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dạng bài tập môn động lực học công trình biển

dạng bài tập môn động lực học công trình biển

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  • 1. http://www.ebook.edu.vn 1§Ò BμI:S2 S3S1EJ=ConstM2M MABaaa a aY£U CÇU:1, X¸c ®Þnh tÇn sè vμ d¹ng dao ®éng riªng?2, X¸c ®Þnh ma trËn xung khai triÓn theo c¸c d¹ng dao ®éng riªng? Khi hÖ chÞuvÐct¬ xung bªn ngoμi t¸c dông vμo hÖ lμ:{ } S5,111SSSS321⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧=3, X¸c ®Þnh vÐct¬ chuyÓn vÞ t¹i c¸c khèi l−îng?4, X¸c ®Þnh vÐct¬ lùc ®μn håi (t−¬ng øng víi tr¹ng th¸i ®éng) ?5, X¸c ®Þnh m« men uèn t¹i A, B theo thêi gian?6, VÏ ®å thÞ m« men uèn theo thêi gian t¹i ®iÓm A: MA(t) khi hÖ chÞu t¸c dôngxung trong kho¶ng thêi gian t = 2T1 (T1 lμ chu kú dao ®éng riªng d¹ng thø nhÊt)Sè liÖu:S = 1 KNsE = 2,1.104KN/cm2J = 8880 cm4M = 1 KNs2/m = 1.10-2KNs2/cma = 2 m= 2.102cm
  • 2. http://www.ebook.edu.vn 2Bμi lμm1. X¸c ®Þnh tÇn sè vµ d¹ng dao ®éng riªng:a, X¸c ®Þnh ma trËn khèi l−îng [ ]M vμ ma trËn mÒm [ ]F :Ta thÊy, hÖ cã 3 bËc tù do nªn:Ma trËn khèi l−îng [ ]M sÏ lμ:[ ]⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=100020001Mm000m000mMM321Vμ ma trËn mÒm [ ]F :[ ]⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−−=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡δδδδδδδδδ=1161,01025,00294,01025,01682,00587,00294,00587,01226,0EJaF3333231232221131211Trong ®ã: C¸c chuyÓn vÞ ®¬n vÞ kiδ ®−îc x¸c ®Þnh theo c«ng thøc nh©nbiÓu ®å Vªrªsaghin tõ c¸c biÓu ®å m« men ®¬n vÞ (H×nh 1):( )( )⎟⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎜⎝⎛−+++++++−+===δ)a0011,0*a0011,0a0832,0*32*a0832,0(a21a0836,0*32*a0836,0*a21a0011,0*a*a0011,0a0857,032*a0857,0*a21a0857,0*a0857,0*a)a1704,0*32*a1704,0a04148*32*a04148,0(a21a04148*32*a04148,0*a21EJ1MM 1111=0,1226EJa3( )( )=⎟⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎜⎝⎛−++++−++++===δ)a0358,0*32*a0358,0a3290,0*32*a3290,0(a21a0358,0*a0358,0*aa3664,0*32*a3664,0*a21)a2314,032*a2314,0a4022,0*32*a4022,0(a21a2314,0*a2314,0*aa1157,0*32*a1157,0,0*a21a1157,0*32*a1157,0*a21EJ1MM 2222=0,1682EJa3
  • 3. http://www.ebook.edu.vn 3( )( )⎟⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎜⎝⎛−+++++−+===δ)a2970,032*a2970,0a4955,0*32*a04955,0(a21a00896,0*a*a0896,0a0274,032*a2074,0*a21)a0896,0*32*a0896,0a1176,0*32*a1176,0(a21a1176,0*32*a1176,0*a2*21EJ1MM 3333=0,1161EJa3( )( )⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎛+−−−+−++−−−−==δ=δa3290,0*a0832,0*a21a0358,0*a0857,0*a21a0857,0*32*a3664,0*a21a0857,0*a4022,0*a21a0587,0*a2314,0*a*21a4022,0*31*a0857,0*a21)a3664,0*32*a0857,0*a21()a1157,0*a1704,0*a21a1157,0*a4148,0*a21()a1704,032*a1157,0a4148,0*31*a1157,0(a21a4148,0*32*a1157,0*a21EJ1MM 212112=-0,0587EJa3( )( )⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎛−+−−−+−+−++−−==δ=δa0832,0*32*a4955,0*a21a0832,0*31*a2970,0*a21a0896,0*a0857,0*a210857,0*31*a2074,0*a21a0857,0*a0896,0*a21a0857,0*a1176,0*a21a1704,0*31*a0896,0*a21a0847,0*21*a1176,0*a32)a0589,0*a1704,0a0589,0*a4148,0(a21a1704,032*a0589,0*a21a4148,0*31*a0589,0*a21a0589,0*32*a4148,0*a21EJ1MM 313113= - 0,0294EJa3
