩘ㍈)䛜ゅ࿘Ἴᩘ  ` ⦪㍈䛿䝀䜲䞁䜢db⾲グ (20log₁₀__G(j )__)䛧䛯䜒䛾 ` (䝀䜲䞁⥺ᅗ) + (఩┦⥺ᅗ) 䛜Bode⥺ᅗ ` ᙉไ᣺ື䛾ሙྜ䛾䝀䜲䞁 ` 䛺䜙䜀䚸__G(j )__䛿 䛻䛶᭱኱್ 䜢䜒䛴

ࢍСਰѣኒƷDzǤȳዴ׋

(1)

` m䛸 <sub>n</sub>䛿ᅛᐃ䚹 䜢ኚ໬䛥䛫䜛䚹 ` : ኱ Ѝ ᣺ᖜῶᑡ䚹≉䛻ᑠ䛥䛔䛿䛂ඹ᣺䛃䜢㉳䛣䛩䚹

ࢍСਰѣኒƷDzǤȳዴ׋

(2)

2020ᖺᗘ ` m 䛸 䜢ᅛᐃ䚹 <sub>n</sub>䜢ኚ໬䛥䛫䜛䚹 `  _n䛜኱э䝀䜲䞁⥺ᅗ䛜ྑୗ䛻ᖹ⾜⛣ື䚹( _n䛜኱䛝䛔䜋䛖䛜᣺ື䜢ᢚไ䚹) 25

ؕӨǁƷщƷˡᢋ

2020ᖺᗘ ` 䛂䛚䜒䜚䛻ຍ䜟䜛እຊ䛃䛛䜙䛂ᇶྎ䛻ఏ䜟䜛ຊ䛃䜎䛷䛾ఏ㐩㛵ᩘ䜢ồ䜑䜛䚹 ` ຊ䛾ఏ㐩䛾䝀䜲䞁 э ຊఏ㐩⋡ 䛯䛰䛧䚸p =  / _n ` p = 1 䛺䜙䜀䚸ຊఏ㐩⋡䛿ᚲ䛪䠍௨ୖ䚹 э 䝀䜲䞁䛜䠍䜘䜚኱䛝䛟䛺䜛࿘Ἴᩘ䛜䛒䜛䚹 ` 䛺䜙䜀䚸__G_F(j)__ 䛿䚸 䛻䜘䜙䛪䛻䠍䚹 э ᐃⅬ 26

щƷˡᢋྙƷDzǤȳዴ׋

` 䜢ቃ䛻䚸㧗࿘Ἴ㡿ᇦ䛷䛿ᑠ䛥䛔 䛜ຊఏ㐩⋡䛾ᢚไ䛻䛺䜚䚸ప ࿘Ἴ㡿ᇦ(≉䛻ඹ᣺࿘Ἴᩘ௜㏆)䛷䛿኱䛝䛔 䛜ຊఏ㐩⋡䛾ᢚไ䛻䛺䜛䚹 `  _n䛜኱䛝䛟䛺䜛䛸ྑ䛻ᖹ⾜⛣ືэඹ᣺䛧䛺䛔ሙྜ䛿䚸 _n䛜ᑠ䛥䛔䜋䛖䛜䛸 ຊఏ㐩⋡䛾ᢚไ䛻䛺䜛䚹(᣺ືᢚไ䛸㏫)

ؕӨƕਰѣƢǔئӳ

(ࢍС٭ˮਰѣ)

` ᇶྎ䛜㏻ᖖ䛾఩⨨䜘䜚 z(t) 䛰䛡ື䛟ሙྜ䜢⪃䛘䜛䚹эධຊ ` Z(s) = _{L[z(t)] 䛛䜙 X(s) = L[x(t)] 䜎䛷䛾ఏ㐩㛵ᩘ䜢ồ䜑䜛䚹} эຊ䛾ఏ㐩䛸ྠ䛨ఏ㐩㛵ᩘ(཮ᑐၥ㢟) ` z(t)䛜᣺ື䛩䜛ሙྜ䛾䝀䜲䞁= ຊఏ㐩⋡䛸ྠ䛨 ` ᣺ືᢚไ䛾䛯䜑䛾᪉⟇ 䛚䜒䜚䜈䛾እຊ䛜᣺ື z(t)䛜᣺ື  _n ኱䛝䛔䜋䛖䛜䜘䛔 ୍⯡ⓗ䛻ᑠ䛥䛔䜋䛖䛜Ⰻ䛔 ኱䛝䛔䜋䛖䛜Ⰻ䛔 ࿘Ἴᩘ䛻䜘䛳䛶ኚ䜟䜛

ɧᤊӳƍόႴǛஊƢǔ࢐ࣱ᠆Ʒׅ᠃

(1)

2020ᖺᗘ ` ೫㔜ᚰ e 䜢ᣢ䛴෇┙䛜ᅇ㌿ G S e  G ᙎᛶ㍈䛾ᙎᛶಀᩘ䜢 k ῶ⾶ಀᩘ䜢 c 䛸䛚䛟 S ධຊ: 29

ɧᤊӳƍόႴǛஊƢǔ࢐ࣱ᠆Ʒׅ᠃

(2)

2020ᖺᗘ ` ᇶᮏ䛾㐠ື᪉⛬ᘧ ` G䛾㐠ື᪉⛬ᘧ ` S䛾㐠ື᪉⛬ᘧ 30

ɧᤊӳƍόႴǛஊƢǔ࢐ࣱ᠆Ʒׅ᠃

(3)

` ୍⯡䛻㔠ᒓෆ㒊䛾ῶ⾶ಀᩘ䛿ᑠ䛥䛔 ` 䜘䛳䛶䚸 =  _n௜㏆䛷ඹ᣺䛩䜛䚹э༴㝤㏿ᗘ ` ㍈䛾ᶓ᣺ື䛾ᅛ᭷᣺ືゅ࿘Ἴᩘ䛷㍈䜢ᅇ㌿䛥䛫䛶䛿䛔䛡䛺䛔䚹 `  䛜኱䛝䛔䛺䜙䜀__G_G(j )__ 䛿䝊䝻䛻₞㏆䚹䛴䜎䜚䚸ᅇ㌿ゅ࿘Ἴᩘ䛜኱䛝 䛔䛸䛝䛿ⅬG䜢୰ᚰ䛻䛧䛶ᅇ䜛(⮬ືㄪᚰస⏝)䚹 `  䛜ᑠ䛥䛔䛺䜙䜀__G_S(j )__ 䛿䝊䝻䛻₞㏆䚹䛴䜎䜚䚸ᅇ㌿ゅ࿘Ἴᩘ䛜ᑠ䛥 䛔䛸䛝䛿ⅬS䜢୰ᚰ䛻䛧䛶ᅇ䜛䚹

ȕǡȳȷȇȫȷȝȸȫ૾ᆉࡸ

(1)

` 䝣䜯䞁䞉䝕䝹䞉䝫䞊䝹(Van der Pol)᪉⛬ᘧ

` 㠀⥺ᙧ䛾᪉⛬ᘧ ` ⢓ᛶᦶ᧿㡯: 䜢ቃ䛻➢ྕ཯㌿ э ཎⅬ䛿୙Ᏻᐃ, እഃ䛷䛿᣺ᖜᢚไ ` ኚ᥮ 䛾ୗ䛷䛿 эඖ䛾䝣䜯䞁䞉䝕䝹䞉䝫䞊䝹᪉⛬ᘧ䛿䛒䜛ព࿡䛷䛂ᶆ‽ᙧ䛃 ` ኚ᥮ ኚ᥮ э э

ȕǡȳȷȇȫȷȝȸȫ૾ᆉࡸ

(2)

2020ᖺᗘ ` ߳ 䛜䝊䝻䛺䜙ゅ࿘Ἴᩘͳ䛾ㄪ࿴᣺ືᏊ ߳ 䛜ᑠ䛥䛔䛸䛝䛾㏆ఝゎ 1. ゅ࿘Ἴᩘ_{1䛷ᅇ㌿䛩䜛ᗙᶆ䜢䛸䜛} 2. ྑ㎶䛜ᑠ䛥䛔䛾䛷ݑ,ݒ 䛾ኚ໬䜘䜚ゅ࿘Ἴᩘ ͳ 䛾᣺ື䛾᪉䛜㏿䛔э ᖹᆒ ໬(averaging)ᡭἲ (ゅ࿘Ἴᩘ ͳ 䛾᣺ື䛾㡯䜢0䡚2ߨ䛷ᖹᆒ್䜢䛸䜛) 3. ༙ᚄ 䛾ᣲື э 4. ༙ᚄݎ 䛿2䛻཰᮰ э߳ 䛜ᑠ䛥䛔䛸䛝 ݔ, ݕ 䛿༙ᚄ2䛾෇䛻཰᮰(㏆ఝⓗ) 33

