Luận văn Phương pháp nguyên lý cực trị gauss đối với các bài toán động lực học công trình

høc l­îng c­ìng bøc Z. C¸c b­íc tiÕn hµnh gièng tr­êng hîp (a). Tõ kÕt qu¶ tÝnh to¸n, nhËn ®­îc: * Tr­êng hîp (c): ; ; (3.61) Thay (3.55), (3.56), (3.61) vµo (3.57), nhËn ®­îc biÓu thøc l­îng c­ìng bøc Z. Tõ kÕt qu¶ tÝnh to¸n, nhËn ®­îc: * NhËn xÐt: c¸c kÕt qu¶ nhËn ®­îc còng chÝnh lµ c¸c kÕt qu¶ ë vÝ dô (3.1.2). 3.3. Bµi to¸n dao ®éng c­ìng bøc cña hÖ h÷u h¹n bËc tù do: Víi hÖ h÷u h¹n bËc tù do chÞu lùc P(t) = Psinrt th× lêi gi¶i cã thÓ ®­îc tiÕn hµnh theo c¸c b­íc ®· ®­îc tr×nh bµy ë môc (1.4.2.2). Trong ®ã, tÇn sè dao ®éng riªng vµ d¹ng dao ®éng riªng ®­îc x¸c ®Þnh dùa vµo nguyªn lý cùc trÞ Gauss víi c¸c b­íc thùc hiÖn nh­ ë môc (2.6) hoÆc (2.7). VÝ dô 11: Cho dÇm ®µn håi mang hai khèi l­îng tËp trung m1 = m = 1800 kg; m2 = 2m (bá qua khèi l­îng cña dÇm). DÇm cã EJ = 150.106 Nm2, chiÒu dµi l = 12 (m). T¸c dông lªn khèi l­îng m1 lùc ®iÒu hoµ cã biªn ®é P = 18KN, tÇn sè vßng r = 108 (s-1). H·y x¸c ®Þnh chuyÓn vÞ vµ néi lùc ®éng cña dÇm. Lêi gi¶i: * X¸c ®Þnh tÇn sè riªng vµ d¹ng dao ®éng riªng: ViÕt biÓu thøc ®­êng ®é vâng cho c¸c ®o¹n d­íi d¹ng ®a thøc nh­ sau: ; ; (3.62) C¸c ®iÒu kiÖn rµng buéc: g1 = ; g2 = ; g3 = ; g4 = ; g5 = ; g6 = (3.63) Chän hÖ so s¸nh gièng dÇm cho nh­ng kh«ng cã liªn kÕt. L­îng c­ìng bøc ®­îc viÕt nh­ sau: (3.64) Cho Z ® min, ta cã hÖ ph­¬ng tr×nh: (3.65) Sau khi ®· t×m cùc trÞ cña phiÕm hµm Z theo c¸c hÖ sè ai vµ bj , ta thay: ; vµo c¸c ph­¬ng tr×nh (3.65). Gi¶i hÖ ph­¬ng tr×nh tuyÕn tÝnh (3.65), x¸c ®Þnh ®­îc c¸c hÖ sè ch­a biÕt ai, bj, cn vµ c¸c nh©n tö Lagrange lk. Tõ kÕt qu¶ tÝnh to¸n cã ®­îc, cho =0, cã kÕt qu¶ sau: Thay c¸c gi¸ trÞ ai, bj, cn t×m ®­îc vµo (3.62), ta cã: khi khèi l­îng m1 dao ®éng víi biªn ®é b»ng 1 th× khèi l­îng m2 dao ®éng víi biªn ®é b»ng () Thay c¸c gi¸ trÞ EJ, m vµ l ®· cho, ta cã: Vecto tÇn sè dao ®éng riªng: Ma trËn c¸c d¹ng chÝnh: ChuÈn ho¸ c¸c d¹ng dao ®éng riªng theo (1.11): + TÝnh c¸c hÖ sè ai: + D¹ng chuÈn ®­îc x¸c ®Þnh: Ma trËn d¹ng chuÈn: * X¸c ®Þnh t¶i träng khai triÓn theo c¸c d¹ng riªng: dùa vµo (1.14), ta cã VËy T¶i träng khai triÓn theo c¸c d¹ng riªng ®­îc thÓ hiÖn ë h×nh (3.23). * X¸c ®Þnh chuyÓn vÞ ®éng cña hÖ; C¸c phÇn tö cña ma trËn c¸c hÖ sè ¶nh h­ëng ®éng häc ®­îc tÝnh theo (1.17): (3.66) Thay c¸c gi¸ trÞ wi vµ r vµo (3.66) nhËn ®­îc ma trËn c¸c hÖ sè ¶nh h­ëng ®éng häc nh­ sau: ChuyÓn vÞ cña hÖ ®­îc tÝnh theo (1.18): * X¸c ®Þnh lùc ®µn håi : Theo (1.20), ta cã: (3.67) Thay c¸c gi¸ trÞ wi vµ r vµo (3.67) nhËn ®­îc ma trËn: Lùc ®µn håi ®­îc tÝnh theo (1.21): * X¸c ®Þnh néi lùc ®éng: tõ biÓu ®å trªn h×nh (3.24), MA do P = 1 ®Æt lÇn l­ît t¹i A vµ t¹i B g©y ra lµ:MA,1 = 3l/16 vµ MA,2 = l/16. M«men uèn t¹i A: M«men uèn t¹i B: BiÓu ®å m«men ®éng nh­ h×nh (3.25) KÕt luËn vµ KiÕn NghÞ 1) Nguyªn lý cùc trÞ Gauss ®­îc K.F.Gauss ph¸t biÓu cho c¬ hÖ chÊt ®iÓm. Dùa trªn nguyªn lý nµy, GS.TSKH. Hµ Huy C­¬ng ®· ®Ò xuÊt ph­¬ng ph¸p gi¶i bµi to¸n c¬ häc vËt r¾n biÕn d¹ng. Khi sö dông ph­¬ng ph¸p nguyªn lý cùc trÞ Gauss vµo bµi to¸n ®éng lùc häc, nã ®· gi¶i quyÕt ®­îc vÊn ®Ò quan träng cña bµi to¸n dao ®éng, ®ã lµ t×m tÇn sè riªng vµ d¹ng dao ®éng riªng víi c¸ch lµm hoµn toµn míi so víi nh÷ng ph­¬ng ph¸p ®· biÕt tr­íc ®©y. 2) C¸c ph­¬ng ph¸p ®· biÕt tùu trung l¹i ®Òu xuÊt ph¸t tõ nguyªn lý n¨ng l­îng. Cßn ph­¬ng ph¸p nguyªn lý cùc trÞ Gauss ®· tho¸t ra khái nguyªn lý n¨ng l­îng. 3) T¸c gi¶ ®· x©y dùng ®­îc c¸c b­íc tiÕn hµnh ®Ó x¸c ®Þnh tÇn sè dao ®éng riªng vµ d¹ng dao ®éng riªng cña hÖ dao ®éng. B»ng viÖc t×m hiÓu vµ ¸p dông tÝnh to¸n cho c¸c bµi to¸n cô thÓ cña hÖ dÇm, khung cã mét sè bËc tù do hoÆc v« sè bËc tù do vµ cã c¸c liªn kÕt kh¸c nhau, t¸c gi¶ ®· chøng tá ®­îc sù ®óng ®¾n vµ hiÖu qu¶ cña ph­¬ng ph¸p nµy. C¸c kÕt qu¶ nhËn ®­îc phï hîp víi nh÷ng kÕt qu¶ ®· cã khi gi¶i b»ng c¸c ph­¬ng ph¸p kh¸c. 4) Ph­¬ng ph¸p nguyªn lý cùc trÞ Gauss x©y dùng vµ ®­a ra lêi gi¶i cho bµi to¸n ®éng nh­ bµi to¸n tÜnh, do ®ã cã c¸ch nh×n ®¬n gi¶n vµ tá ra cã hiÖu qu¶ ®èi víi c¸c bµi to¸n ®éng lùc häc. * KiÕn nghÞ: 1) Cã thÓ sö dông ph­¬ng ph¸p nguyªn lý cùc trÞ Gauss nh­ mét ph­¬ng ph¸p míi trong gi¶ng d¹y vµ häc tËp, nghiªn cøu. 2) Ph­¬ng ph¸p nguyªn lý cùc trÞ Gauss më ra mét h­íng míi vÒ thùc nghiÖm, thay v× thÝ nghiÖm trªn cÊu kiÖn nµy (hÖ cho) th× cã thÓ tiÕn hµnh thÝ nghiÖm trªn cÊu kiÖn kh¸c (hÖ so s¸nh). ViÖc cùc tiÓu ho¸ l­îng c­ìng bøc cho phÐp ®i ®Õn kÕt qu¶ thÝ nghiÖm ®èi víi cÊu kiÖn ph¶i xÐt. Tµi liÖu tham kh¶o Ph¹m §×nh Ba, NguyÔn V¨n Héi, Gi¸o tr×nh ®éng lùc häc c«ng tr×nh, Häc viÖn kü thuËt qu©n sù, Hµ néi, 1994. Hµ Huy C­¬ng, Ph­¬ng ph¸p nguyªn lý cùc trÞ Gauss, T¹p chÝ Khoa häc vµ kü thuËt, IV/2005 Tr. 112 ¸118. Ninh Quang H¶i, C¬ häc lý thuyÕt, Nhµ xuÊt b¶n X©y dùng, Hµ néi, 1999. TrÇn ThÞ Kim HuÕ, Ph­¬ng ph¸p nguyªn lÝ cùc trÞ Gauss ®èi víi c¸c bµi to¸n c¬ häc kÕt cÊu, LuËn v¨n th¹c sÜ kü thuËt, Hµ néi, 2005. NguyÔn V¨n Khang, Dao ®éng kü thuËt, Nhµ xuÊt b¶n Khoa häc vµ kü thuËt, Hµ néi, 1998. NguyÔn Xu©n Hïng, §éng lùc häc c«ng tr×nh biÓn, Nhµ xuÊt b¶n khoa häc vµ kü thuËt, Hµ néi, 1999. Vò §×nh Lai, NguyÔn Xu©n Lùu, Bïi §×nh Nghi, Søc bÒn vËt liÖu, Nhµ xuÊt b¶n Giao th«ng vËn t¶i, Hµ néi, 2002. NguyÔn V¨n Liªn, §inh Träng B»ng, NguyÔn Ph­¬ng Thµnh, Søc bÒn vËt liÖu, Nhµ xuÊt b¶n X©y dùng, Hµ néi, 2003. NguyÔn V¨n Liªn, NguyÔn Ph­¬ng Thµnh, Xö lý d÷ liÖu ®éng ®Ó x¸c ®Þnh dao ®éng c¸c c«ng tr×nh, T¹p chÝ X©y dùng, 11/2001 Tr. 48 ¸ 56. NguyÔn V¨n Ph­îng, §éng lùc häc c«ng tr×nh, Nhµ xuÊt b¶n Khoa häc vµ kü thuËt. Hoµng Nh­ S¸u, TÝnh to¸n kÕt cÊu x©y dùng b»ng ph­¬ng ph¸p sai ph©n h÷u h¹n, biÕn ph©n vµ hçn hîp sai ph©n h÷u h¹n – biÕn ph©n, Nhµ xuÊt b¶n X©y dùng, Hµ néi, 1982. NguyÔn Ph­¬ng Thµnh, Nghiªn cøu tr¹ng th¸i øng suÊt – biÕn d¹ng tÊm nhiÒu líp chÞu t¶i träng ®éng cã xÐt lùc ma s¸t ë c¸c mÆt tiÕp xóc, LuËn ¸n tiÕn sÜ khoa häc, Hµ néi, 2002. NguyÔn Ph­¬ng Thµnh, Nghiªn cøu ph¶n øng ®éng tÊm nhiÒu líp cã xÐt lùc ma s¸t ë c¸c mÆt tiÕp xóc, T¹p chÝ Khoa häc vµ C«ng nghÖ, Trung t©m Khoa häc tù nhiªn vµ C«ng nghÖ Quèc gia, TËp XXXI - 2001 - 2, Tr. 48 ¸ 56. LÒu Thä Tr×nh, C¬ häc kÕt cÊu, TËp I – HÖ tÜnh ®Þnh, Nhµ xuÊt b¶n Khoa häc vµ kü thuËt, Hµ néi, 2003. LÒu Thä Tr×nh, C¬ häc kÕt cÊu, TËp II – HÖ siªu tÜnh, Nhµ xuÊt b¶n Khoa häc vµ kü thuËt, Hµ néi, 2003. LÒu Thä Tr×nh, æn ®inh vµ ®éng lùc häc c«ng tr×nh, Nhµ xuÊt b¶n Khoa häc vµ kü thuËt. NguyÔn V¨n V­îng, Lý thuyÕt ®µn håi øng dông, Nhµ xuÊt b¶n Gi¸o dôc, Hµ néi, 1999. Bath K.J, Numerical methods in finite elements analysis, Prentice Hall, INC, Englewood Cliffs, New Jersey, 1976. William T.Thomson, Theory of Vibration with Applications, Stanley Thornes Ray W.Clough, Joseph Penzien, Dynamics of structures. Ha Huy Cuong, Nguyen Phuong Thanh, Application du principe d’ obligation minimale dans la rÐsolution des problÌmes de la mÐcanique des fluids, Structures and Interactions, Nha Trang, Vietnam August 14 - 18. 