  • 4. http://www.ebook.edu.vn 4( )( )⎟⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎜⎝⎛−+++−+−+==δ=δa2970,0*a31a4955,0*32(a329,0*a21a2074,0*31*a4022,0*a21a896,0*a4022,0*a21)a896,0*a2314,0*31a0896,0*a4022,0*32a11176,031*a4022,0a1176,0*32*a2314,0(a21a2314,0*32*a1176,0*a2*21EJ1MM 323223= 0,1025EJa3P1=1P2=1P3=1O,23140,11570,40220,32900,11760,08960,29700,49550.05890,17040,41480,08570,0011 0,08320,0358M1M2M3H×nh 1: BiÓu ®å m« men ®¬n vÞ
  • 5. http://www.ebook.edu.vn 5b, X¸c ®Þnh tÇn sè dao ®éng riªng ω :Ph−¬ng tr×nh tÇn sè viÕt ë d¹ng ma trËn:[ ] [ ] [ ] 0*1* 2=− EMFωThay sè vμo cho ta:u116143,0205080,00293788,0102540,0u336349,00586946,0029378,0117389,0u122576,0EJMa3−−−−−−−= 0 ;Víi: 23MaEJuω= (1)Khai triÓn ®Þnh thøc ta ®−îc:-u3+ 0,57507u2- 0,065747u + 0,0018275 = 0Gi¶i ph−¬ng tr×nh nμy cho ta nghiÖm:⎪⎩⎪⎨⎧===0423,0u0998,0u4330,0u321ThÕ c¸c gi¸ trÞ u võa t×m ®−îc vμo biÓu thøc (1) cho ta c¸c gi¸ trÞ ω t−¬ngøng cÇn t×m:⎪⎩⎪⎨⎧=ω=ω=ω3746,738126,1527919,234321(s-1)Víi:E = 2,1.104KN/cm2J = 8880 cm4M = 1 KNs2/m = 1.10-2KNs2/cma = 2m = 2.102cmc, X¸c ®Þnh d¹ng dao ®éng riªng [ ]Φ :+ D¹ng dao ®éng riªng ®−îc x¸c ®Þnh d−íi d¹ng ma trËn theo c«ng thøc:{ } [ ] { }1111*BBi−−=ϕThay sè vμo cho ta:
  • 6. http://www.ebook.edu.vn 6{ }⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−−⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−=ϕ−0293788,00586946,0u116143,0205080,0102540,0u336349,01*i(2)* i=1; thay u1= 0,9111 vμo biÓu thøc (2), cho ta:{ }⎭⎬⎫⎩⎨⎧−−=ϕ5527,12555,2*1⇒ { }⎪⎪⎭⎪⎪⎬⎫⎪⎪⎩⎪⎪⎨⎧−−=ϕ5527,12555,211 (*)* i=2; thay u2= 0,3048 vμo biÓu thøc (2), cho ta:{ }⎭⎬⎫⎩⎨⎧=ϕ2964,01197,0*2⇒ { }⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧=ϕ2964,01197,012 (**)* i=3; thay u3 = 0,049 vμo biÓu thøc (2), cho ta:{ }⎭⎬⎫⎩⎨⎧−=ϕ9211,49156,1*3⇒ { }⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−=ϕ9211,49156,113 (***)+ Ma trËn c¸c d¹ng dao ®éng:Gép vμo ma trËn c¸c d¹ng dao ®éng riªng cña hÖ;Tõ (*),(**),(***):[ ] { } { } { }[ ]==Φ 321 ϕϕϕ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−9211,42964,05527,19156,11197,02555,20000,10000,10000,1
  • 7. http://www.ebook.edu.vn 7+ KiÓm tra ®iÒu kiÖn trùc giao c¸c d¹ng dao ®éng riªng:{ } [ ]{ }21 ϕϕ MT={ } 0000008724,02964,01197,00000,1M1000200015527,12555,20000,1 ≈−=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−{ } [ ]{ }31 ϕϕ MT={ } 000005624,09211,49156,10000,1M1000200015527,12555,20000,1 ≈−=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−{ } [ ]{ }1T2 M ϕϕ ={ } 0000008724,05527,12555,20000,1M1000200012964,01197,0000,1 ≈=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−−⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡{ } [ ]{ }1T3 M ϕϕ ={ } 000005624,05527,12555,20000,1M1000200019211,49156,1000,1 ≈−=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−−⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−{ } [ ]{ }3T2 M ϕϕ ={ } 0000005018,09211,49156,10000,1M1000200012964,01197,0000,1 ≈=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡{ } [ ]{ }2T3 M ϕϕ ={ } 0000005018,02964,01197,00000,1M1000200019211,49156,1000,1 ≈=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−=> Nh− vËy ®iÒu kiÖn trùc giao ®−îc tháa m·n.+ ThÓ hiÖn c¸c d¹ng dao ®éng riªng b»ng h×nh vÏ (H×nh 2):