ȕǡȳȷȇȫȷȝȸȫ૾ᆉࡸ

(3)

2020ᖺᗘ ߳ 䛜኱䛝䛔䛸䛝䛾㏆ఝゎ ` ߳ 䛜኱ э ݔ 䛻ẚ䜉ݖ䛾ື䛝䛜㐜䛔 ` 㐜䛔ື䛝 э ཯᫬ィᅇ䜚 ` ᭱⤊ⓗ䛻㉥䛔⥺ୖ䜢ື䛟 ᐃᩘ䛸䜏䛺䛩 ㏿䛔ື䛝 䛻╔┠ ᖹ⾮Ⅼ䛾ศ㢮 Ѝ Ᏻᐃ䛺ᖹ⾮Ⅼ䛜1䛴 Ᏻᐃ䛺ᖹ⾮Ⅼ䛜2䛴 Ѝ 䠇 ୙Ᏻᐃ䛺ᖹ⾮Ⅼ䛜1䛴 ㏿䛔ື䛝 䛾᪉ྥ 34

ȕǡȳȷȇȫȷȝȸȫ૾ᆉࡸ

(4)

` ߳ = 0.1 Time

ȕǡȳȷȇȫȷȝȸȫ૾ᆉࡸ

(5)

` ߳ ൌ ͳ ඲䛶䛾ṇ䛾߳ 䛻䛚䛔䛶䚸 䛒䜛㛢㌶㐨(߳ 䛻䜘䛳䛶␗䛺䜛)䛻ᕳ䛝௜䛟 䝸䝭䝑䝖䞉䝃䜲䜽䝹(Limit cycle) Ꮡᅾᛶ: 䝸䜶䝘䞊䝹(Liénard)䛾ᐃ⌮ Time

ȕǡȳȷȇȫȷȝȸȫ૾ᆉࡸ

(6)

2020ᖺᗘ ` ߳ = 15 37 Time

ȕǡȳȷȇȫȷȝȸȫ૾ᆉࡸ

(7)

2020ᖺᗘ 38 ` 䝖䞁䝛䝹䝎䜲䜸䞊䝗䜢⏝䛔䛯Ⓨ᣺ᅇ㊰

ȪǨȊȸȫƷܭྸ

(Liénard’s Theorem)

` 䝸䜶䝘䞊䝹᪉⛬ᘧ 䛾䝸䝭䝑䝖䝃䜲䜽䝹䛾Ꮡᅾ䜢♧䛩ᐃ⌮ ` [ᐃᐃ⌮] 䝸䜶䝘䞊䝹᪉⛬ᘧ䛜௨ୗ䛾4᮲௳䜢‶䛯䛩䛺䜙䜀䠈၏୍䛾Ᏻᐃ䛺 䝸䝭䝑䝖䞉䝃䜲䜽䝹䜢ᣢ䛴䠊 1. 䛿㐃⥆䛺അ㛵ᩘ 2. 䛿㐃⥆䛺ወ㛵ᩘ䠊 3. 䛸 䛾➢ྕ䛜୍⮴ 4. ṇᐃᩘ 䛜Ꮡᅾ䛧䠈 䛜ḟ䛾䜘䛖䛺ᛶ㉁䜢‶䛯䛩 䝣䜯䞁䞉䝕䝹䞉䝫䞊䝹᪉⛬ᘧ䛿ᚲ䛪䝸䝭䝑䝖䝃䜲䜽䝹䜢ᣢ䛴

ᡲ঺ਰѣ

(1)

` 㐃ᡂ᣺ື(coupled vibration; coupled osillation) = 2⮬⏤ᗘ௨ୖ䛾᣺ື ` 䛯䛸䛘䜀䚸௨ୗ䛾౛䛷⪃䛘䜘䛖䚹 m₁ k₁ x₁ f = acos t c₁ m₂ k₂ x₂ c₂

ᡲ঺ਰѣ

(2)

2020ᖺᗘ ` ఏ㐩㛵ᩘ 41

ᡲ঺ਰѣ

(3)

2020ᖺᗘ ` 㐃ᡂ᣺ື䜢ṇ☜䛻⌮ゎ䛩䜛䛻䛿䚸≧ែ✵㛫䛷⪃䛘䜛ᚲせ䛜䛒䜛䚹 ` 䛯䛸䛘䜀䚸䛣䛣䛷䛾౛䛷䛿௨ୗ䛾䜘䛖䛻䛺䜛䚹 ` 㐠ື᪉⛬ᘧ ` ≧ែ᪉⛬ᘧ 42

Ȣȸȉᚐௌ

(1)

` እຊ f = 0 䛸䛧䛶⪃䛘䜛䚹 䜎䛯䚸㠀ῶ⾶䛾ሙྜC = 0 䜢⪃䛘䜛䚹 ` ⾜ิ A 䛾ᅛ᭷್䛸ᅛ᭷䝧䜽䝖䝹䛜㔜せ ` 䛣䛾ሙྜ䚸⾜ิ A 䛿⣧⹫ᩘ䛾ᅛ᭷್䛧䛛ᣢ䛯䛺䛔䚹 [⡆༢䛺ド᫂] 䛾ᅛ᭷್䛩䛺䜟䛱 䛾ᅛ᭷್䜢 `_i(i = 1,«,n)䛸䛩䜛䛸䚸`<sub>i</sub>䛿ṇ䛷䛒䜛䚹䛣䛾䛸䛝䚸⾜ิ A 䛾ᅛ᭷್䛿䚸 䛸䛺䜛䚹 ` 䛾ᅛ᭷䝧䜽䝖䝹䜢 p_i(㛗䛥䠍䛻ṇつ໬)䛸䛩䜛䛸䚸A 䛾ᅛ᭷䝧䜽䝖 䝹䛿䚸 ` 䛴䜎䜚䚸 p_i(i = 1,«,n) 䛾 n ಶ䛾᪉ྥ䛾᣺ື䛻ศゎ䛷䛝䜛䚹

Ȣȸȉᚐௌ

(2)

` ᗙᶆኚ᥮ ` 䝰䞊䝗ศゎ䛥䜜䛯䝅䝇䝔䝮(n ಶ䛾ㄪ࿴᣺ືᏊ䛻ศゎ) ඖ䛾ᗙᶆ䛷⾲⌧䛧䛯ゎ

͞ࣇЎ૾ᆉࡸưᘙྵƞǕǔǷǹȆȠ

2020ᖺᗘ 45 ` 䛣䜜䛛䜙䠈೫ᚤศ᪉⛬ᘧ䛷⾲⌧䛥䜜䜛䝅䝇䝔䝮䛻㛵䛧䛶Ꮫ䜃䜎䛩䠊 ` ೫ᚤศ᪉⛬ᘧ䛿↓㝈ḟඖ⣔䠊 э ከ⮬⏤ᗘ⣔䛾ḟඖᩘ䛜↓㝈኱䛾ሙྜ䛻┦ᙜ䠊 ` ୺䛻ᢅ䛖䛾䛿䠈䝙䝳䞊䝖䞁䛾㐠ື᪉⛬ᘧ䛻ᨭ㓄䛥䜜䜛ᑐ㇟ ` ≉䛻↓ῶ⾶⣔(ㄝ᫂䛾⡆༢໬䛾䛯䜑эῶ⾶䛒䜚䛷䜒Ṥ䛹ྠ䛨) ` ↓ῶ⾶᭷㝈ḟඖ⣔䛾ゎ(ᡂศ䛤䛸䛾⾲⌧) э ྛ䝰䞊䝗䛾ゎ䛾࿴ ` ↓ῶ⾶↓㝈ḟඖ⣔(㞳ᩓ䝇䝨䜽䝖䝹䜢ᣢ䛴⣔)䛾ゎ 䝰䞊䝗ᩘ䛜↓㝈኱䛷䠈ᅛ᭷䝧䜽䝖䝹䛾௦䜟䜚䛻㛵ᩘ 䝰䞊䝗ẖ䛻ぢ䜜䜀ኚᩘศ㞳

ࢅƷਰѣ૾ᆉࡸ

(1)