2000, P. 693 - 702. phô lôc tÝnh to¸n 3.1. X¸c ®Þnh tÇn sè dao déng riªng vµ d¹ng dao ®éng riªng A. DÇm h÷u h¹n bËc tù do. VÝ dô 1 : dÇm ®¬n gi¶n cã hai bËc tù do. syms a1 a2 a3 a4; syms b0 b1 b2 b3 b4; syms c0 c1 c2 c3 c4; syms z t l1 l2 l3 l ej m f1 f2; %f1 f2 la luc quan tinh syms lamda1 lamda2 lamda3 lamda4 lamda5 lamda6; syms omega; y1=(a1*z+a2*z^2+a3*z^3+a4*z^4)*sin(omega*t);% duong do vong cua doan 1 y2=(b0+b1*z+b2*z^2+b3*z^3+b4*z^4)*sin(omega*t);% duong do vong cua doan 2 y3=(c0+c1*z+c2*z^2+c3*z^3+c4*z^4)*sin(omega*t);% duong do vong cua doan 3 u1=diff(y1,'z');%goc xoay cua doan 1 u2=diff(y2,'z');%goc xoay cua doan 2 u3=diff(y3,'z');%goc xoay cua doan 3 n1=diff(y1,'z',2)%dao ham bac hai cua duong do vong doan 1 n2=diff(y2,'z',2)%dao ham bac hai cua duong do vong doan 2 n3=diff(y3,'z',2)%dao ham bac hai cua duong do vong doan 3 t1=ej*int(n1^2,'z',0,l1); t2=ej*int(n2^2,'z',0,l2); t3=ej*int(n3^2,'z',0,l3); t4=2*f1*subs(y1,'z',l1);%luong cuong buc do luc quan tinh t5=2*f2*subs(y2,'z',l2);%luong cuong buc do luc quan tinh t6=lamda1*subs(y3,'z',l3);%dieu kien bien t7=lamda2*(subs(y1,'z',l1)-subs(y2,'z',0));%dieu kien bien cua doan 1 va 2 t8=lamda3*(subs(u1,'z',l1)-subs(u2,'z',0));%dieu kien lbien cua doan 1 va 2 t9=lamda4*(subs(y2,'z',l2)-subs(y3,'z',0));%dieu kien bien cua doan 2 va 3 t10=lamda5*(subs(u2,'z',l2)-subs(u3,'z',0))%dieu kien bien cua doan 2 va do¹n 3 t11=lamda6*(subs(y1,'z',l1)-1);%dk co nghiem lcb=t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+t11 h1=diff(lcb,'a1'); h2=diff(lcb,'a2'); h3=diff(lcb,'a3'); h4=diff(lcb,'a4'); h5=diff(lcb,'b0'); h6=diff(lcb,'b1'); h7=diff(lcb,'b2'); h8=diff(lcb,'b3'); h9=diff(lcb,'b4'); h10=diff(lcb,'c0'); h11=diff(lcb,'c1'); h12=diff(lcb,'c2'); h13=diff(lcb,'c3'); h14=diff(lcb,'c4'); h15=diff(lcb,'lamda1'); h16=diff(lcb,'lamda2'); h17=diff(lcb,'lamda3'); h18=diff(lcb,'lamda4'); h19=diff(lcb,'lamda5'); h20=diff(lcb,'lamda6'); k1=-m*omega^2*subs(y1,'z',l1);%luc quan tinh k2=-m*omega^2*subs(y2,'z',l2);%luc quan tinh g1=subs(h1,f1,k1); g2=subs(h2,f1,k1); g3=subs(h3,f1,k1); g4=subs(h4,f1,k1); g5=subs(h5,f2,k2); g6=subs(h6,f2,k2); g7=subs(h7,f2,k2); g8=subs(h8,f2,k2); g9=subs(h9,f2,k2); r=solve(g1,g2,g3,g4,g5,g6,g7,g8,g9,h10,h11,h12,h13,h14,h15,h16,h17,h18,h19,h20,'a1','a2','a3','a4','b0','b1','b2','b3','b4','c0','c1','c2','c3','c4','lamda1','lamda2','lamda3','lamda4','lamda5','lamda6') a1=r.a1 a2=r.a2 a3=r.a3 a4=r.a4 b0=r.b0 b1=r.b1 b2=r.b2 b3=r.b3 b4=r.b4 c0=r.c0 c1=r.c1 c2=r.c2 c3=r.c3 c4=r.c4 lamda1=r.lamda1 lamda2=r.lamda2 lamda3=r.lamda3 lamda4=r.lamda4 lamda5=r.lamda5 lamda6=r.lamda6 x=solve(lamda6,'omega') x1=subs(x,'l1',l/3)%thay l1 bang l/3 x2=subs(x1,'l2',l/3)%thay l2 bang l/3 x3=subs(x2,'l3',l/3)%thay l3 bang l/3 VÝ dô 2 : dÇm ®¬n gi¶n cã ba bËc tù do. syms a1 a2 a3 a4; syms b0 b1 b2 b3 b4; syms c0 c1 c2 c3 c4; syms d1 d2 d3 d4; syms z t l ej m f1 f2 f3; %f1 f2 f3 la luc quan tinh syms lamda1 lamda2 lamda3 lamda4 lamda5 lamda6 lamda7; syms omega; y1=(a1*z+a2*z^2+a3*z^3+a4*z^4)*sin(omega*t);% duong do vong cua doan 1 y2=(b0+b1*z+b2*z^2+b3*z^3+b4*z^4)*sin(omega*t);% duong do vong cua doan 2 y3=(c0+c1*z+c2*z^2+c3*z^3+c4*z^4)*sin(omega*t);% duong do vong cua doan 3 y4=(d1*z+d2*z^2+d3*z^3+d4*z^4)*sin(omega*t);% duong do vong cua doan 4 u1=diff(y1,'z');%goc xoay cua doan 1 u2=diff(y2,'z');%goc xoay cua doan 2 u3=diff(y3,'z');%goc xoay cua doan 3 u4=diff(y4,'z');%goc xoay cua doan 4 n1=diff(y1,'z',2);%dao ham bac hai cua duong do vong doan 1 n2=diff(y2,'z',2);%dao ham bac hai cua duong do vong doan 2 n3=diff(y3,'z',2);%dao ham bac hai cua duong do vong doan 3 n4=diff(y4,'z',2);%dao ham bac hai cua duong do vong doan 4 t1=ej*int(n1^2,'z',0,l/4); t2=ej*int(n2^2,'z',0,l/4); t3=ej*int(n3^2,'z',0,l/4); t4=ej*int(n4^2,'z',0,l/4); t5=2*f1*subs(y1,'z',l/4);%luong cuong buc do luc quan tinh t6=2*f2*subs(y2,'z',l/4);%luong cuong buc do luc quan tinh t7=2*f3*subs(y3,'z',l/4);%luong cuong buc do luc quan tinh t8=lamda1*(subs(y1,'z',l/4)-subs(y2,'z',0));%dieu kien bien cua doan 1 va 2 t9=lamda2*(subs(u1,'z',l/4)-subs(u2,'z',0)); t10=lamda3*(subs(y2,'z',l/4)-subs(y3,'z',0));%dieu kien bien cua doan 2 va 3 t11=lamda4*(subs(u2,'z',l/4)-subs(u3,'z',0)); t12=lamda5*(subs(y3,'z',l/4)-subs(y4,'z',l/4));%dieu kien bien cua doan 3 va 4 t13=lamda6*(subs(u3,'z',l/4)-subs((-u4),'z',l/4)); t14=lamda7*(subs(y1,'z',l/4)-1);%dk co nghiem lcb=t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+t11+t12+t13+t14 h1=diff(lcb,'a1'); h2=diff(lcb,'a2'); h3=diff(lcb,'a3'); h4=diff(lcb,'a4'); h5=diff(lcb,'b0'); h6=diff(lcb,'b1'); h7=diff(lcb,'b2'); h8=diff(lcb,'b3'); h9=diff(lcb,'b4'); h10=diff(lcb,'c0'); h11=diff(lcb,'c1'); h12=diff(lcb,'c2'); h13=diff(lcb,'c3'); h14=diff(lcb,'c4'); h15=diff(lcb,'d1'); h16=diff(lcb,'d2'); h17=diff(lcb,'d3'); h18=diff(lcb,'d4'); h19=diff(lcb,'lamda1'); h20=diff(lcb,'lamda2'); h21=diff(lcb,'lamda3'); h22=diff(lcb,'lamda4'); h23=diff(lcb,'lamda5'); h24=diff(lcb,'lamda6'); h25=diff(lcb,'lamda7'); k1=-m*omega^2*subs(y1,'z',l/4);%luc quan tinh k2=-m*omega^2*subs(y2,'z',l/4);%luc quan tinh k3=-m*omega^2*subs(y3,'z',l/4);%luc quan tinh