  • 8. http://www.ebook.edu.vn 8ϕ11,002,2551,552BBϕ21,000,119 0,296Bϕ31,001,9154,921H×nh 2: BiÓu ®å c¸c d¹ng dao ®éng riªngd/ ChuÈn hãa c¸c d¹ng dao ®éng riªng:{ } [ ]{ } { } M5851,135527,12555,20000,1M1000200015527,12555,20000,1Ma 1T121 =⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−−⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−=ϕϕ=⇒ M6858,3a1 ={ } [ ]{ } { } M1143,12964,01197,00000,1M1000200012964,01197,0000,1Ma 2T222 =⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=ϕϕ=⇒ M0566,1a2 =
  • 9. http://www.ebook.edu.vn 9{ } [ ]{ } { } M5562,329211,49156,10000,1M1000200019211,49156,1000,1Ma 3T323 =⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−=ϕϕ=⇒ M7058,5a3 =+ D¹ng dao ®éng riªng chuÈn:{ }⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−−=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−−=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−−=ϕ4213,06119,02713,0M15527,12555,20000,1M6858,315527,12555,20000,1a11ch,1{ }⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧=ϕ2805,01133,09464,0M12964,01197,00000,1M0566,112964,01197,00000,1a12ch,2{ }⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−=ϕ8625,03357,01753,0M19211,49156,10000,1M7058,519211,49156,10000,1a13ch,3[ ]⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−=Φ⇒8625,02805,04213,03357,01133,06119,01753,09464,02713,0M1ch+ KiÓm tra ®iÒu kiÖn trùc chuÈn:[ ] [ ][ ]M1000200018625,02805,04213,03357,01133,06119,01753,09464,02713,0M18625,03357,01753,02805,01133,09464,04213,06119,02713,0M1M chTch⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−==ΦΦ=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡10001000100000,100000,0000000,000000,000000,100000,000000,000000,000000,1VËy ®iÒu kiÖn trùc chuÈn ®−îc tháa m·n.
  • 10. http://www.ebook.edu.vn 102. X¸c ®Þnh ma trËn xung khai triÓn theo c¸c d¹ng dao ®éng riªng:Khi hÖ chÞu vÐct¬ xung bªn ngoμi t¸c dông vμo:{ } S5,111SSSS321⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧= Víi S = 1 KNsX¸c ®Þnh vÐct¬ xung khai triÓn tõ ma trËn c¸c d¹ng dao ®éng riªng ®·®−îc chuÈn hãa.a, VÐct¬ xung khai triÓn vμo c¸c d¹ng dao ®éng riªng:§−îc x¸c ®Þnh theo c«ng thøc:{ } { } { }[ ]{ }chiTchii MSS ,, ϕϕ=- Víi i = 1 ta cã:{ } { } { }[ ]{ }chTch MSS ,1,11 ϕϕ={ }⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−−⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−−=4213,06119,02713,0M1M100020001S5,1114213,06119,02713,0M1Víi S = 1 KNs th×: { }1S⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−=4097,01902,12639,0- Víi i = 2 ta cã:{ } { } { }[ ]{ }chTch MSS ,2,22 ϕϕ={ }⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧=2805,01133,09464,0M1M100020001S5,1112805,01133,09464,0M1Víi S = 1 KNs th×: { }2S⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧=4152,03353,04010,1- Víi i = 3 ta cã:{ } { } { }[ ]{ }chTch MSS ,3,33 ϕϕ={ }⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−=8625,03357,01753,0M1M100020001S5,1118625,03357,01753,0M1Víi S = 1 KNs th×: { }3S⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−−=6751,05256,01372,0