2020ᖺᗘ 46 ᘻ(string)䛾㐠ື᪉⛬ᘧ䜢ồ䜑䜛䚹 ` ᙇຊ: T, ⥺ᐦᗘ: ' 䛸䛩䜛䚹 ` ᘻ䛾ሙᡤ䜢♧䛩䝟䝷䝯䞊䝍: x, ᫬้䜢⾲䛩䝟䝷䝯䞊䝍㻦㻌t ` ሙᡤ x, ᫬้ t 䛻䛚䛡䜛ᘻ䛾(ᶓ᪉ྥ䛾)ኚ఩: ݑ(x, t ) x 䛛䜙 x +ߜx 䜎䛷䛾ᚤᑠ⥺⣲䛻╔┠ 䛣䜜䛜ኚᩘ! 䛣䜜䜢ồ䜑䜛 э 䝔䞊䝷䞊⣭ᩘᒎ㛤 ୖ᪉䛜ṇ ㈇᪉ྥ ᙇຊ

ࢅƷਰѣ૾ᆉࡸ

(2)

` ᚤᑠ⥺⣲䛾㉁㔞: ` ᣺ᖜ᪉ྥ䛾ຊ: ` 䝙䝳䞊䝖䞁䛾㐠ື᪉⛬ᘧ: ୧㎶䜢ߜx 䛷๭䛳䛶䚸ߜx→0 ᘻ䛾Ἴື᪉⛬ᘧ(Wave equation)

э ೫ᚤศ᪉⛬ᘧ(partial differential equation; PDE)

Ƴƥඬѣ૾ᆉࡸƱǑƹǕǔƷƔ

` ᪉⛬ᘧ

` ୍⯡ゎ

э ௦ධ䛩䜛䛸୧㎶䛜 䛷୍⮴

` 㐍⾜Ἴ(Traveling wave) u_fwd(x‒ ct): ㏿䛥 c 䛷ṇ䛾᪉ྥ䛻㐍䜐

` ᚋ㏥Ἴ(Backward wave) u_back(x + ct): ㏿䛥 c 䛷㈇䛾᪉ྥ䛻㐍䜐

Ἴ䛾ື䛝䜢⾲⌧䛩䜛᪉⛬ᘧ э Ἴື᪉⛬ᘧ

` ቃ⏺᮲௳䛸ึᮇ᮲௳䛷䚸㛵ᩘ u_fwd(«) 䛚䜘䜃 u_back(«) 䛜Ỵ䜎䜛䚹

` ቃ⏺᮲௳䛻䜘䛳䛶䛿䚸㐍⾜Ἴ䛸ᚋ㏥Ἴ䛾㔜䛽ྜ䜟䛫䛷ᐃᅾἼ(Standing

ࢅƷਰѣƷؾမவˑƱ٭ૠЎᩉ

2020ᖺᗘ 49 ` 䜼䝍䞊䛾ᘻ䛾䜘䛖䛻䚸୧➃䛜ᅛᐃ䛥䜜䛶䜛㛗䛥L䛾ᘻ䜢⪃䛘䜛䚹 ୧➃ᅛᐃ䛾ቃ⏺᮲௳(boundary conditions) ኚᩘศ㞳䛥䜜䛯ゎ䜢௬ᐃ э ୍䛴䛾䝰䞊䝗䛰䛡䜢⪃䛘䜛 ௦ධ: 䛚䜘䜃 ᕥ㎶䛿 x ྑ㎶䛿 t 䛾㛵ᩘ䛷 ᜏ➼ⓗ䛻➼䛧䛔эᐃᩘ௨እ䛒䜚䛘䛺䛔

ࢅƷਰѣƷᚐǛ൭NJǔ

(1)

2020ᖺᗘ 50 ` E< 0䛸௬ᐃ䛧䚸(x)䛻㛵䛩䜛ゎ䜢ồ䜑䜛 ቃ⏺᮲௳ (0) =  (L) = 0䜢‶䛯䛩ゎ䛿ᜏ➼ⓗ䛻䝊䝻䛺ゎ䛾䜏 э୙㐺 ` E= 0䛾䛸䛝䜒୙㐺䛷䛒䜛䛣䛸䛿䚸 ྠᵝ䛻♧䛩䛣䛸䛜䛷䛝䜛䠊 E > 0 䛾ሙྜ䛾䜏ព࿡䛾䛒䜛ゎ䛜Ꮡᅾ ` 䛸䛚䛟䛸䚸ゎ䛿 ` ቃ⏺᮲௳ (0) = 0䜘䜚A_c= 0 ` ቃ⏺᮲௳ (L) = 0䜘䜚 ᅛ᭷ゅ᣺ືᩘ(ᅛ᭷ゅ࿘Ἴᩘ;

natural angular frequency; angular eigenfrequency)

ࢅƷਰѣƷᚐǛ൭NJǔ

(2)

` ༢୍䝰䞊䝗ゎ(ᇶ‽䝰䞊䝗) 䛣䛣䛷䚸 䛸䛚䛔䛯䚹 ` ᇶᮏ᣺ື(fundamental vibration)䛾ゅ࿘Ἴᩘ ᙇຊ䛜኱䛝䛔䜋䛹䚸⥺ᐦᗘ䛜㍍䛔(ྠ୍ᮦᩱ䛺䜙⣽䛔)䜋䛹䚸ᘻ䛜▷䛔 䜋䛹䚸ᇶᮏ᣺ືᩘ䛿㧗䛟䛺䜛䚹 ` ᅛ᭷ゅ᣺ືᩘ䛿ᇶᮏ᣺ືゅ᣺ືᩘ䛾⮬↛ᩘಸ ᇶ‽䝰䞊䝗䛿 ↓㝈ಶ䛒䜛

ࢅƷؕแȢȸȉ

` ᇶ‽䝰䞊䝗䛾Ἴᙧ i = 1 (ᇶ‽᣺ື) i = 2 (2ಸ᣺ື) i = 3 (3ಸ᣺ື) ⠇: node ⭡: anti-node ⭡ ⭡ ⭡ _{⠇ ⠇}

ӵǓɦƛƨᦋƷਰѣ

(1)

2020ᖺᗘ 53 ` ྞ䜚ୗ䛢䛯㙐(ྑᅗ) ` ᙇຊ䛜ሙᡤ䛻䜘䛳䛶␗䛺䜛(ୗ➃䛷䛿ᙇຊ䝊䝻) ` ᙇຊ䛸㉁㔞䛜⥺ᐦᗘ䛻ẚ౛䛩䜛䛾䛷䚸 ᪉⛬ᘧ䛿⥺ᐦᗘ䛻䜘䜙䛺䛔䚹 ` ྞ䜚ୗ䛢䛯㙐䛾᣺ື䛾᪉⛬ᘧ ` ቃ⏺᮲௳ ` ኚᩘศ㞳 㔜ຊຍ㏿ᗘ g L x u(x,t)

ӵǓɦƛƨᦋƷਰѣ

(2)

2020ᖺᗘ 54 ` 䝟䝷䝯䞊䝍ኚ᥮: x = (g / 4) ( / )2 ` 0ḟ䝧䝑䝉䝹(Bessel)ᚤศ᪉⛬ᘧ ` ቃ⏺᮲௳ э ᅛ᭷ゅ᣺ືᩘ䜢ồ䜑䜛᪉⛬ᘧ = 2.4048, 5.5201, 8.6537, 11.7915, « ` ᭱⤊ⓗ䛺ᇶ‽䝰䞊䝗 ୍⯡ゎ J₀( ) : 0ḟ䛾➨1✀䝧䝑䝉䝹㛵ᩘ ( : 0ḟ䝧䝑䝉䝹㛵ᩘ䛾䝊䝻Ⅼ) ( _i: 0ḟ䝧䝑䝉䝹㛵ᩘ䛾i␒┠䛾䝊䝻Ⅼ) (ཎⅬ䛷Ⓨᩓ䛧䛺䛔᮲௳䜘䜚 Neumann㛵ᩘ䛿୙㐺)

ӵǓɦƛƨᦋƷਰѣ

(3)