g1=subs(h1,f1,k1); g2=subs(h2,f1,k1); g3=subs(h3,f1,k1); g4=subs(h4,f1,k1); g5=subs(h5,f2,k2); g6=subs(h6,f2,k2); g7=subs(h7,f2,k2); g8=subs(h8,f2,k2); g9=subs(h9,f2,k2); g10=subs(h10,f3,k3); g11=subs(h11,f3,k3); g12=subs(h12,f3,k3); g13=subs(h13,f3,k3); g14=subs(h14,f3,k3); r=solve(g1,g2,g3,g4,g5,g6,g7,g8,g9,g10,g11,g12,g13,g14,h15,h16,h17,h18,h19,h20,h21,h22,h23,h24,h25,'a1','a2','a3','a4','b0','b1','b2','b3','b4','c0','c1','c2','c3','c4','d1','d2','d3','d4','lamda1','lamda2','lamda3','lamda4','lamda5','lamda6','lamda7') a1=r.a1 a2=r.a2 a3=r.a3 a4=r.a4 b0=r.b0 b1=r.b1 b2=r.b2 b3=r.b3 b4=r.b4 c0=r.c0 c1=r.c1 c2=r.c2 c3=r.c3 c4=r.c4 d1=r.d1 d2=r.d2 d3=r.d3 d4=r.d4 lamda1=r.lamda1 lamda2=r.lamda2 lamda3=r.lamda3 lamda4=r.lamda4 lamda5=r.lamda5 lamda6=r.lamda6 lamda7=r.lamda7 x=solve(lamda7,'omega') VÝ dô 3 : dÇm ®¬n gi¶n cã ®Çu thõa. syms a1 a2 a3 a4; syms b0 b1 b2 b3 b4; syms c1 c2 c3 c4; syms z t l ej m f1 f2; %f1 f2 la luc quan tinh syms lamda1 lamda2 lamda3 lamda4 lamda5; syms omega; y1=(a1*z+a2*z^2+a3*z^3+a4*z^4)*sin(omega*t);% duong do vong cua doan 1 y2=(b0+b1*z+b2*z^2+b3*z^3+b4*z^4)*sin(omega*t);% duong do vong cua doan 2 y3=(c1*z+c2*z^2+c3*z^3+c4*z^4)*sin(omega*t);% duong do vong cua doan 3 u1=diff(y1,'z');%goc xoay cua doan 1 u2=diff(y2,'z');%goc xoay cua doan 2 u3=diff(y3,'z');%goc xoay cua doan 3 n1=diff(y1,'z',2)%dao ham bac hai cua duong do vong doan 1 n2=diff(y2,'z',2)%dao ham bac hai cua duong do vong doan 2 n3=diff(y3,'z',2)%dao ham bac hai cua duong do vong doan 3 t1=ej*int(n1^2,'z',0,l/2); t2=ej*int(n2^2,'z',0,l/2); t3=ej*int(n3^2,'z',0,l/2); t4=2*f1*subs(y1,'z',l/2);%luong cuong buc do luc quan tinh t5=2*f2*subs(y3,'z',l/2);%luong cuong buc do luc quan tinh t6=lamda1*subs(y2,'z',l/2);%dieu kien bien t7=lamda2*(subs(y1,'z',l/2)-subs(y2,'z',0));%dieu kien bien cua doan 1 va 2 t8=lamda3*(subs(u1,'z',l/2)-subs(u2,'z',0));%dieu kien bien cua doan 1 va 2 t9=lamda4*(subs(u2,'z',l/2)-subs(u3,'z',0));%dieu kien bien cua doan 2 va 3 t10=lamda5*(subs(y1,'z',l/2)-1);%dk co nghiem lcb=t1+t2+t3+t4+t5+t6+t7+t8+t9+t10 h1=diff(lcb,'a1'); h2=diff(lcb,'a2'); h3=diff(lcb,'a3'); h4=diff(lcb,'a4'); h5=diff(lcb,'b0'); h6=diff(lcb,'b1'); h7=diff(lcb,'b2'); h8=diff(lcb,'b3'); h9=diff(lcb,'b4'); h10=diff(lcb,'c1'); h11=diff(lcb,'c2'); h12=diff(lcb,'c3'); h13=diff(lcb,'c4'); h14=diff(lcb,'lamda1'); h15=diff(lcb,'lamda2'); h16=diff(lcb,'lamda3'); h17=diff(lcb,'lamda4'); h18=diff(lcb,'lamda5'); k1=-m*omega^2*subs(y1,'z',l/2);%luc quan tinh k2=-m*omega^2*subs(y3,'z',l/2);%luc quan tinh g1=subs(h1,f1,k1); g2=subs(h2,f1,k1); g3=subs(h3,f1,k1); g4=subs(h4,f1,k1); g5=subs(h10,f2,k2); g6=subs(h11,f2,k2); g7=subs(h12,f2,k2); g8=subs(h13,f2,k2); r=solve(g1,g2,g3,g4,g5,g6,g7,g8,h5,h6,h7,h8,h9,h14,h15,h16,h17,h18,'a1','a2','a3','a4','b0','b1','b2','b3','b4','c1','c2','c3','c4','lamda1','lamda2','lamda3','lamda4','lamda5') a1=r.a1 a2=r.a2 a3=r.a3 a4=r.a4 b0=r.b0 b1=r.b1 b2=r.b2 b3=r.b3 b4=r.b4 c1=r.c1 c2=r.c2 c3=r.c3 c4=r.c4 lamda1=r.lamda1 lamda2=r.lamda2 lamda3=r.lamda3 lamda4=r.lamda4 lamda5=r.lamda5 x=solve(lamda5,'omega') VÝ dô 4 : dÇm liªn tôc cã hai bËc tù do. syms a1 a2 a3 a4; syms b0 b1 b2 b3 b4; syms c1 c2 c3 c4; syms d0 d1 d2 d3 d4; syms z t l ej m f1 f2;%f1 f2 la luc quan tinh syms lamda1 lamda2 lamda3 lamda4 lamda5 lamda6 lamda7 lamda8; syms omega; y1=(a1*z+a2*z^2+a3*z^3+a4*z^4)*sin(omega*t);% duong do vong cua doan 1 y2=(b0+b1*z+b2*z^2+b3*z^3+b4*z^4)*sin(omega*t);% duong do vong cua doan 2 y3=(c1*z+c2*z^2+c3*z^3+c4*z^4)*sin(omega*t);% duong do vong cua doan 3 y4=(d0+d1*z+d2*z^2+d3*z^3+d4*z^4)*sin(omega*t);% duong do vong cua doan 4 u1=diff(y1,'z');%goc xoay cua doan 1 u2=diff(y2,'z');%goc xoay cua doan 2 u3=diff(y3,'z');%goc xoay cua doan 3 u4=diff(y4,'z');%goc xoay cua doan 4 n1=diff(y1,'z',2);%dao ham bac hai cua duong do vong doan 1 n2=diff(y2,'z',2);%dao ham bac hai cua duong do vong doan 2 n3=diff(y3,'z',2);%dao ham bac hai cua duong do vong doan 3 n4=diff(y4,'z',2);%dao ham bac hai cua duong do vong doan 4 t1=ej*int(n1^2,'z',0,l/2); t2=ej*int(n2^2,'z',0,l/2); t3=ej*int(n3^2,'z',0,l/2); t4=ej*int(n4^2,'z',0,l/2); t5=2*f1*subs(y1,'z',l/2);%luong cuong buc do luc quan tinh t6=2*f2*subs(y3,'z',l/2);%luong cuong buc do luc quan tinh t7=lamda1*subs(y2,'z',l/2);%dieu kien bien t8=lamda2*subs(y4,'z',l/2);%dieu kien bien t9=lamda3*(subs(y1,'z',l/2)-subs(y2,'z',0));%dieu kien bien cua doan 1 va 2 t10=lamda4*(subs(u1,'z',l/2)-subs(u2,'z',0));%dieu kien bien cua doan 1 va doan 2 t11=lamda5*(subs(u2,'z',l/2)-subs(u3,'z',0));%dieu kien bien cua doan 2 va doan 3 t12=lamda6*(subs(y3,'z',l/2)-subs(y4,'z',0));%dieu kien bien cua doan 3 va doan 4 t13=lamda7*(subs(u3,'z',l/2)-subs(u4,'z',0));%dieu kien bien cua doan 3 va doan 4 t14=lamda8*(subs(y1,'z',l/2)-1);%dk co nghiem lcb=t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+t11+t12+t13+t14 h1=diff(lcb,'a1'); h2=diff(lcb,'a2'); h3=diff(lcb,'a3'); h4=diff(lcb,'a4'); h5=diff(lcb,'b0'); h6=diff(lcb,'b1'); h7=diff(lcb,'b2'); h8=diff(lcb,'b3'); h9=diff(lcb,'b4'); h10=diff(lcb,'c1'); h11=diff(lcb,'c2'); h12=diff(lcb,'c3'); h13=diff(lcb,'c4'); h14=diff(lcb,'d0'); h15=diff(lcb,'d1'); h16=diff(lcb,'d2'); h17=diff(lcb,'d3'); h18=diff(lcb,'d4'); h19=diff(lcb,'lamda1'); h20=diff(lcb,'lamda2'); h21=diff(lcb,'lamda3'); h22=diff(lcb,'lamda4'); h23=diff(lcb,'lamda5'); h24=diff(lcb,'lamda6'); h25=diff(lcb,'lamda7'); h26=diff(lcb,'lamda8'); k1=-m*omega^2*subs(y1,'z',l/2);%luc quan tinh k2=-m*omega^2*subs(y3,'z',l/2);%luc quan tinh g1=subs(h1,f1,k1); g2=subs(h2,f1,k1); g3=subs(h3,f1,k1); g4=subs(h4,f1,k1); g5=subs(h10,f2,k2); g6=subs(h11,f2,k2); g7=subs(h12,f2,k2); g8=subs(h13,f2,k2); r=solve(g1,g2,g3,g4,g5,g6,g7,g8,h5,h6,h7,h8,h9,h14,h15,h16,h17,h18,h19,h20,h21,h22,h23,h24,h25,h26,'a1','a2','a3','a4','b0','b1','b2','b3','b4','c1','c2','c3','c4','d0','d1','d2','d3','d4','lamda1','lamda2','lamda3','lamda4','lamda5','lamda6','lamda7','lamda8') a1=r.a1 a2=r.a2 a3=r.a3 a4=r.a4 b0=r.b0 b1=r.b1 b2=r.b2 b3=r.b3 b4=r.b4 c1=r.c1 c2=r.c2 c3=r.c3 c4=r.c4 d0=r.d0 d1=r.d1 d2=r.d2 d3=r.d3 d4=r.d4 lamda1=r.lamda1 lamda2=r.lamda2 lamda3=r.lamda3 lamda4=r.lamda4 lamda5=r.lamda5 lamda6=r.lamda6 lamda7=r.lamda7 lamda8=r.lamda8 x=solve(lamda8,'omega') VÝ dô 5 : dÇm cã liªn kÕt ®Çu ngµm, ®Çu gèi di ®éng. syms a2 a3 a4; syms b0 