  • 11. http://www.ebook.edu.vn 11b, Ma trËn xung khai triÓn:[ ]⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=6751,05256,01372,04152,03353,04010,14097,01902,12639,0SSSSSSSSSS333231232221131211KTrc, KiÓm tra ma trËn xung khai triÓn theo hμng k:32131kkkikik SSSSS ++== ∑=Víi i = 1:199999,0)1372,04010,12639,0(SSSS 1312111 ≈=−+−=++=Víi i = 2:00000,1)5256,03353,01902,1(SSSS 2322212 =−+=++=Víi i = 3:5,1)6751,04152,04097,0(SSSS 3332313 =++=++=Nh− vËy lμ ma trËn xung khai triÓn ®óng.3, X¸c ®Þnh vÐct¬ chuyÓn vÞ t¹i c¸c khèi l−îng:VÐct¬ chuyÓn vÞ t¹i c¸c khèi l−îng ®−îc x¸c ®Þnh theo c«ng thøc:( ){ } [ ] [ ] ( ){ }tKSMy aiKTrt1−=Trong ®ã:+ [ ]M lμ ma trËn khèi l−îng:[ ]⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=100020001Mm000m000mMM321+ [ ]KTrS lμ ma trËn xung khai triÓn:[ ]⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=6751,05256,01372,04152,03353,04010,14097,01902,12639,0SSSSSSSSSS333231232221131211KTr+ ( ){ }tKai lμ vÐct¬ cã c¸c phÇn tö lμ Kai(t), i = 1, 2, 3Kai(t)=iitωωsin* i = 1; 11 s7919,234 −=ω ta cã:Ka1(t)=7919,234t7919,234sintsin11=ωω
  • 12. http://www.ebook.edu.vn 12* i = 2; 12 s8186,152 −=ω ta cã:Ka2(t)=8186,152t8186,152sintsin22=ωω* i = 3; 13 s3746,73 −=ω ta cã:Ka3(t)=3746,73t3746,73sintsin33=ωω⇒ Kai(t) =( )( )( )⎪⎪⎪⎭⎪⎪⎪⎬⎫⎪⎪⎪⎩⎪⎪⎪⎨⎧=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧3746,73t3746,73sin8186,152t8186,152sin7919,234t7919,234sintKtKtK3a2a1aTõ ®ã: ( ){ }ty =1100020001−⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−6751,05256,01372,04152,03353,04010,14097,01902,12639,0⎪⎪⎪⎭⎪⎪⎪⎬⎫⎪⎪⎪⎩⎪⎪⎪⎨⎧3746,73t3746,73sin8186,152t8186,152sin7919,234t7919,234sin⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧++−+−+−=t792,234sin287524,0t813,152sin271722,0t3746,73sin558358,0t792,234sin111920,0t813,152sin109716,0t3746,73sin811071,0t792,234sin058271,0t813,152sin916836,0t3746,73sin359603,0KÕt luËn: VÐct¬ chuyÓn vÞ t¹i c¸c khèi l−îng lμ:( ){ }ty =( )( )( )⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧tytyty321⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧++−+−+−=t792,234sin287524,0t813,152sin271722,0t3746,73sin558358,0t792,234sin111920,0t813,152sin109716,0t3746,73sin811071,0t792,234sin058271,0t813,152sin916836,0t3746,73sin359603,04, X¸c ®Þnh vÐct¬ lùc ®µn håi:VÐct¬ lùc ®μn håi ®−îc x¸c ®Þnh theo c«ng thøc:( ){ }tPd = [ ]KTrS ( ){ }tKiTrong ®ã:+ [ ]KTrS lμ ma trËn xung khai triÓn:
  • 13. http://www.ebook.edu.vn 13[ ]⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=6751,05256,01372,04152,03353,04010,14097,01902,12639,0SSSSSSSSSS333231232221131211KTr+ ( ){ }tKi lμ vÐct¬ cã c¸c phÇn tö lμ Ki(t), i = 1, 2, 3Ki(t) = tii ωω sin* i = 1; 11 ts7919,234 −=ω ta cã:K1(t) = t11 sinωω = 7919,234 sin t7919,234* i = 2; 12 s8186,152 −=ω ta cã:K2(t) = t22 sinωω = 8186,152 sin 8186,152 t* i = 3; 13 s3746,73 −=ω ta cã:K3(t) = t33 sinωω = 3746,73 sin 3746,73 t⇒ Ki(t) =( )( )( ) ⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧t3746,73sin3746,73t8186,152sin8186,152t7919,234sin7919,234tKtKtK321Tõ ®ã: ( ){ }tPd =⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−6751,05256,01372,04152,03353,04010,14097,01902,12639,0⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧t3746,73sin3746,73t8186,152sin8186,152t7919,234sin7919,234⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧++−+−+−=t792,234sin304,158t813,152sin4518,63t3746,73sin0611,30t792,234sin397,123t813,152sin2411,51t3746,73sin3334,87t792,234sin2092,32t813,152sin097,214t3746,73sin3604,19KÕt