` ྞ䜚ୗ䛢䛯㙐䛻䛚䛡䜛 ᇶ‽䝰䞊䝗䛾Ἴᙧ ` 0ḟ䛾➨୍✀䝧䝑䝉䝹㛵ᩘ䛸ᖹ᪉᰿ 䛷⾲⌧䛷䛝䜛 1ḟ 2ḟ 3ḟ

ొǛˡǘǔጏඬ

` Წ䜢ఏ䜟䜛⦪Ἴ(Longitudinal wave) x 䛾఩⨨䛾ṍ䜏(ᘬ䛳ᙇ䜚): x + dx 䛾఩⨨䛾ṍ䜏: x 䛾఩⨨䛾᩿㠃䛜 x + u(x, t) 䛻ኚ఩ э u(x, t) 䛿ኚ఩㔞 ` ⦪Ἴ䛾Ἴື᪉⛬ᘧ(E: 䝲䞁䜾⋡, ': ᐦᗘ, S: ᩿㠃✚) ` _{୧㎶䜢 dx 䛷๭䛳䛶 dx  0} x u(x, t) x+dx ᘻ䛾Ἴື᪉⛬ᘧ䛸ྠ䛨 ྑ➃䛛䜙䛾ຊ䛸ᕥ➃ 䛛䜙䛾ຊ䛷᪉ྥ䛜㏫

ጏඬƷؾမவˑ

2020ᖺᗘ 57 ` ᅛᐃ➃ ` ⮬⏤➃ ` Ẽᰕ䛾ሙྜ䛿䚸 ኚ఩Ἴ䛸⪃䛘䛯ሙྜ䚹ᐦᗘἼ䛸⪃䛘䛯ሙྜ䛿ቃ⏺᮲௳ 䛜ධ䜜᭰䜟䜛䚹 (Ẽᰕ䛾ሙྜ) (Ẽᰕ䛾ሙྜ = 㛤ཱྀ➃) ᮏᙜ䛿㛤ཱྀ➃⿵ṇ䛜ᚲせ

׍ܭᇢȷᐯဌᇢƷƱƖƷ׍ஊਰѣඬ࢟

2020ᖺᗘ 58 ` ᕥᅛᐃ➃䚸ྑ⮬⏤➃䛾ሙྜ䛾ᅛ᭷᣺ື(ᇶ‽䝰䞊䝗)Ἴᙧ ᅛᐃ➃ ⮬⏤➃

ᐏƷਰѣ

` ⭷䛾᣺ື䛿ᘻ䛾᣺ື䛾2ḟඖ∧ ` Ἴື᪉⛬ᘧ T: ⾲㠃ᙇຊ (ḟඖ䛿ຊ㾂㛗䛥), ': 㠃ᐦᗘ ` ѭ䜢౑䛳䛯⾲⌧ ѭ䛾ᐃ⩏(3ḟඖ∧䛜ᬑ㏻䛰䛜䛣䛣䛷䛿2ḟඖ) x y u 䝷䝥䝷䝇(Laplace)స⏝⣲

ᧈ૾࢟ᐏƷਰѣ

(1)

` 㛗᪉ᙧ⭷: ` ቃ⏺᮲௳: э 㛗᪉ᙧ䛾ྛ㎶ୖ䛷䝊䝻 ` ༢୍䝰䞊䝗ゎ䛾ኚᩘศ㞳 ` ௦ධ䛧䛶ᩚ⌮ эᕥ㎶䛿t䛾䜏䚸ྑ㎶➨1㡯䛿x䛰䛡䚸ྑ㎶➨2㡯䛿y䛰䛡䛾㛵ᩘ э䛭䜜䛮䜜ᜏ➼ⓗ䛻ᐃᩘ

ᧈ૾࢟ᐏƷਰѣ

(2)

2020ᖺᗘ 61 ` ศゎ䛥䜜䛯᪉⛬ᘧ ᘻ䛾ሙྜ䛸ྠᵝ䛻㈇䛾 <sub>x</sub>, <sub>y</sub>䛿୙㐺 ` ቃ⏺᮲௳䜢⪃䛘䜛䛸

ᧈ૾࢟ᐏƷਰѣ

(3)

2020ᖺᗘ 62 ` 㛗᪉ᙧ⭷䛾᣺ື䛾ゎ ` L_y→ ∞䛾䛸䛝䚸ᘻ䛾᣺ື䛾ᘧ䛻䛺䜛 ` ᅛ᭷ゅ᣺ືᩘ䛿x䛸y䛾୧᪉䛾Ἴ䛾ᩘ䛻౫Ꮡ

ᧈ૾࢟ᐏƷਰѣ

(3)

` 䛥䜎䛦䜎䛺᣺ື䝰䞊䝗(⦪/ᶓ = 2/3䛾䛸䛝)

ό࢟ᐏƷਰѣ

(1)

` ༙ᚄR䛾෇䛾ᙧ≧䛾⭷䛾᣺ື䜢⪃䛘䜛 ` ᴟᗙᶆ⾲⌧: ` ቃ⏺᮲௳: ` ᴟᗙᶆ⾲⌧䛻䛚䛡䜛䝷䝥䝷䝇స⏝⣲ ` ᴟᗙᶆ⾲⌧䛷䛾Ἴື᪉⛬ᘧ u(x, y, t) эu(r, , t)䛸⾲䛩 ࿘ᮇቃ⏺᮲௳ (Periodic boundary condition; Cyclic symmetry condition)

ό࢟ᐏƷਰѣ

(2)

2020ᖺᗘ 65 ` ኚᩘศ㞳: ᘻ䛾᣺ື䛾ሙྜ䛸ྠᵝ䛻䚸 䛣䜜䜢䛥䜙䛻ኚᙧ䛧䛶䚸 ෇࿘ୖ䛾ቃ⏺᮲௳䜘䜚䛣䜜䛜ṇ䜎䛯䛿䝊䝻䛾ሙྜ䛿୙㐺(ド᫂┬␎) ࿘ᮇቃ⏺᮲௳䜘䜚䛣䜜䛜㈇䛾ሙྜ䛿୙㐺(ド᫂┬␎)

ό࢟ᐏƷਰѣ

(3)

2020ᖺᗘ 66 ` ෇࿘᪉ྥ( ᪉ྥ) 䛾᪉⛬ᘧ ` ༙ᚄ᪉ྥ(r᪉ྥ) 䛾᪉⛬ᘧ ゎ: ቃ⏺᮲௳䜘䜚 ࿘ᮇቃ⏺᮲௳䜘䜚n䛿ᩚᩘ(㈇䛾n䛿෕㛗䛺䛾䛷⪃䛘䛺䛔) nḟ䝧䝑䝉䝹᪉⛬ᘧ nḟ䝧䝑䝉䝹㛵ᩘ : nḟ䝧䝑䝉䝹㛵ᩘ䛾䛒䜛䝊䝻Ⅼ

ό࢟ᐏƷਰѣ

(4)

` ␗䛺䜛*䛿ู䝰䞊䝗эsin䛸cos䛻ศゎ ` ᭱⤊ⓗ䛺ゎ 䛣䛣䛷䚸 ( : nḟ䝧䝑䝉䝹㛵ᩘ䛾j␒┠䛾䝊䝻Ⅼ)

ό࢟ᐏƷਰѣ

(4)

` <sub>n,j</sub>䛸 <sub>n,j</sub>䛿ู ` 䛾䛸䛝 ෇࿘᪉ྥ䛾㐍⾜Ἴ

నƷਰѣƷ૾ᆉࡸ

(1)

2020ᖺᗘ 69 ` ᱱ(䛿䜚; beam) 䛾᣺ື (E: 䝲䞁䜾⋡, I: ᩿㠃஧ḟ䝰䞊䝯䞁䝖, ': ⥺ᐦᗘ) ` ᅗ䛾▮༳᪉ྥ䛜ṇ(ྑ䛛䜙ാ䛟䝰䞊䝯䞁䝖䞉ຊ䛿䚸ᕥ䛛䜙ാ䛟ሙྜ䛸㏫) ` ᭤䛢䝰䞊䝯䞁䝖䛸ᱱ䛾ᙧ≧䛾㛵ಀ ` 䝰䞊䝯䞁䝖䛾䛴䜚ྜ䛔᪉⛬ᘧ స⏝䞉཯స⏝ ୧㎶x䛷๭䛳䛶x→ 0

నƷਰѣƷ૾ᆉࡸ

(1)