b1 b2 b3 b4; syms z t l1 l2 l ej m f;%f la luc quan tinh syms lamda1 lamda2 lamda3 lamda4; syms omega; y1=(a2*z^2+a3*z^3+a4*z^4)*sin(omega*t);% duong do vong cua doan 1 y2=(b0+b1*z+b2*z^2+b3*z^3+b4*z^4)*sin(omega*t);% duong do vong cua doan 2 u1=diff(y1,'z');%goc xoay cua doan 1 u2=diff(y2,'z');%goc xoay cua doan 2 n1=diff(y1,'z',2);%dao ham bac hai cua duong dan hoi doan 1 n2=diff(y2,'z',2);%dao ham bac hai cua duong dan hoi doan 2 t1=ej*int(n1^2,'z',0,l1); t2=ej*int(n2^2,'z',0,l2); t3=2*f*subs(y1,'z',l1);%luong cuong buc do luc quan tinh t4=lamda1*subs(y2,'z',l2);%dieu kien bien t5=lamda2*(subs(y1,'z',l1)-subs(y2,'z',0));%dieu kien bien cua doan 1 va 2 t6=lamda3*(subs(u1,'z',l1)-subs(u2,'z',0));%dieu kien bien cua doan 1 va 2 t7=lamda4*(subs(y1,'z',l1)-1);%dk co nghiem lcb=t1+t2+t3+t4+t5+t6+t7 h1=diff(lcb,'a2'); h2=diff(lcb,'a3'); h3=diff(lcb,'a4'); h4=diff(lcb,'b0'); h5=diff(lcb,'b1'); h6=diff(lcb,'b2'); h7=diff(lcb,'b3'); h8=diff(lcb,'b4'); h9=diff(lcb,'lamda1'); h10=diff(lcb,'lamda2'); h11=diff(lcb,'lamda3'); h12=diff(lcb,'lamda4'); k=-m*omega^2*subs(y1,'z',l1);%luc quan tinh g1=subs(h1,f,k); g2=subs(h2,f,k); g3=subs(h3,f,k); r=solve(g1,g2,g3,h4,h5,h6,h7,h8,h9,h10,h11,h12,'a2','a3','a4','b0','b1','b2','b3','b4','lamda1','lamda2','lamda3','lamda4') a2=r.a2 a3=r.a3 a4=r.a4 b0=r.b0 b1=r.b1 b2=r.b2 b3=r.b3 b4=r.b4 lamda1=r.lamda1 lamda2=r.lamda2 lamda3=r.lamda3 lamda4=r.lamda4 x=solve(lamda4,'omega') x1=subs(x,'l1',l/2)%thay l1 bang l/2 x2=subs(x1,'l2',l/2)%thay l2 bang l/2 B. Khung h÷u h¹n bËc tù do. VÝ dô 6 : khung cã mét bËc tù do. Khèi l­îng m dao ®éng theo ph­¬ng ®øng (dao ®éng ®èi xøng): syms a2 a3 a4; syms b0 b1 b2 b3 b4; syms c0 c1 c2 c3 c4; syms d2 d3 d4; syms z t l ej m f1;%f1 la luc quan tinh syms lamda1 lamda2 lamda3 lamda4 lamda5 lamda6 lamda7 lamda8 lamda9 lamda10; syms omega; y1=(a2*z^2+a3*z^3+a4*z^4)*sin(omega*t);% duong do vong cua doan 1 y2=(b0+b1*z+b2*z^2+b3*z^3+b4*z^4)*sin(omega*t);% duong do vong cua doan 2 y3=(c0+c1*z+c2*z^2+c3*z^3+c4*z^4)*sin(omega*t);% duong do vong cua doan 3 y4=(d2*z^2+d3*z^3+d4*z^4)*sin(omega*t);% duong do vong cua doan 4 u1=diff(y1,'z');%goc xoay cua doan 1 u2=diff(y2,'z');%goc xoay cua doan 2 u3=diff(y3,'z');%goc xoay cua doan 3 u4=diff(y4,'z');%goc xoay cua doan 4 n1=diff(y1,'z',2);%dao ham bac hai cua duong do vong doan 1 n2=diff(y2,'z',2);%dao ham bac hai cua duong do vong doan 2 n3=diff(y3,'z',2);%dao ham bac hai cua duong do vong doan 3 n4=diff(y4,'z',2);%dao ham bac hai cua duong do vong doan 4 t1=ej*int(n1^2,'z',0,3*l/2); t2=2*ej*int(n2^2,'z',0,l/2); t3=2*ej*int(n3^2,'z',0,l/2); t4=ej*int(n4^2,'z',0,3*l/2); t5=2*f1*subs(y2,'z',l/2);%luong cuong buc do luc quan tinh t6=lamda1*subs(y1,'z',3*l/2);%dieu kien bien cua doan 1 t7=lamda2*subs(y4,'z',3*l/2);%dieu kien bien cua doan 4 t8=lamda3*subs(y2,'z',0);%dieu kien bien cua doan 2 t9=lamda4*subs(y3,'z',l/2);%dieu kien bien cua doan 3 t10=lamda5*(subs(u1,'z',3*l/2)-subs(u2,'z',0));%dieu kien bien cua doan 1 va 2 t11=lamda6*(subs(y2,'z',l/2)-subs(y3,'z',0));%dieu kien bien cua doan 2 va 3 t12=lamda7*(subs(u2,'z',l/2)-subs(u3,'z',0)); t13=lamda8*(subs(u2,'z',0)-subs((-u3),'z',l/2)); t14=lamda9*(subs(u3,'z',l/2)-subs(u4,'z',3*l/2)); t15=lamda10*(subs(y2,'z',l/2)-1);%dk co nghiem lcb=t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+t11+t12+t13+t14+t15 h1=diff(lcb,'a2'); h2=diff(lcb,'a3'); h3=diff(lcb,'a4'); h4=diff(lcb,'b0'); h5=diff(lcb,'b1'); h6=diff(lcb,'b2'); h7=diff(lcb,'b3'); h8=diff(lcb,'b4'); h9=diff(lcb,'c0'); h10=diff(lcb,'c1'); h11=diff(lcb,'c2'); h12=diff(lcb,'c3'); h13=diff(lcb,'c4'); h14=diff(lcb,'d2'); h15=diff(lcb,'d3'); h16=diff(lcb,'d4'); h17=diff(lcb,'lamda1'); h18=diff(lcb,'lamda2'); h19=diff(lcb,'lamda3'); h20=diff(lcb,'lamda4'); h21=diff(lcb,'lamda5'); h22=diff(lcb,'lamda6'); h23=diff(lcb,'lamda7'); h24=diff(lcb,'lamda8'); h25=diff(lcb,'lamda9'); h26=diff(lcb,'lamda10'); k1=-m*omega^2*subs(y2,'z',l/2);%luc quan tinh g1=subs(h4,f1,k1); g2=subs(h5,f1,k1); g3=subs(h6,f1,k1); g4=subs(h7,f1,k1); g5=subs(h8,f1,k1); r=solve(g1,g2,g3,g4,g5,h1,h2,h3,h9,h10,h11,h12,h13,h14,h15,h16,h17,h18,h19,h20,h21,h22,h23,h24,h25,h26,'a2','a3','a4','b0','b1','b2','b3','b4','c0','c1','c2','c3','c4','d2','d3','d4','lamda1','lamda2','lamda3','lamda4','lamda5','lamda6','lamda7','lamda8','lamda9','lamda10') a2=r.a2 a3=r.a3 a4=r.a4 b0=r.b0 b1=r.b1 b2=r.b2 b3=r.b3 b4=r.b4 c0=r.c0 c1=r.c1 c2=r.c2 c3=r.c3 c4=r.c4 d2=r.d2 d3=r.d3 d4=r.d4 lamda1=r.lamda1 lamda2=r.lamda2 lamda3=r.lamda3 lamda4=r.lamda4 lamda5=r.lamda5 lamda6=r.lamda6 lamda7=r.lamda7 lamda8=r.lamda8 lamda9=r.lamda9 lamda10=r.lamda10 x=solve(lamda10,'omega') Khèi l­îng m dao ®éng theo ph­¬ng ngang (dao ®éng ph¶n xøng): syms a2 a3 a4; syms b0 b1 b2 b3 b4; syms c0 c1 c2 c3 c4; syms d2 d3 d4; syms z t l ej m f1;%f1 la luc quan tinh syms lamda1 lamda2 lamda3 lamda4 lamda5 lamda6 lamda7 lamda8; syms omega; y1=(a2*z^2+a3*z^3+a4*z^4)*sin(omega*t);% duong do vong cua doan 1 y2=(b0+b1*z+b2*z^2+b3*z^3+b4*z^4)*sin(omega*t);% duong do vong cua doan 2 y3=(c0+c1*z+c2*z^2+c3*z^3+c4*z^4)*sin(omega*t);% duong do vong cua doan 3 y4=(d2*z^2+d3*z^3+d4*z^4)*sin(omega*t);% duong do vong cua doan 4 u1=diff(y1,'z');%goc xoay cua doan 1 u2=diff(y2,'z');%goc xoay cua doan 2 u3=diff(y3,'z');%goc xoay cua doan 3 u4=diff(y4,'z');%goc xoay cua doan 4 n1=diff(y1,'z',2);%dao ham bac hai cua duong do vong doan 1 n2=diff(y2,'z',2);%dao ham bac hai cua duong do vong doan 2 n3=diff(y3,'z',2);%dao ham bac hai cua duong do vong doan 3 n4=diff(y4,'z',2);%dao ham bac hai cua duong do vong doan 4 t1=ej*int(n1^2,'z',0,3*l/2); t2=2*ej*int(n2^2,'z',0,l/2); t3=2*ej*int(n3^2,'z',0,l/2); t4=ej*int(n4^2,'z',0,3*l/2); t5=2*f1*subs(y1,'z',3*l/2);%luong cuong buc do luc quan tinh t6=lamda1*subs(y2,'z',l/2);% chuyen vi dung=0 t7=lamda2*subs(y3,'z',0);%chuyen vi dung=0 t8=lamda3*(subs(u1,'z',3*l/2)-subs(u2,'z',0));%dieu kien bien doan 1 va 2 t9=lamda4*(subs(u2,'z',l/2)-subs(u3,'z',0));%dieu kien bien doan 2 va 3 t10=lamda5*(subs(u3,'z',l/2)-subs(u4,'z',3*l/2));%dieu kien bien doan 3 va 4 