luËn: VÐct¬ lùc ®μn håi lμ:( ){ }tPd =( )( )( )⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧tPtPtP3d2d1d⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧++−+−+−=t792,234sin304,158t813,152sin4518,63t3746,73sin0611,30t792,234sin397,123t813,152sin2411,51t3746,73sin3334,87t792,234sin2092,32t813,152sin097,214t3746,73sin3604,19
  • 14. http://www.ebook.edu.vn 145, X¸c ®Þnh m«men uèn t¹i A,B:P1=1P2=1P3=1O,23140,11570,40220,32900,11760,08960,29700,49550.05890,17040,41480,08570,0011 0,08320,0358M1M2M3H×nh 3: BiÓu ®å m« men ®¬n vÞa, M« men uèn t¹i A:( ) { }( )( )( )⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧=tPtPtPMMMtM3d2d1d3A2A1AATrong ®ã:* ( ){ }tPd =( )( )( )⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧tPtPtP3d2d1d⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧++−+−+−=t792,234sin304,158t813,152sin4518,63t3746,73sin0611,30t792,234sin397,123t813,152sin2411,51t3746,73sin3334,87t792,234sin2092,32t813,152sin097,214t3746,73sin3604,19* C¸c gi¸ trÞ ( )1AM , ( )2AM , ( )3AM lμ c¸c gi¸ trÞ m« men uèn t¹i ®iÓm A ®−îc x¸c®Þnh tõ biÓu ®å m« men ®¬n vÞ (H×nh 3):
  • 15. http://www.ebook.edu.vn 15( ) a0857,0M 1A −= ; ( ) a4022,0M 2A = ; ( ) a0896,0M 3A =( ) { }*a0896.0a4022,0a0857,0tMA −=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧++−+−+−t792,234sin304,158t813,152sin4518,63t3746,73sin0611,30t792,234sin397,123t813,152sin2411,51t3746,73sin3334,87t792,234sin2092,32t813,152sin097,214t3746,73sin3604,19Thay a = 200 vμo cho ta kÕt qu¶:( ) t792,234sin59,6533t813,152sin27,1589t3746,73sin63,7895tMA +−−=b, M« men uèn t¹i B:Hoμn toμn t−¬ng tù t¹i A, m« men uèn t¹i B ®−îc x¸c ®Þnh nh− sau:( ) { }( )( )( )⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧=tPtPtPMMMtM3d2d1d3B2B1BBTrong ®ã:* ( ){ }tPd =( )( )( )⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧tPtPtP3d2d1d⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧++−+−+−=t792,234sin304,158t813,152sin4518,63t3746,73sin0611,30t792,234sin397,123t813,152sin2411,51t3746,73sin3334,87t792,234sin2092,32t813,152sin097,214t3746,73sin3604,19* C¸c gi¸ trÞ ( )1BM , ( )2BM , ( )3BM lμ c¸c gi¸ trÞ m« men uèn t¹i ®iÓm B còng ®−îcx¸c ®Þnh tõ biÓu ®å m« men ®¬n vÞ (H×nh 3):( ) a0832,0M 1B = ; ( ) a3290,0M 2B −= ; ( ) a4955,0M 3B −=( ) { }*a4955,0a3290,0a0832,0tMB −−=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧++−+−+−t792,234sin304,158t813,152sin4518,63t3746,73sin0611,30t792,234sin397,123t813,152sin2411,51t3746,73sin3334,87t792,234sin2092,32t813,152sin097,214t3746,73sin3604,19Thay a = 200 vμo cho ta kÕt qu¶:( ) t)*2sin(234.79*8124.18t)*3sin(152.81*6097.16t)*6sin(73.374*9047.76tMB ++=
  • 16. http://www.ebook.edu.vn 166, VÏ ®å thÞ m« men uèn MA(t) theo thêi gian t¹i ®iÓm A:Khi hÖ chÞu t¸c dông xung trong kho¶ng thêi gian t = 2T1 (T1 lμ chu kúdao ®éng riªng d¹ng thø nhÊt) t¹i ®iÓm A cã §å thÞ m« men uèn ®éng MAT(t)®−îc thÓ hiÖn nh− h×nh sau (H×nh 4)H×nh 4: M« men MA (t) t¹i A theo thêi gian