2020ᖺᗘ 70 ` ᚤᑠせ⣲䛾⦪᪉ྥ䛾㐠ື᪉⛬ᘧ(ᘬ䛳ᙇ䜚䞉ᅽ⦰ຊ䛜䛺䛔䛸௬ᐃ) ` ୧㎶x䛷๭䛳䛶x→ 0 ` ᱱ䛾᪉⛬ᘧ ` ୍ᵝ䛺ᱱ䛾ሙྜ

నƷਰѣᲧ٭ૠЎᩉ

` ᘻ䛾᣺ື䛸ྠᵝ䛻䚸༢୍䝰䞊䝗䜢⪃䛘䛶ኚᩘศ㞳 ` ௦ධ䛧䛶ᩚ⌮(ୗグ䛾ᘧ䛿ᐃᩘ䛷䛺䛟䛶䛿䛺䜙䛺䛔) ` ศ㞳䛥䜜䛯᪉⛬ᘧ䛸ゎ(୍ᵝ䛺ᱱ䛾ሙྜ; L: ᱱ䛾㛗䛥) ⌧ᐇ䛾ቃ⏺䛾⤌䜏ྜ䜟䛫䛷䛿䚸䛣䜜䛜ṇ䜎䛯䛿䝊䝻䛾ሙྜ䛿୙㐺(ド᫂┬␎)

నƷؾမவˑ

` ༢⣧ᨭᣢ(䛯䜟䜏0, 䝰䞊䝯䞁䝖0) ` ᅛᐃ(䛯䜟䜏0, ഴᩳ0) ` ᙎᛶᨭᣢ(䝰䞊䝯䞁䝖0, ຊ㔮ྜ) ` ⮬⏤(䝰䞊䝯䞁䝖0, ๧᩿ຊ0) ` 㞟୰㉁㔞(䝰䞊䝯䞁䝖0, ຊ㔮ྜ) ` ๛య(䝰䞊䝯䞁䝖㔮ྜ, ຊ㔮ྜ) ➢ྕྠ㡰䛷䚸ୖ䛜ᕥ➃䚸ୗ䛜ྑ➃ k m m J

ҥኝૅਤనƷਰѣᚐ

2020ᖺᗘ 73 ` ༢⣧ᨭᣢᱱ䛾ቃ⏺᮲௳ ` ୍⯡ゎ䛻௦ධ ᕥ➃: ྑ➃(ᕥ➃᮲௳䜢⪃៖): 䜘䜚 э ` 䛜㠀䝊䝻䛾 䜢チᐜ䛩䜛᮲௳э ` ୍⯡ゎ: э ࿘Ἴᩘ䛿䚸䝰䞊䝗␒ྕ䛾2஌䛻ẚ౛䛧䠈ᱱ䛾㛗䛥䛾2஌䛻཯ẚ౛䛩䜛䚹

ɲᇢ׍ܭనƷਰѣᚐ

(1)

2020ᖺᗘ 74 ` ቃ⏺᮲௳: ` ᕥ➃䜘䜚䚸 ` ྑ➃䜘䜚䚸 э 㠀䝊䝻ゎ䛜Ꮡᅾ䛩䜛᮲௳: ` ୍⯡ゎ

ɲᇢ׍ܭనƷਰѣᚐ

(2)

` 䛾ゎ: ` ྛᅛ᭷䝰䞊䝗䛾Ἴᙧ(᭱኱᣺ᖜ䛷ṇつ໬) i = 1 i = 2 i = 3 i = 4

έᇢឋ᣽໯Ơ༾ਤƪనƷਰѣᚐ

(1)

` ᕥᅛᐃ, ྑ⮬⏤䛾ቃ⏺᮲௳: ` ᕥ➃䜘䜚䚸 ` ྑ➃䜘䜚䚸 э 㠀䝊䝻ゎ䛜Ꮡᅾ䛩䜛᮲௳: ` ୍⯡ゎ

έᇢឋ᣽໯Ơ༾ਤƪనƷਰѣᚐ

(2)

2020ᖺᗘ 77 ` 䛾ゎ: ` ྛᅛ᭷䝰䞊䝗䛾Ἴᙧ(᭱኱᣺ᖜ䛷ṇつ໬) i = 1 i = 2 i = 3 i = 4

ᚐௌщܖ

ᇹ2ᇘ

ˎेˁʙƷҾྸ

(1)

` n ㉁Ⅼ⣔䛾ຊ䛾㔮䜚ྜ䛔䜢⪃䛘䜛䚹 ` ྛ㉁Ⅼ i 䛾఩⨨䜢 x_{i; <}3_{䛸䛩䜛䚹䛭䜜䜙䜢඲䛶䜎䛸䜑䛯䝧䜽䝖䝹䜢 x 䛸䛚䛟䚹} ` 㔮䜚ྜ䛔≧ែ䛷䛿䚸ྛ㉁Ⅼ i 䛻䛚䛔䛶䛾ຊ䛾ྜຊ f<sub>i</sub>(x)<sub>; <</sub>3<sub>䛿䝊䝻䚹</sub> ` 䛣䛾ᘧ䛿䚸

䛸➼౯䚹 (௬᝿௙஦䛾ཎ⌮; virtual work principle)

䛂ᜏ➼ᘧ䛜ᡂ䜚❧䛴䛃 э 䛂ಀᩘ= 0䛃 䛾㛵ಀ䜢ᛮ䛔ฟ䛩䛣䛸䚹 [ὀព] 䛣䛾ẁ㝵䛷䛿δx_i䛿㐺ᙜ䛻ᣢ䛱ฟ䛧䛯ᩘᏛୖ䛾ኚᩘ䚹 (ຊ䛾䛴䜚䛒䛔ᘧ)

ˎेˁʙƷҾྸ

(2)

` δx<sub>i</sub>䜢䚸㉁Ⅼ i 䛾䛂௬᝿ⓗ䛺ᚤᑠ䛺ኚ఩䛃(௬᝿ኚ఩; virtual displacement) 䛸䜏䛺䛩䚹 䛣䜜䛿䛩䛺䜟䛱䚸x<sub>i</sub>䛜 x<sub>i</sub>+δx<sub>i</sub>䛻ኚ఩䛧䛯䛸⪃䛘䜛䚹 ` 䛂ຊ㽢ኚ఩䛃䛿䛂௙஦䛃䚹䜘䛳䛶 䛿௬᝿௙஦(virtual work)䚹 ` ௬᝿௙஦䛾ཎ⌮ 䛂㔮䜚ྜ䛔≧ែ䛷䛿䚸඲䛶䛾ᚤᑠ䛺௬᝿ኚ఩䛻ᑐ䛧௬᝿௙஦䛾࿴䛿䝊䝻䛃 ` ኚ఩䛸ຊ䛿ྠ䛨ྥ䛝䛜ṇ э ௬᝿௙஦䛿䚸ຊ䛜㉁Ⅼ䛻ᑐ䛧䛶⾜䛖௙஦䚹 ㉁Ⅼ䛜እ⏺䛻ᑐ䛧䛶⾜䛖௙஦䛷䛿䛺䛔䚹 ` [␲ၥ] 䛺䛬௬᝿ኚ఩䛿ᚤᑠ䛷䛺䛟䛶䛿䛺䜙䛺䛔䛾䛛? [ᅇ⟅] ᚤᑠ䛨䜓䛺䛔䛸 䛿௙஦㔞䛸࿧䜉䛺䛟䛺䜛䛛䜙䚹 䛺䛬䛺䜙䜀䚸x_i䛜 x_i+δx_i䛻ኚ఩䛩䜛䛸 f_i(x) 䛾್䜒ኚ໬䛩䜛䚹

ӈ݅Ƴˁʙ᣽ưᙸƯLjǔƱ

2020ᖺᗘ ` 䝫䝔䞁䝅䝱䝹ሙ+ ୍ᐃእຊ䜢௬ᐃ (ୗ䛾✚ศ䛜✚ศ⤒㊰䛻䜘䜙䛺䛔) ` ௬᝿ኚ఩䜢ᚤᑠ䛸䛧䛺䛔ሙྜ䛾ཝᐦ䛺௙஦㔞 ` 䛂_δx_i䛻㛵䛩䜛୍ḟ䛾㡯䛜䝊䝻䛃䛜௬᝿௙஦䛾ཎ⌮ 81

਒ளщƕƋǔئӳ

(1)