t11=lamda6*(subs(u2,'z',0)-subs(u3,'z',l/2)); t12=lamda7*(subs(y1,'z',3*l/2)-subs((-y4),'z',3*l/2)); t13=lamda8*(subs(y1,'z',3*l/2)-1); lcb=t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+t11+t12+t13 h1=diff(lcb,'a2'); h2=diff(lcb,'a3'); h3=diff(lcb,'a4'); h4=diff(lcb,'b0'); h5=diff(lcb,'b1'); h6=diff(lcb,'b2'); h7=diff(lcb,'b3'); h8=diff(lcb,'b4'); h9=diff(lcb,'c0'); h10=diff(lcb,'c1'); h11=diff(lcb,'c2'); h12=diff(lcb,'c3'); h13=diff(lcb,'c4'); h14=diff(lcb,'d2'); h15=diff(lcb,'d3'); h16=diff(lcb,'d4'); h17=diff(lcb,'lamda1'); h18=diff(lcb,'lamda2'); h19=diff(lcb,'lamda3'); h20=diff(lcb,'lamda4'); h21=diff(lcb,'lamda5'); h22=diff(lcb,'lamda6'); h23=diff(lcb,'lamda7'); h24=diff(lcb,'lamda8'); k1=-m*omega^2*subs(y1,'z',3*l/2);%luc quan tinh g1=subs(h1,f1,k1); g2=subs(h2,f1,k1); g3=subs(h3,f1,k1); r=solve(g1,g2,g3,h4,h5,h6,h7,h8,h9,h10,h11,h12,h13,h14,h15,h16,h17,h18,h19,h20,h21,h22,h23,h24,'a2','a3','a4','b0','b1','b2','b3','b4','c0','c1','c2','c3','c4','d2','d3','d4','lamda1','lamda2','lamda3','lamda4','lamda5','lamda6','lamda7','lamda8') a2=r.a2 a3=r.a3 a4=r.a4 b0=r.b0 b1=r.b1 b2=r.b2 b3=r.b3 b4=r.b4 c0=r.c0 c1=r.c1 c2=r.c2 c3=r.c3 c4=r.c4 d2=r.d2 d3=r.d3 d4=r.d4 lamda1=r.lamda1 lamda2=r.lamda2 lamda3=r.lamda3 lamda4=r.lamda4 lamda5=r.lamda5 lamda6=r.lamda6 lamda7=r.lamda7 lamda8=r.lamda8 x=solve(lamda8,'omega') VÝ dô 7 : khung cã hai bËc tù do. syms a1 a2 a3 a4; syms b0 b1 b2 b3 b4; syms c2 c3 c4; syms d0 d1 d2 d3 d4; syms e1 e2 e3 e4; syms z t ej m f1 f2;%f1 f2 la luc quan tinh syms lamda1 lamda2 lamda3 lamda4 lamda5 lamda6 lamda7 lamda8 lamda9 lamda10 lamda11; syms omega; y1=(a1*z+a2*z^2+a3*z^3+a4*z^4)*sin(omega*t);% duong do vong cua doan 1 y2=(b0+b1*z+b2*z^2+b3*z^3+b4*z^4)*sin(omega*t);% duong do vong cua doan 2 y3=(c2*z^2+c3*z^3+c4*z^4)*sin(omega*t);% duong do vong cua doan 3 y4=(d0+d1*z+d2*z^2+d3*z^3+d4*z^4)*sin(omega*t);% duong do vong cua doan 4 y5=(e1*z+e2*z^2+e3*z^3+e4*z^4)*sin(omega*t);% duong do vong cua doan 5 u1=diff(y1,'z');%goc xoay cua doan 1 u2=diff(y2,'z');%goc xoay cua doan 2 u3=diff(y3,'z');%goc xoay cua doan 3 u4=diff(y4,'z');%goc xoay cua doan 4 u5=diff(y5,'z');%goc xoay cua doan 5 n1=diff(y1,'z',2);%dao ham bac hai cua duong do vong doan 1 n2=diff(y2,'z',2);%dao ham bac hai cua duong do vong doan 2 n3=diff(y3,'z',2);%dao ham bac hai cua duong do vong doan 3 n4=diff(y4,'z',2);%dao ham bac hai cua duong do vong doan 4 n5=diff(y5,'z',2);%dao ham bac hai cua duong do vong doan 5 t1=ej*int(n1^2,'z',0,3); t2=ej*int(n2^2,'z',0,3); t3=ej*int(n3^2,'z',0,4); t4=ej*int(n4^2,'z',0,4); t5=ej*int(n5^2,'z',0,4); t6=2*f1*subs(y1,'z',3);%luong cuong buc do luc quan tinh t7=2*f2*subs(y4,'z',4);%luong cuong buc do luc quan tinh t8=lamda1*(subs(y1,'z',3)-subs(y2,'z',0));%dieu kien bien cua doan 1 va 2 t9=lamda2*(subs(u1,'z',3)-subs(u2,'z',0)); t10=lamda3*(subs(y4,'z',4)-subs(y5,'z',4));%dieu kien bien cua doan 4 va 5 t11=lamda4*(subs(u4,'z',4)-subs((-u5),'z',4)); t12=lamda5*(subs(u2,'z',3)-subs(u3,'z',4));%dieu kien bien cua doan 2 va 3 t13=lamda6*(subs(u2,'z',3)-subs(u4,'z',0));%dieu kien bien cua doan 2 va 4 t14=lamda7*subs(y3,'z',4);%chuyen vi ngang=0 t15=lamda8*subs(y2,'z',3);%chuyen vi dung=0 t16=lamda9*subs(y4,'z',0);%chuyen vi dung=0 t17=lamda10*(subs(y1,'z',3)-1); lcb=t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+t11+t12+t13+t14+t15+t16+t17 h1=diff(lcb,'a1'); h2=diff(lcb,'a2'); h3=diff(lcb,'a3'); h4=diff(lcb,'a4'); h5=diff(lcb,'b0'); h6=diff(lcb,'b1'); h7=diff(lcb,'b2'); h8=diff(lcb,'b3'); h9=diff(lcb,'b4'); h10=diff(lcb,'c2'); h11=diff(lcb,'c3'); h12=diff(lcb,'c4'); h13=diff(lcb,'d0'); h14=diff(lcb,'d1'); h15=diff(lcb,'d2'); h16=diff(lcb,'d3'); h17=diff(lcb,'d4'); h18=diff(lcb,'e1'); h19=diff(lcb,'e2'); h20=diff(lcb,'e3'); h21=diff(lcb,'e4'); h22=diff(lcb,'lamda1'); h23=diff(lcb,'lamda2'); h24=diff(lcb,'lamda3'); h25=diff(lcb,'lamda4'); h26=diff(lcb,'lamda5'); h27=diff(lcb,'lamda6'); h28=diff(lcb,'lamda7'); h29=diff(lcb,'lamda8'); h30=diff(lcb,'lamda9'); h31=diff(lcb,'lamda10'); k1=-m*omega^2*subs(y1,'z',3);%luc quan tinh k2=-m*omega^2*subs(y4,'z',4);%luc quan tinh g1=subs(h1,f1,k1); g2=subs(h2,f1,k1); g3=subs(h3,f1,k1); g4=subs(h4,f1,k1); g5=subs(h13,f2,k2); g6=subs(h14,f2,k2); g7=subs(h15,f2,k2); g8=subs(h16,f2,k2); g9=subs(h17,f2,k2); r=solve(g1,g2,g3,g4,g5,g6,g7,g8,g9,h5,h6,h7,h8,h9,h10,h11,h12,h18,h19,h20,h21,h22,h23,h24,h25,h26,h27,h28,h29,h30,h31,'a1','a2','a3','a4','b0','b1','b2','b3','b4','c2','c3','c4','d0','d1','d2','d3','d4','e1','e2','e3','e4','lamda1','lamda2','lamda3','lamda4','lamda5','lamda6','lamda7','lamda8','lamda9','lamda10') a1=r.a1 a2=r.a2 a3=r.a3 a4=r.a4 b0=r.b0 b1=r.b1 b2=r.b2 b3=r.b3 b4=r.b4 c2=r.c2 c3=r.c3 c4=r.c4 d0=r.d0 d1=r.d1 d2=r.d2 d3=r.d3 d4=r.d4 e1=r.e1 e2=r.e2 e3=r.e3 e4=r.e4 lamda1=r.lamda1 lamda2=r.lamda2 lamda3=r.lamda3 lamda4=r.lamda4 lamda5=r.lamda5 lamda6=r.lamda6 lamda7=r.lamda7 lamda8=r.lamda8 lamda9=r.lamda9 lamda10=r.lamda10 x=solve(lamda10,'omega') 3.2. T×m tÇn sè dao ®éng riªng tõ d¹ng dao ®éng riªng, VÝ dô 9 : Tr­êng hîp (a): syms a1 a2 a3 a4; syms b0 b1 b2 b3 b4; syms c0 c1 c2 c3 c4; syms z t l ej m f1 f2; %f1 f2 la luc quan tinh syms lamda1 lamda2 lamda3 lamda4 lamda5 lamda6 lamda7; syms omega; y1=(a1*z+a2*z^2+a3*z^3+a4*z^4)*sin(omega*t);% duong do vong cua doan 1 y2=(b0+b1*z+b2*z^2+b3*z^3+b4*z^4)*sin(omega*t);% duong do vong cua doan 2 y3=(c0+c1*z+c2*z^2+c3*z^3+c4*z^4)*sin(omega*t);% duong do vong cua doan 3 n1=diff(y1,'z',2)%dao ham bac hai cua duong do vong doan 1 n2=diff(y2,'z',2)%dao ham bac hai cua duong do vong doan 2 n3=diff(y3,'z',2)%dao ham bac hai cua duong do vong doan 3 t1=ej*int(n1^2,'z',0,l/3) t2=ej*int(n2^2,'z',0,l/3) t3=ej*int(n3^2,'z',0,l/3) u1=diff(y1,'z');%goc xoay cua doan 1 u2=diff(y2,'z');%goc xoay cua doan 2 u3=diff(y3,'z');%goc xoay cua doan 3 t4=2*f1*subs(y1,'z',l/3);%luong cuong buc do luc quan tinh t5=2*f2*subs(y2,'z',l/3);%luong cuong buc do luc quan tinh t6=lamda1*subs(y3,'z',l/3);%dieu kien bien t7=lamda2*(subs(y1,'z',l/3)-subs(y2,'z',0));%dieu kien lien tuc cua doan 1 va 2 t8=lamda3*(subs(u1,'z',l/3)-subs(u2,'z',0));%dieu kien lien tuc cua doan 1 va 2 t9=lamda4*(subs(y2,'z',l/3)-subs(y3,'z',0));%dieu