2020ᖺᗘ ` ඲䛶䛾㉁Ⅼ䛜඲䛟⊂❧䛻ື䛔䛶䛔䜛䜟䛡䛷䛿䛺䛔䚹 ` 3n ಶ䜘䜚ᑡ䛺䛔 m ಶ䛾⮬⏤ᗘ䛧䛛䛺䛔ሙྜ䜢⪃䛘䜛䚹 x = (q), q = (q₁,«,q_m)T ` ᮏ᮶䛾௬᝿ኚ఩䛾ព࿡䛷䛿䚸x(䛒䜛䛔䛿 q)䜢ᅛᐃ䛧䛯ሙྜ䚸௬᝿ኚ఩ 䛻䜒 m ಶ䛾⮬⏤ᗘ䛧䛛䛺䛔䚹(༢䛺䜛ᮍᐃ஌ᩘ䛾ព࿡䛷䛿 3n ಶ䛒䜛) ` 䛂඲䛶䛾ᚤᑠ䛺௬᝿ኚ఩䛃䜢⪃䛘䜛䛸䚸ຊ䛾㔮䜚ྜ䛔䛻ᣊ᮰ຊ= ෆຊ䜢⪃ 䛘䛺䛡䜜䜀䛺䜙䛺䛔䚹 m₁ m2 㔜ຊ 䝞䝛ຊ 䝞䝛ຊ ᙇຊ= ᣊ᮰ຊ ᣊ᮰:⣒䛾㛗䛥䛜ኚ䜟䜙䛺䛔 ᣊ᮰ຊ: ⣒䛾ᙇຊ 82

਒ளщƕƋǔئӳ

(2)

` ຊ䜢䛂ᣊ᮰ຊ䛃䛸䛂䛭䜜௨እ䛾ຊ䛃䛻ศ䛡䜛䚹 f<sub>i</sub>(x) = f<sub>ci</sub>(x) + f<sub>ei</sub>(x) (i = 1,«,n) ᣊ᮰ຊ ` 2䛴䛾ຊ䛻ศ䛡䛯௬᝿௙஦䛾ཎ⌮䠖 ` ᣊ᮰ຊ䛜䛒䜛ሙྜ䛾௬᝿௙஦䛾ཎ⌮: (ᣊ᮰ຊ䜢⪃䛘䛺䛟䛶Ⰻ䛔!!!) 䛩䛺䜟䛱 q 䛰䛡䛷⪃䛘䛶䚸 ᣊ᮰᮲௳䜢‶䛯䛩௬᝿ኚ఩䛻ᑐ䛧䛶䛿䚸 ᣊ᮰ຊ䛻ᑐ䛧䛶ᡂ䛩௙஦䛾⥲࿴䛿䝊䝻䚹 ᣊ᮰᮲௳䜢‶䛯䛩௬᝿ኚ఩䛻ᑐ䛧䛶䛾䜏ᡂ❧

ϋᢿǨȍȫǮȸưᎋƑƨˎेˁʙƷҾྸ

` ᣊ᮰ຊ௨እ䛾ຊ䛻ᑐ䛩䜛௬᝿௙஦䛜䚸䛂䝫䝔䞁䝅䝱䝹䜶䝛䝹䜼䞊 D(x)䛃 䛾ቑຍ㔞䛻䛺䜛ሙྜ䜢⪃䛘䜛䚹 ` 䝫䝔䞁䝅䝱䝹䜶䝛䝹䜼䞊(potential energy)䛸ຊ䛾㛵ಀ䛾౛ ‡ 䝞䝛䛻⵳✚䛥䜜䜛䜶䝛䝹䜼䞊 ‡ ఩⨨䜶䝛䝹䜼䞊 ` 䝫䝔䞁䝅䝱䝹䛻䜘䜛ຊ䛸ᣊ᮰ຊ䛧䛛䛺䛔ሙྜ䛿䚸⥲䝫䝔䞁䝅䝱䝹䜶䝛䝹 䜼䞊 D(x) 䜢⏝䛔䛶䚸௬᝿௙஦䛾ཎ⌮䛿䚸ḟ䛾䜘䛖䛻᭩䛡䜛 䝫䝔䞁䝅䝱䝹䛻䜘䜛ຊ䛸ᣊ᮰ຊ䛾䜏䛾ሙྜ䚸ᚤᑠ䛺௬᝿ኚ఩䛻ᑐ䛧䝫䝔 䞁䝅䝱䝹䜶䝛䝹䜼䞊䛜೵␃䛩䜛䛣䛸䛜䚸㔮䜚ྜ䛔䛾᮲௳䚹

ѣႎƳئӳ

Ɇ ȀȩȳșȸȫƷҾྸ

2020ᖺᗘ ` 㟼ⓗ䛺㔮䜚ྜ䛔䛷䛿䛺䛟䚸≀䛜ື䛟≧ἣ(ືⓗ䛺ሙྜ)䜢⪃䛘䜛䚹 (ືຊᏛ = Dynamics) ` ືⓗ䛺ሙྜ䛻ᑐ䛩䜛௬᝿௙஦䛾ཎ⌮䛾ᣑᙇ э 䝎䝷䞁䝧䞊䝹䛾ཎ⌮ ` 䝙䝳䞊䝖䞁䛾㐠ື᪉⛬ᘧ(Newton’s law): 䛂័ᛶຊ䛃䛸䛧䛶⪃䛘䜛 ` 䝎䝷䞁䝧䞊䝹䛾ཎ⌮ (d'Alembert's principle): э ௬᝿௙஦䛾ཎ⌮䛸ྠᵝ䛻䚸ᣊ᮰䜢‶䛯䛩ᚤᑠ䛺௬᝿ኚ఩䛰䛡䜢⪃䛘䜛䛣 䛸䛷䚸ᣊ᮰ຊ䜢↓ど䛷䛝䜛䚹 85

٭Ў

2020ᖺᗘ ` ௬᝿ኚ఩δx<sub>i</sub>䛿䚸≀䛾ື䛝䛾㌶㐨䛻ἢ䛳䛶ኚ䜟䜛䜉䛝䜒䛾䚹эδx<sub>i</sub>(t) ` δ䛿䚸᫬㛫䛻㛵䛩䜛ᚤศ d 䛸䛿ู≀䚹 эኚศ(variation) ` ㌶㐨 x (t) 䛜 x (t) +δx(t) 䛻ኚ䜟䜛䚸䛸⪃䛘䜛䚹 ` _{⪃䛘䛶䛔䜛᫬㛫༊㛫 {t | t1  t  t2} 䛾➃䛷ኚศ䛿䝊䝻䛸⪃䛘䜛䚹} x (t) t₁ t₂ x x (t) +δx(t) 86

ᢃѣǨȍȫǮȸƱƷ᧙̞

` 㐠ື䜶䝛䝹䜼䞊(kinetic energy): ` 㐠ື䜶䝛䝹䜼䞊䛾ኚศ䛾᫬㛫✚ศ: 㒊ศ✚ศ ቃ⏺䛷௬᝿ኚ఩䝊䝻

ȀȩȳșȸȫƷҾྸƷ୿Ɩ੭Ƒ

` ຊ䜢ᣊ᮰ຊ䛸䝫䝔䞁䝅䝱䝹ຊ䛰䛡䛻㝈ᐃ䚹

` ᭱ᑠస⏝䛾ཎ⌮(principle of minimum action): (᭱ᑠస⏝䛾ཎ⌮䛸࿧䜀

䜜䜛䜒䛾䛻䛿2䛴䛒䛳䛶䚸௨ୗ䛿䝝䝭䝹䝖䞁䛾ཎ⌮(Hamilton principle)䚹) ቃ⏺᮲௳䛸ᣊ᮰䜢‶䛯䛩ᚤᑠ䛺 䛻ᑐ䛧䛶䚸 ` 䝷䜾䝷䞁䝆䜰䞁(Lagrangian): ` స⏝(Action): T ± D 䜢⊂❧䛺⮬⏤ᗘ䛷⾲⌧䛧䛯䜒䛾 ᭱ᑠస⏝䛾ཎ⌮: ຊᏛἲ๎䛻ᚑ䛖⮬↛䛺ື䛝䛾㌶㐨䛻ἢ䛳䛶䚸 స⏝䛿೵␃䛩䜛

இݱ˺ဇƷҾྸƔǒᢃѣ૾ᆉࡸǛݰƘ

` ୧➃䛷௬᝿ኚ఩䝊䝻䛺䛾䛷 ` ඲䛶䛾ᚤᑠ䛺δq(t) 䛻㛵䛧䛶δS = 0 䛺䛾䛷䚸 䜸䜲䝷䞊䞉䝷䜾䝷䞁䝆䝳᪉⛬ᘧ(Euler-Lagrange equation): 2020ᖺᗘ 89

ǪǤȩȸȷȩǰȩȳǸȥ૾ᆉࡸƴ᧙Ƣǔදॖ

2020ᖺᗘ ` 䝫䝔䞁䝅䝱䝹ຊ䞉័ᛶຊ䞉ᣊ᮰ຊ௨እ䛾ຊ䛜ຍ䜟䜛䛸䛝䛿䚸ྑ㎶䛿䝊䝻䛷 䛿䛺䛔䚹 ` F = (F₁,«,F _m)T_{䛿 q 䛛䜙䜏䛯እຊ䚹} ‡ 䛯䛸䛘䜀 q_i䛜ᅇ㌿䛺䜙䜀䚸F_i䛿䛭䛾ᅇ㌿㍈䛻ຍ䜟䜛䝖䝹䜽 ‡ 䛯䛸䛘䜀 q_i䛜┤⥺≧䛾ኚ఩䛺䜙䜀䚸F_i䛿䛭䛾⮬⏤ᗘ䛻ຍ䜟䜛ຊ(ຊ䛾 ື䛝䛾᪉ྥ䛻ἢ䛳䛯ศຊ) ‡ ᦶ᧿ຊ䞉䝎䞁䝢䞁䜾ຊ䜒䚸䛣䛾እຊ䛻ศ㢮䛥䜜䜛䚹 ` 䝷䜾䝷䞁䝆䜰䞁 L 䛛䜙㐠ື䜶䝛䝹䜼䞊 T 䛸䝫䝔䞁䝅䝱䝹䜶䝛䝹䜼䞊 D 䜢 ศ㞳䛷䛝䜛䚹эᘬ䛝⟬䛧䛯䛣䛸䛷᝟ሗ䛿ኻ䜟䜜䛶䛔䛺䛔䚹 90

ɟᑍ҄ࡈ೅ȷɟᑍ҄ᡮࡇ

` ୍⯡໬ᗙᶆ(generalized coordinate): q ` ୍⯡໬㏿ᗘ(generalized velocity): ` ୍⯡໬ຍ㏿ᗘ(generalized acceleration): ` ୍⯡໬ຊ(generalized force): F ` ᅇ㌿䛾ሙྜ䚸୍⯡໬ᗙᶆ䛿ᅇ㌿ゅ, ୍⯡໬㏿ᗘ䛿ゅ㏿ᗘ, ୍⯡໬ຍ㏿ᗘ 䛿ゅຍ㏿ᗘ, ୍⯡໬ຊ䛿䝖䝹䜽 ` ㉁Ⅼ䛾㞟ྜ䛷䛿䛺䛟䚸㉁㔞䛜ศᕸ䛧䛶䛔䜛ሙྜ䛷䜒䚸୍⯡໬ᗙᶆ䛜᭷㝈 ḟඖ䛺䜙䜀䚸䛂㉁Ⅼ䛻㛵䛩䜛⥲࿴䛃䜢䛂㉁㔞䛾ศᕸ䛻㛵䛩䜛✚ศ䛃䛻ኚ䛘䜛 䛰䛡䛷䚸䜸䜲䝷䞊䞉䝷䜾䝷䞁䝆䝳᪉⛬ᘧ䛿ᡂ䜚❧䛴䚹 э ↓㝈ಶ䛾(ศᕸ䛧䛯)ᣊ᮰ຊ䜢䜎䛸䜑䛶↓ど䛷䛝䚸᭷㝈ḟඖ䛾ᚤศ᪉⛬ ᘧ䛜⡆༢䛻ᚓ䜙䜜䜛䚹

E-L૾ᆉࡸƷ̊ Ɇ ҥኝƳᡲ঺ਰѣ

` እຊ䛸ῶ⾶㡯䜢㝖䛔䛶⪃䛘䜛 ` 㐠ື䜶䝛䝹䜼䞊䛸䝫䝔䞁䝅䝱䝹䜶䝛䝹䜼䞊 ` 䜸䜲䝷䞊䞉䝷䜾䝷䞁䝆䝳᪉⛬ᘧ k₁ m₂ k₂ m₁ 㐠ື᪉⛬ᘧ

ሁᡮႺዴᢃѣǛׅ᠃ࡈ೅ưᙸǔ

2020ᖺᗘ ` 㟼Ṇᗙᶆ⣔ (x,y) 䛷䛾㐠ື䜢䚸➼ゅ㏿ᗘ  䛷ᅇ㌿䛧䛶䛔䜛ᅇ㌿ᗙᶆ (X,Y)䛷䚸ぢ䜛䛸䛹䛖䛺䜛䛛䚹 ` ㏿ᗘ䛾㛵ಀ ` 䝷䜾䝷䞁䝆䜰䞁… t 䜢㝧䛻ྵ䜎䛪䚸᫬୙ኚ䚹䛯䛰䛧䚸㐠ື䜶䝛䝹䜼䞊䛜୍ ⯡໬㏿ᗘ䛾ṇᐃ㛵ᩘ䛻䛺䜙䛺䛔䛾䛷䚸≉Ṧ䛺ሙྜ䛸䛔䛘䜛䚹 ` 䜸䜲䝷䞊䞉䝷䜾䝷䞁䝆䝳᪉⛬ᘧ 䛾༙ศ= 䝁䝸䜸䝸ຊ(Coriolis force) 䛾༙ศ+ =㐲ᚰຊ (centrifugal force) 93

଺ɧ٭Ƴࡈ೅٭੭ưƸ

(1)

2020ᖺᗘ ` ᫬୙ኚ䛺ᗙᶆኚ᥮䛷䛿䚸 ` 䛣䛾䛸䛝䛾㐠ື䜶䝛䝹䜼䞊 ` M(q): ័ᛶ⾜ิ (inertia matrix) эṇᐃᑐ⛠⾜ิ䛻䛺䜛 ୍⯡໬㏿ᗘ䛾2ḟᙧᘧ (quadratic form) = ྠḟ2ḟᘧ 94

଺ɧ٭Ƴࡈ೅٭੭ưƸ

(2)

` ័ᛶ⾜ิ䛻䜘䜛E-L᪉⛬ᘧ䛾⾲⌧: ័ᛶຊ 䝁䝸䜸䝸ຊ 㐲ᚰຊ 䝫䝔䞁䝅䝱䝹ຊ እຊ

̊᫆ Ყ Ȑȍ˄ƖƷਰ܇

` r: 䝞䝛䛾⦰䜏,  : ᣺Ꮚ䛾᣺ゅ ` +: ᣺Ꮚ䜈䛾䝖䝹䜽(እຊ) ` `: Წ䛾㛗䛥 – 䝞䝛䛾⮬↛㛗 ` k: 䝞䝛ᐃᩘ, m : 䛚䜒䜚䛾㉁㔞 ` g: 㔜ຊຍ㏿ᗘ ` ୍⯡໬ᗙᶆ q = (r, )T ` 䝷䜾䝷䞁䝆䜰䞁: ` 䜸䜲䝷䞊䞉䝷䜾䝷䞁䝆䝳᪉⛬ᘧ(㐠ື᪉⛬ᘧ): +

ǸȣǤȭјௐ

(1)

2020ᖺᗘ ` ᅗ䛻䛚䛔䛶䚸䝁䝬䛻ᅛᐃ䛧䛯ᗙᶆ⣔䛷䛾ゅ㏿ᗘ䝧䜽䝖䝹䛿䚸 ` 䜘䛳䛶㐠ື䜶䝛䝹䜼䞊䛿  *  + 97

ǸȣǤȭјௐ

(2)

2020ᖺᗘ ` 䜸䜲䝷䞊䞉䝷䜾䝷䞁䝆䝳᪉⛬ᘧ: ` ≉䛻䚸 䛾䛸䛝䚸 ` 䝁䝬䛻ᅛᐃ䛧䛯ᗙᶆ䛻䛚䛡䜛ゅ㐠ື㔞䝧䜽䝖䝹 䝁䝬䛜ᅇ㌿䛧䛶䛔䜛䛸䛝䛻䚸 ㍈䛻ゅ㏿ᗘ䜢୚䛘䜛䛸䚸 ┤஺䛩䜛*㍈ᅇ䜚䛻䝖䝹䜽䛜Ⓨ⏕ 䝆䝱䜲䝻䝰䞊䝯䞁䝖 98

3ȪȳǯƷȭȜȃȈǢȸȠ(1)