kien lien tuc cua doan 2 va 3 t10=lamda5*(subs(u2,'z',l/3)-subs(u3,'z',0));%dieu kien lien tuc cua doan 2 va 3 t11=lamda6*(subs(y1,'z',l/3)-1); t12=lamda7*(subs(y2,'z',l/3)-1); lcb=t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+t11+t12; h1=diff(lcb,'a1'); h2=diff(lcb,'a2'); h3=diff(lcb,'a3'); h4=diff(lcb,'a4'); h5=diff(lcb,'b0'); h6=diff(lcb,'b1'); h7=diff(lcb,'b2'); h8=diff(lcb,'b3'); h9=diff(lcb,'b4'); h10=diff(lcb,'c0'); h11=diff(lcb,'c1'); h12=diff(lcb,'c2'); h13=diff(lcb,'c3'); h14=diff(lcb,'c4'); h15=diff(lcb,'lamda1'); h16=diff(lcb,'lamda2'); h17=diff(lcb,'lamda3'); h18=diff(lcb,'lamda4'); h19=diff(lcb,'lamda5'); h20=diff(lcb,'lamda6'); h21=diff(lcb,'lamda7'); k1=-m*omega^2*subs(y1,'z',l/3);%luc quan tinh k2=-m*omega^2*subs(y2,'z',l/3);%luc quan tinh g1=subs(h1,f1,k1); g2=subs(h2,f1,k1); g3=subs(h3,f1,k1); g4=subs(h4,f1,k1); g5=subs(h5,f2,k2); g6=subs(h6,f2,k2); g7=subs(h7,f2,k2); g8=subs(h8,f2,k2); g9=subs(h9,f2,k2); r=solve(g1,g2,g3,g4,g5,g6,g7,g8,g9,h10,h11,h12,h13,h14,h15,h16,h17,h18,h19,h20,h21,'a1','a2','a3','a4','b0','b1','b2','b3','b4','c0','c1','c2','c3','c4','lamda1','lamda2','lamda3','lamda4','lamda5','lamda6','lamda7') a1=r.a1 a2=r.a2 a3=r.a3 a4=r.a4 b0=r.b0 b1=r.b1 b2=r.b2 b3=r.b3 b4=r.b4 c0=r.c0 c1=r.c1 c2=r.c2 c3=r.c3 c4=r.c4 lamda1=r.lamda1 lamda2=r.lamda2 lamda3=r.lamda3 lamda4=r.lamda4 lamda5=r.lamda5 lamda6=r.lamda6 lamda7=r.lamda7 x=solve(lamda7,omega) Tr­êng hîp (b): syms a1 a2 a3 a4; syms b0 b1 b2 b3 b4; syms c0 c1 c2 c3 c4; syms z t l ej m f1 f2; %f1 f2 la luc quan tinh syms lamda1 lamda2 lamda3 lamda4 lamda5 lamda6 lamda7; syms omega; y1=(a1*z+a2*z^2+a3*z^3+a4*z^4)*sin(omega*t);% duong do vong cua doan 1 y2=(b0+b1*z+b2*z^2+b3*z^3+b4*z^4)*sin(omega*t);% duong do vong cua doan 2 y3=(c0+c1*z+c2*z^2+c3*z^3+c4*z^4)*sin(omega*t);% duong do vong cua doan 3 n1=diff(y1,'z',2)%dao ham bac hai cua duong do vong doan 1 n2=diff(y2,'z',2)%dao ham bac hai cua duong do vong doan 2 n3=diff(y3,'z',2)%dao ham bac hai cua duong do vong doan 3 t1=ej*int(n1^2,'z',0,l/3) t2=ej*int(n2^2,'z',0,l/3) t3=ej*int(n3^2,'z',0,l/3) u1=diff(y1,'z');%goc xoay cua doan 1 u2=diff(y2,'z');%goc xoay cua doan 2 u3=diff(y3,'z');%goc xoay cua doan 3 t4=2*f1*subs(y1,'z',l/3);%luong cuong buc do luc quan tinh t5=2*f2*subs(y2,'z',l/3);%luong cuong buc do luc quan tinh t6=lamda1*subs(y3,'z',l/3);%dieu kien bien t7=lamda2*(subs(y1,'z',l/3)-subs(y2,'z',0));%dieu kien lien tuc cua doan 1 va 2 t8=lamda3*(subs(u1,'z',l/3)-subs(u2,'z',0));%dieu kien lien tuc cua doan 1 va 2 t9=lamda4*(subs(y2,'z',l/3)-subs(y3,'z',0));%dieu kien lien tuc cua doan 2 va 3 t10=lamda5*(subs(u2,'z',l/3)-subs(u3,'z',0));%dieu kien lien tuc cua doan 2 va 3 t11=lamda6*(subs(y1,'z',l/3)-1); t12=lamda7*(subs(y2,'z',l/3)+1); lcb=t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+t11+t12; h1=diff(lcb,'a1'); h2=diff(lcb,'a2'); h3=diff(lcb,'a3'); h4=diff(lcb,'a4'); h5=diff(lcb,'b0'); h6=diff(lcb,'b1'); h7=diff(lcb,'b2'); h8=diff(lcb,'b3'); h9=diff(lcb,'b4'); h10=diff(lcb,'c0'); h11=diff(lcb,'c1'); h12=diff(lcb,'c2'); h13=diff(lcb,'c3'); h14=diff(lcb,'c4'); h15=diff(lcb,'lamda1'); h16=diff(lcb,'lamda2'); h17=diff(lcb,'lamda3'); h18=diff(lcb,'lamda4'); h19=diff(lcb,'lamda5'); h20=diff(lcb,'lamda6'); h21=diff(lcb,'lamda7'); k1=-m*omega^2*subs(y1,'z',l/3);%luc quan tinh k2=-m*omega^2*subs(y2,'z',l/3);%luc quan tinh g1=subs(h1,f1,k1); g2=subs(h2,f1,k1); g3=subs(h3,f1,k1); g4=subs(h4,f1,k1); g5=subs(h5,f2,k2); g6=subs(h6,f2,k2); g7=subs(h7,f2,k2); g8=subs(h8,f2,k2); g9=subs(h9,f2,k2); r=solve(g1,g2,g3,g4,g5,g6,g7,g8,g9,h10,h11,h12,h13,h14,h15,h16,h17,h18,h19,h20,h21,'a1','a2','a3','a4','b0','b1','b2','b3','b4','c0','c1','c2','c3','c4','lamda1','lamda2','lamda3','lamda4','lamda5','lamda6','lamda7') a1=r.a1 a2=r.a2 a3=r.a3 a4=r.a4 b0=r.b0 b1=r.b1 b2=r.b2 b3=r.b3 b4=r.b4 c0=r.c0 c1=r.c1 c2=r.c2 c3=r.c3 c4=r.c4 lamda1=r.lamda1 lamda2=r.lamda2 lamda3=r.lamda3 lamda4=r.lamda4 lamda5=r.lamda5 lamda6=r.lamda6 lamda7=r.lamda7 x=solve(lamda7,omega) VÝ dô 10 : *Bµi to¸n tÜnh: x¸c ®Þnh tØ sè cña c¸c chuyÓn vÞ th«ng qua d¹ng dao ®éng. - Tr­êng hîp (a), Tr­êng hîp (c): syms a1 a2 a3 a4; syms b0 b1 b2 b3 b4; syms c0 c1 c2 c3 c4; syms d1 d2 d3 d4; syms z t l ej m; syms lamda1 lamda2 lamda3 lamda4 lamda5 lamda6; syms omega; y1=(a1*z+a2*z^2+a3*z^3+a4*z^4); y2=(b0+b1*z+b2*z^2+b3*z^3+b4*z^4) y3=(c0+c1*z+c2*z^2+c3*z^3+c4*z^4); y4=(d1*z+d2*z^2+d3*z^3+d4*z^4); u1=diff(y1,'z'); u2=diff(y2,'z'); u3=diff(y3,'z'); u4=diff(y4,'z'); n1=diff(y1,'z',2); n2=diff(y2,'z',2); n3=diff(y3,'z',2); n4=diff(y4,'z',2); t1=ej*int(n1^2,'z',0,l/4); t2=ej*int(n2^2,'z',0,l/4); t3=ej*int(n3^2,'z',0,l/4); t4=ej*int(n4^2,'z',0,l/4); t5=lamda1*(subs(y1,'z',l/4)-subs(y2,'z',0));%dieu kien bien cua doan 1 va 2 t6=lamda2*(subs(u1,'z',l/4)-subs(u2,'z',0)); t7=lamda3*(subs(y2,'z',l/4)-subs(y3,'z',0));%dieu kien bien cua doan 2 va 3 t8=lamda4*(subs(u2,'z',l/4)-subs(u3,'z',0)); t9=lamda5*(subs(y3,'z',l/4)-subs(y4,'z',l/4));%dieu kien bien cua doan 3 va 4 t10=lamda6*(subs(u3,'z',l/4)-subs((-u4),'z',l/4)); lcb=t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+2*subs(y1,'z',l/4)+2*subs(y2,'z',l/4)+ +2*subs(y3,'z',l/4); h1=diff(lcb,'a1'); h2=diff(lcb,'a2'); h3=diff(lcb,'a3'); h4=diff(lcb,'a4'); h5=diff(lcb,'b0'); h6=diff(lcb,'b1'); h7=diff(lcb,'b2'); h8=diff(lcb,'b3'); h9=diff(lcb,'b4'); h10=diff(lcb,'c0'); h11=diff(lcb,'c1'); h12=diff(lcb,'c2'); h13=diff(lcb,'c3'); h14=diff(lcb,'c4'); h15=diff(lcb,'d1'); h16=diff(lcb,'d2'); h17=diff(lcb,'d3'); h18=diff(lcb,'d4'); h19=diff(lcb,'lamda1'); h20=diff(lcb,'lamda2'); h21=diff(lcb,'lamda3'); h22=diff(lcb,'lamda4'); h23=diff(lcb,'lamda5'); h24=diff(lcb,'lamda6'); r=solve(h1,h2,h3,h4,h5,h6,h7,h8,h9,h10,h11,h12,h13,h14,h15,h16,h17,h18,h19,h20,h21,h22,h23,h24,'a1','a2','a3','a4','b0','b1','b2','b3','b4','c0','c1','c2','c3','c4','d1','d2','d3','d4','lamda1','lamda2','lamda3','lamda4','lamda5','lamda6') a1=r.a1 a2=r.a2 a3=r.a3 a4=r.a4 b0=r.b0 b1=r.b1 b2=r.b2 b3=r.b3 b4=r.b4 c0=r.c0 c1=r.c1 c2=r.c2 c3=r.c3 c4=r.c4 d1=r.d1 d2=r.d2 d3=r.d3 d4=r.d4 lamda1=r.lamda1 lamda2=r.lamda2 lamda3=r.lamda3 lamda4=r.lamda4 lamda5=r.lamda5 lamda6=r.lamda6 *Bµi to¸n ®éng: x¸c ®Þnh tÇn sè dao ®éng riªng. -Tr­êng hîp (a): syms a1 a2 a3 a4; syms b0 b1 b2 b3 b4; syms c0 c1 c2 c3 c4; syms d1 d2 d3 d4; syms z t l ej m f1 f2 f3; %f1 f2 f3 la luc quan tinh syms lamda1 lamda2 lamda3 lamda4 lamda5 lamda6 lamda7 lamda8 lamda9; syms omega; y1=(a1*z+a2*z^2+a3*z^3+a4*z^4)*sin(omega*t);% duong do vong cua doan 1 y2=(b0+b1*z+b2*z^2+b3*z^3+b4*z^4)*sin(omega*t);% duong do vong cua doan 2 y3=(c0+c1*z+c2*z^2+c3*z^3+c4*z^4)*sin(omega*t);% duong do vong cua doan 3 y4=(d1*z+d2*z^2+d3*z^3+d4*z^4)*sin(omega*t);% duong do vong cua doan 4 u1=diff(y1,'z');%goc xoay cua doan 1 u2=diff(y2,'z');%goc xoay cua doan 2 u3=diff(y3,'z');%goc xoay cua doan 3 u4=diff(y4,'z');%goc xoay cua doan 4 n1=diff(y1,'z',2);%dao ham bac hai cua duong do vong doan 1 n2=diff(y2,'z',2);%dao ham bac hai cua duong do vong doan 2 n3=diff(y3,'z',2);%dao ham bac hai cua duong do vong doan 3 n4=diff(y4,'z',2);%dao ham bac hai cua duong do vong doan 4 t1=ej*int(n1^2,'z',0,l/4); t2=ej*int(n2^2,'z',0,l/4); t3=ej*int(n3^2,'z',0,l/4); t4=ej*int(n4^2,'z',0,l/4); t5=2*f1*subs(y1,'z',l/4);%luong cuong buc do luc quan tinh t6=2*f2*subs(y2,'z',l/4);%luong cuong buc do luc quan tinh t7=2*f3*subs(y3,'z',l/4);%luong cuong buc do luc quan tinh t8=lamda1*(subs(y1,'z',l/4)-subs(y2,'z',0));%dieu kien bien cua doan 1 va 2 t9=lamda2*(subs(u1,'z',l/4)-subs(u2,'z',0)); t10=lamda3*(subs(y2,'z',l/4)-subs(y3,'z',0));%dieu kien bien cua doan 2 va 3 t11=lamda4*(subs(u2,'z',l/4)-subs(u3,'z',0)); t12=lamda5*(subs(y3,'z',l/4)-subs(y4,'z',l/4));%dieu kien bien cua doan 3 va 4 t13=lamda6*(subs(u3,'z',l/4)-subs((-u4),'z',l/4)); t14=lamda7*(subs(y1,'z',l/4)-1); t15=lamda8*(subs(y2,'z',l/4)-1.41); t16=lamda9*(subs(y3,'z',l/4)-1); lcb=t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+t11+t12+t13+t14+t15+t16; h1=diff(lcb,'a1'); h2=diff(lcb,'a2'); h3=diff(lcb,'a3'); h4=diff(lcb,'a4'); h5=diff(lcb,'b0'); h6=diff(lcb,'b1'); h7=diff(lcb,'b2'); h8=diff(lcb,'b3'); h9=diff(lcb,'b4'); h10=diff(lcb,'c0'); h11=diff(lcb,'c1'); h12=diff(lcb,'c2'); h13=diff(lcb,'c3'); h14=diff(lcb,'c4'); h15=diff(lcb,'d1'); h16=diff(lcb,'d2'); h17=diff(lcb,'d3'); h18=diff(lcb,'d4'); h19=diff(lcb,'lamda1'); h20=diff(lcb,'lamda2'); h21=diff(lcb,'lamda3'); h22=diff(lcb,'lamda4'); h23=diff(lcb,'lamda5'); h24=diff(lcb,'lamda6'); h25=diff(lcb,'lamda7'); h26=diff(lcb,'lamda8'); h27=diff(lcb,'lamda9'); k1=-m*omega^2*subs(y1,'z',l/4);%luc quan tinh k2=-m*omega^2*subs(y2,'z',l/4);%luc quan tinh k3=-m*omega^2*subs(y3,'z',l/4);%luc quan tinh g1=subs(h1,f1,k1); g2=subs(h2,f1,k1); g3=subs(h3,f1,k1); g4=subs(h4,f1,k1); g5=subs(h5,f2,k2); g6=subs(h6,f2,k2); g7=subs(h7,f2,k2); g8=subs(h8,f2,k2); g9=subs(h9,f2,k2); g10=subs(h10,f3,k3); g11=subs(h11,f3,k3); g12=subs(h12,f3,k3); g13=subs(h13,f3,k3); g14=subs(h14,f3,k3); r=solve(g1,g2,g3,g4,g5,g6,g7,g8,g9,g10,g11,g12,g13,g14,h15,h16,h17,h18,h19,h20,h21,h22,h23,h24,h25,h26,h27,'a1','a2','a3','a4','b0','b1','b2','b3','b4','c0','c1','c2','c3','c4','d1','d2','d3','d4','lamda1','lamda2','lamda3','lamda4','lamda5','lamda6','lamda7','lamda8','lamda9') a1=r.a1 a2=r.a2 a3=r.a3 a4=r.a4 b0=r.b0 b1=r.b1 b2=r.b2 b3=r.b3 b4=r.b4 c0=r.c0 c1=r.c1 c2=r.c2 c3=r.c3 c4=r.c4 d1=r.d1 d2=r.d2 d3=r.d3 d4=r.d4 lamda1=r.lamda1 lamda2=r.lamda2 lamda3=r.lamda3 lamda4=r.lamda4 lamda5=r.lamda5 lamda6=r.lamda6 lamda7=r.lamda7 lamda8=r.lamda8 lamda9=r.lamda9 x=solve(lamda7,'omega') -Tr­êng hîp (b): syms a1 a2 a3 a4; syms b0 b1 b2 b3 b4; syms c0 c1 c2 c3 c4; syms d1 d2 d3 d4; syms z t l ej m f1 f2 f3; %f1 f2 f3 la luc quan tinh syms lamda1 lamda2 lamda3 lamda4 lamda5 lamda6 lamda7 lamda8 lamda9; syms omega; y1=(a1*z+a2*z^2+a3*z^3+a4*z^4)*sin(omega*t);% duong do vong cua doan 1 y2=(b0+b1*z+b2*z^2+b3*z^3+b4*z^4)*sin(omega*t);% duong do vong cua doan 2 y3=(c0+c1*z+c2*z^2+c3*z^3+c4*z^4)*sin(omega*t);% duong do vong cua doan 3 y4=(d1*z+d2*z^2+d3*z^3+d4*z^4)*sin(omega*t);% duong do vong cua doan 4 u1=diff(y1,'z');%goc xoay cua doan 1 u2=diff(y2,'z');%goc xoay cua doan 2 u3=diff(y3,'z');%goc xoay cua doan 3 u4=diff(y4,'z');%goc xoay cua doan 4 n1=diff(y1,'z',2);%dao ham bac hai cua duong do vong doan 1 n2=diff(y2,'z',2);%dao ham bac hai cua duong do vong doan 2 n3=diff(y3,'z',2);%dao ham bac hai cua duong do vong doan 3 n4=diff(y4,'z',2);%dao ham bac hai cua duong do vong doan 4 t1=ej*int(n1^2,'z',0,l/4); t2=ej*int(n2^2,'z',0,l/4); t3=ej*int(n3^2,'z',0,l/4); t4=ej*int(n4^2,'z',0,l/4); t5=2*f1*subs(y1,'z',l/4);%luong cuong buc do luc quan tinh t6=2*f2*subs(y2,'z',l/4);%luong cuong buc do luc quan tinh t7=2*f3*subs(y3,'z',l/4);%luong cuong buc do luc quan tinh t8=lamda1*(subs(y1,'z',l/4)-subs(y2,'z',0));%dieu kien bien cua doan 1 va 2 t9=lamda2*(subs(u1,'z',l/4)-subs(u2,'z',0)); t10=lamda3*(subs(y2,'z',l/4)-subs(y3,'z',0));%dieu kien bien cua doan 2 va 3 t11=lamda4*(subs(u2,'z',l/4)-subs(u3,'z',0)); t12=lamda5*(subs(y3,'z',l/4)-subs(y4,'z',l/4));%dieu kien bien cua doan 3 va 4 t13=lamda6*(subs(u3,'z',l/4)-subs((-u4),'z',l/4)); t14=lamda7*(subs(y1,'z',l/4)-1); t15=lamda8*subs(y2,'z',l/4); t16=lamda9*(subs(y3,'z',l/4)+1); lcb=t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+t11+t12+t13+t14+t15+t16; h1=diff(lcb,'a1'); h2=diff(lcb,'a2'); h3=diff(lcb,'a3'); h4=diff(lcb,'a4'); h5=diff(lcb,'b0'); h6=diff(lcb,'b1'); h7=diff(lcb,'b2'); h8=diff(lcb,'b3'); h9=diff(lcb,'b4'); h10=diff(lcb,'c0'); h11=diff(lcb,'c1'); h12=diff(lcb,'c2'); h13=diff(lcb,'c3'); h14=diff(lcb,'c4'); h15=diff(lcb,'d1'); h16=diff(lcb,'d2'); h17=diff(lcb,'d3'); h18=diff(lcb,'d4'); h19=diff(lcb,'lamda1'); h20=diff(lcb,'lamda2'); h21=diff(lcb,'lamda3'); h22=diff(lcb,'lamda4'); h23=diff(lcb,'lamda5'); h24=diff(lcb,'lamda6'); h25=diff(lcb,'lamda7'); h26=diff(lcb,'lamda8'); h27=diff(lcb,'lamda9'); k1=-m*omega^2*subs(y1,'z',l/4);%luc quan tinh k2=-m*omega^2*subs(y2,'z',l/4);%luc quan tinh k3=-m*omega^2*subs(y3,'z',l/4);%luc quan tinh