` 3䝸䞁䜽䛾䝻䝪䝑䝖䜰䞊䝮䜢⪃䛘䜛䚹 ` 䝸䞁䜽⮬య䛾㉁㔞䛿⪃䛘䛺䛔䚹 m₂ +₁  +₂ +₃ m₁  *

3ȪȳǯƷȭȜȃȈǢȸȠ(2)

` m<sub>1</sub>, m<sub>2</sub>䛾఩⨨ ` m₁, m₂䛾㏿ᗘ ` 䝷䜾䝷䞁䝆䜰䞁

3ȪȳǯƷȭȜȃȈǢȸȠ(3)

2020ᖺᗘ ` ୍⯡໬ᗙᶆ: ` ័ᛶ⾜ิ: 101

3ȪȳǯƷȭȜȃȈǢȸȠ(4)

2020ᖺᗘ ` 䝁䝸䜸䝸ຊ䛸㐲ᚰຊ: ` 㔜ຊ㡯: ` 㐠ື᪉⛬ᘧ: 102

ɟᑍ҄ᢃѣ᣽

` ୍⯡໬㐠ື㔞(generalized momentum) : ` 㐠ື䜶䝛䝹䜼䞊䛜୍⯡໬㏿ᗘ䛾2ḟᙧᘧ䛺䜙䜀䚸 ` ᖹ⾜⛣ື䛻㛵䛧䛶 ` ᅇ㌿㐠ື䛻㛵䛧䛶 ᐃ⩏ ㏻ᖖ䛾㐠ື㔞䛸୍⮴ ゅ㐠ື㔞 ୍⯡໬㐠ື㔞䛷⾲䛧䛯䝅䝇䝔䝮⾲⌧䛿䛹䛾䜘䛖䛻䛺䜛䛾䛷䛒䜝䛖䛛?

ࡈ೅٭੭

` ୍⯡໬ᗙᶆ䞉୍⯡໬㏿ᗘ䛾⤌䜏ྜ䜟䛫䛾⾲⌧ ୍⯡໬ᗙᶆ䞉୍⯡໬㐠ື㔞䛻䜘䜛⾲⌧ ` 䝷䜾䝷䞁䝆䜰䞁䛿䛂୍⯡໬ᗙᶆ䞉୍⯡໬㏿ᗘ䛾⤌䜏ྜ䜟䛫䛃 䝷䜾䝷䞁䝆䜰䞁䛾ᚤᑠኚ఩: ᗙᶆኚ᥮ 䛣䛾㡯䜢ᾘ䛩䜘䛖䛺⾲⌧䛜䜋䛧䛔

ȏȟȫȈȋǢȳ

2020ᖺᗘ ` 䝝䝭䝹䝖䝙䜰䞁(Hamiltonian) : ` 㐠ື䜶䝛䝹䜼䞊䛜୍⯡໬㏿ᗘ䛾2ḟᙧᘧ䛺䜙䜀䚸 ⤖ᒁ䚸 H = T + D䚹 ` 䛴䜎䜚䚸㏻ᖖ䛾ሙྜ䚸䝝䝭䝹䝖䝙䜰䞁䛿䚸(୍⯡໬㐠ື㔞䛷⾲⌧䛧䛯)㐠ື 䜶䝛䝹䜼䞊䛸䝫䝔䞁䝅䝱䝹䜶䝛䝹䜼䞊䛾࿴䛸୍⮴䛩䜛䚹䛩䛺䜟䛱䚸䝅䝇䝔 䝮ෆ㒊䛾඲䜶䝛䝹䜼䞊䛻୍⮴䛩䜛䚹 ᐃ⩏ 105

ȫǸȣȳȉȫ٭੭

2020ᖺᗘ ` 䝝䝭䝹䝖䝙䜰䞁䛾ᚤᑠኚ఩䛸䝷䜾䝷䞁䝆䜰䞁䛾ᚤᑠኚ఩䛸䛾㛵ಀ ` ᜏ➼ᘧ 䛸㡯ูẚ㍑ ` 䝹䝆䝱䞁䝗䝹ኚ᥮䛻䜘䛳䛶ᚓ䜙䜜䜛㛵ಀᘧ: ᾘཤ䛧䛯䛛䛳䛯㡯䛸ྠ䛨 H䛾ᚤᑠኚ఩䛜䚸(ᐃ⩏㏻䜚䛻) p䛸q䛾ᚤᑠኚ ఩䛷⾲⌧䛥䜜䛶䛔䜛䚹 э䝹䝆䝱䞁䝗䝹ኚ᥮(Legendre transformation) ୧㎶䛾೫ᚤศ䛾ព࿡䛜㐪䛖 ᕥ㎶: q 䛸 q 䛷⾲⌧䛧䛯ሙྜ ྑ㎶: p 䛸 q 䛷⾲⌧䛧䛯ሙྜ 106

ദแ૾ᆉࡸ

` ୍⯡໬㐠ື㔞䜢౑䛳䛯䜸䜲䝷䞊䞉䝷䜾䝷䞁䝆䝳᪉⛬ᘧ: ௨ୖ䜎䛸䜑䜛䛸䚸

` Hamilton䛾ṇ‽᪉⛬ᘧ (Hamilton's canonical equation):

ǨȍȫǮȸ̬܍Щ

` 䝝䝭䝹䝖䝙䜰䞁䛿඲䜶䝛䝹䜼䞊

` እຊ䛺䛧(F = 0) 䛾䛸䛝䚸඲䜶䝛䝹䜼䞊䛜ಖᏑ䛥䜜䜛䛣䛸䜢♧䛭䛖䚹

` 䝝䝭䝹䝖䝙䜰䞁䛾᫬㛫ᚤศ

᫬㛫ᚤศ䛜䝊䝻 э ್䛜ኚ໬䛧䛺䛔 э ඲䜶䝛䝹䜼䞊䛜ಖᏑ䛥䜜䜛 䜶䝛䝹䜼䞊ಖᏑ๎(Law of the conservation of energy)

̬܍᣽

2020ᖺᗘ

` 䜶䝛䝹䜼䞊䛾䜘䛖䛻䚸እຊ= 0䛾ୗ䛷䚸⮬↛䛺ື䛝䛻ἢ䛳䛶䛭䛾್䜢ಖ䛱

⥆䛡䜛㔞䜢ಖᏑ㔞(conservative quantity)䛒䜛䛔䛿➨䠍✚ศ(first integral) 䛸䛔䛖䚹 ` ಖᏑ㔞䜢ᣢ䛴ືⓗ䝅䝇䝔䝮䜢ಖᏑ⣔(conservative system)䛸䛔䛖䚹 ` ୍⯡ⓗ䛺ಖᏑ㔞䛾౛ ‡ 䜶䝛䝹䜼䞊(䝝䝭䝹䝖䝙䜰䞁) H(p(t), q(t)) = const. ‡ 㐠ື㔞䝧䜽䝖䝹 ‡ ゅ㐠ື㔞䝧䜽䝖䝹 109

଺ᆰƷ٭੭Ʊݣᆅࣱ

2020ᖺᗘ ` ᫬㛫䛸୍⯡໬ᗙᶆ䛾ኚ᥮(䝟䝷䝯䞊䝍  䛻ἢ䛳䛶⁥䜙䛛䛻ኚ໬䛩䜛): 䛯䛰䛧䚸(q, 0) = q䚹 ` స⏝ S 䛜䛣䛾ኚ᥮䛻㛵䛧䛶୙ኚ䛺䜙䜀䚸⣔䛿ᑐ⛠ᛶ(symmetry)䜢ᣢ䛴䛸 䛔䛖䚹 ኚ᥮ᚋ䛾స⏝✚ศ: ᑐ⛠ᛶ䜢ᣢ䛴䛸䛿: ௵ព䛾 t_F, t_I䛻䛴䛔䛶 110

ݣᆅࣱǛਤƭƳǒƹ

(1)

` <sub>≉䛻ᚤᑠኚື   0 䛾䛸䛝</sub> ` ✚ศ䛾୰䛿䚸䜸䜲䝷䞊䞉䝷䜾䝷䞁䝆䝳᪉⛬ᘧ䜘䜚 䛸୍⮴䚹 ` 䜎䛯䚸

ݣᆅࣱǛਤƭƳǒƹ

(2)

` ᭱⤊ⓗ䛻 ` ௵ព䛾 t_F, t_I䛻䛴䛔䛶ᡂ䜚❧䛴䛾䛷䚸 ಖᏑ㔞

ȍȸǿȸƷܭྸ

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