g1=subs(h1,f1,k1); g2=subs(h2,f1,k1); g3=subs(h3,f1,k1); g4=subs(h4,f1,k1); g5=subs(h5,f2,k2); g6=subs(h6,f2,k2); g7=subs(h7,f2,k2); g8=subs(h8,f2,k2); g9=subs(h9,f2,k2); g10=subs(h10,f3,k3); g11=subs(h11,f3,k3); g12=subs(h12,f3,k3); g13=subs(h13,f3,k3); g14=subs(h14,f3,k3); r=solve(g1,g2,g3,g4,g5,g6,g7,g8,g9,g10,g11,g12,g13,g14,h15,h16,h17,h18,h19,h20,h21,h22,h23,h24,h25,h26,h27,'a1','a2','a3','a4','b0','b1','b2','b3','b4','c0','c1','c2','c3','c4','d1','d2','d3','d4','lamda1','lamda2','lamda3','lamda4','lamda5','lamda6','lamda7','lamda8','lamda9') a1=r.a1 a2=r.a2 a3=r.a3 a4=r.a4 b0=r.b0 b1=r.b1 b2=r.b2 b3=r.b3 b4=r.b4 c0=r.c0 c1=r.c1 c2=r.c2 c3=r.c3 c4=r.c4 d1=r.d1 d2=r.d2 d3=r.d3 d4=r.d4 lamda1=r.lamda1 lamda2=r.lamda2 lamda3=r.lamda3 lamda4=r.lamda4 lamda5=r.lamda5 lamda6=r.lamda6 lamda7=r.lamda7 lamda8=r.lamda8 lamda9=r.lamda9 x=solve(lamda7,'omega') -Tr­êng hîp (c): syms a1 a2 a3 a4; syms b0 b1 b2 b3 b4; syms c0 c1 c2 c3 c4; syms d1 d2 d3 d4; syms z t l ej m f1 f2 f3; %f1 f2 f3 la luc quan tinh syms lamda1 lamda2 lamda3 lamda4 lamda5 lamda6 lamda7 lamda8 lamda9; syms omega; y1=(a1*z+a2*z^2+a3*z^3+a4*z^4)*sin(omega*t);% duong do vong cua doan 1 y2=(b0+b1*z+b2*z^2+b3*z^3+b4*z^4)*sin(omega*t);% duong do vong cua doan 2 y3=(c0+c1*z+c2*z^2+c3*z^3+c4*z^4)*sin(omega*t);% duong do vong cua doan 3 y4=(d1*z+d2*z^2+d3*z^3+d4*z^4)*sin(omega*t);% duong do vong cua doan 4 u1=diff(y1,'z');%goc xoay cua doan 1 u2=diff(y2,'z');%goc xoay cua doan 2 u3=diff(y3,'z');%goc xoay cua doan 3 u4=diff(y4,'z');%goc xoay cua doan 4 n1=diff(y1,'z',2);%dao ham bac hai cua duong do vong doan 1 n2=diff(y2,'z',2);%dao ham bac hai cua duong do vong doan 2 n3=diff(y3,'z',2);%dao ham bac hai cua duong do vong doan 3 n4=diff(y4,'z',2);%dao ham bac hai cua duong do vong doan 4 t1=ej*int(n1^2,'z',0,l/4); t2=ej*int(n2^2,'z',0,l/4); t3=ej*int(n3^2,'z',0,l/4); t4=ej*int(n4^2,'z',0,l/4); t5=2*f1*subs(y1,'z',l/4);%luong cuong buc do luc quan tinh t6=2*f2*subs(y2,'z',l/4);%luong cuong buc do luc quan tinh t7=2*f3*subs(y3,'z',l/4);%luong cuong buc do luc quan tinh t8=lamda1*(subs(y1,'z',l/4)-subs(y2,'z',0));%dieu kien bien cua doan 1 va 2 t9=lamda2*(subs(u1,'z',l/4)-subs(u2,'z',0)); t10=lamda3*(subs(y2,'z',l/4)-subs(y3,'z',0));%dieu kien bien cua doan 2 va 3 t11=lamda4*(subs(u2,'z',l/4)-subs(u3,'z',0)); t12=lamda5*(subs(y3,'z',l/4)-subs(y4,'z',l/4));%dieu kien bien cua doan 3 va 4 t13=lamda6*(subs(u3,'z',l/4)-subs((-u4),'z',l/4)); t14=lamda7*(subs(y1,'z',l/4)-1); t15=lamda8*(subs(y2,'z',l/4)+1.41); t16=lamda9*(subs(y3,'z',l/4)-1); lcb=t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+t11+t12+t13+t14+t15+t16; h1=diff(lcb,'a1'); h2=diff(lcb,'a2'); h3=diff(lcb,'a3'); h4=diff(lcb,'a4'); h5=diff(lcb,'b0'); h6=diff(lcb,'b1'); h7=diff(lcb,'b2'); h8=diff(lcb,'b3'); h9=diff(lcb,'b4'); h10=diff(lcb,'c0'); h11=diff(lcb,'c1'); h12=diff(lcb,'c2'); h13=diff(lcb,'c3'); h14=diff(lcb,'c4'); h15=diff(lcb,'d1'); h16=diff(lcb,'d2'); h17=diff(lcb,'d3'); h18=diff(lcb,'d4'); h19=diff(lcb,'lamda1'); h20=diff(lcb,'lamda2'); h21=diff(lcb,'lamda3'); h22=diff(lcb,'lamda4'); h23=diff(lcb,'lamda5'); h24=diff(lcb,'lamda6'); h25=diff(lcb,'lamda7'); h26=diff(lcb,'lamda8'); h27=diff(lcb,'lamda9'); k1=-m*omega^2*subs(y1,'z',l/4);%luc quan tinh k2=-m*omega^2*subs(y2,'z',l/4);%luc quan tinh k3=-m*omega^2*subs(y3,'z',l/4);%luc quan tinh g1=subs(h1,f1,k1); g2=subs(h2,f1,k1); g3=subs(h3,f1,k1); g4=subs(h4,f1,k1); g5=subs(h5,f2,k2); g6=subs(h6,f2,k2); g7=subs(h7,f2,k2); g8=subs(h8,f2,k2); g9=subs(h9,f2,k2); g10=subs(h10,f3,k3); g11=subs(h11,f3,k3); g12=subs(h12,f3,k3); g13=subs(h13,f3,k3); g14=subs(h14,f3,k3); r=solve(g1,g2,g3,g4,g5,g6,g7,g8,g9,g10,g11,g12,g13,g14,h15,h16,h17,h18,h19,h20,h21,h22,h23,h24,h25,h26,h27,'a1','a2','a3','a4','b0','b1','b2','b3','b4','c0','c1','c2','c3','c4','d1','d2','d3','d4','lamda1','lamda2','lamda3','lamda4','lamda5','lamda6','lamda7','lamda8','lamda9') a1=r.a1 a2=r.a2 a3=r.a3 a4=r.a4 b0=r.b0 b1=r.b1 b2=r.b2 b3=r.b3 b4=r.b4 c0=r.c0 c1=r.c1 c2=r.c2 c3=r.c3 c4=r.c4 d1=r.d1 d2=r.d2 d3=r.d3 d4=r.d4 lamda1=r.lamda1 lamda2=r.lamda2 lamda3=r.lamda3 lamda4=r.lamda4 lamda5=r.lamda5 lamda6=r.lamda6 lamda7=r.lamda7 lamda8=r.lamda8 lamda9=r.lamda9 x=solve(lamda7,'omega') x1=solve(lamda8,'omega') x=solve(lamda9,'omega') VÝ dô 11 : syms a1 a2 a3 a4; syms b0 b1 b2 b3 b4; syms c0 c1 c2 c3 c4; syms z t l ej m f1 f2; %f1 f2 la luc quan tinh syms lamda1 lamda2 lamda3 lamda4 lamda5 lamda6; syms omega; y1=(a1*z+a2*z^2+a3*z^3+a4*z^4)*sin(omega*t);% duong do vong cua doan 1 y2=(b0+b1*z+b2*z^2+b3*z^3+b4*z^4)*sin(omega*t);% duong do vong cua doan 2 y3=(c0+c1*z+c2*z^2+c3*z^3+c4*z^4)*sin(omega*t);% duong do vong cua doan 3 n1=diff(y1,'z',2)%dao ham bac hai cua duong do vong doan 1 n2=diff(y2,'z',2)%dao ham bac hai cua duong do vong doan 2 n3=diff(y3,'z',2)%dao ham bac hai cua duong do vong doan 3 t1=ej*int(n1^2,'z',0,l/4) t2=ej*int(n2^2,'z',0,l/2) t3=ej*int(n3^2,'z',0,l/4) u1=diff(y1,'z');%goc xoay cua doan 1 u2=diff(y2,'z');%goc xoay cua doan 2 u3=diff(y3,'z');%goc xoay cua doan 3 t4=2*f1*subs(y1,'z',l/4)%luong cuong buc do luc quan tinh t5=2*f2*subs(y2,'z',l/2)%luong cuong buc do luc quan tinh t6=lamda1*subs(y3,'z',l/4)%dieu kien bien t7=lamda2*(subs(y1,'z',l/4)-subs(y2,'z',0))%dieu kien lien tuc cua doan 1 va 2 t8=lamda3*(subs(u1,'z',l/4)-subs(u2,'z',0))%dieu kien lien tuc cua doan 1 va 2 t9=lamda4*(subs(y2,'z',l/2)-subs(y3,'z',0))%dieu kien lien tuc cua doan 2 va 3 t10=lamda5*(subs(u2,'z',l/2)-subs(u3,'z',0))%dieu kien lien tuc cua doan 2 va 3 t11=lamda6*(subs(y1,'z',l/4)-1)%dk co nghiem lcb=t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+t11 h1=diff(lcb,'a1') h2=diff(lcb,'a2') h3=diff(lcb,'a3') h4=diff(lcb,'a4') h5=diff(lcb,'b0') h6=diff(lcb,'b1') h7=diff(lcb,'b2') h8=diff(lcb,'b3') h9=diff(lcb,'b4') h10=diff(lcb,'c0') h11=diff(lcb,'c1') h12=diff(lcb,'c2') h13=diff(lcb,'c3') h14=diff(lcb,'c4') h15=diff(lcb,'lamda1') h16=diff(lcb,'lamda2') h17=diff(lcb,'lamda3') h18=diff(lcb,'lamda4') h19=diff(lcb,'lamda5') h20=diff(lcb,'lamda6') k1=-m*omega^2*subs(y1,'z',l/4)%luc quan tinh k2=-2*m*omega^2*subs(y2,'z',l/2)%luc quan tinh g1=subs(h1,f1,k1) g2=subs(h2,f1,k1) g3=subs(h3,f1,k1) g4=subs(h4,f1,k1) g5=subs(h5,f2,k2) g6=subs(h6,f2,k2) g7=subs(h7,f2,k2) g8=subs(h8,f2,k2) g9=subs(h9,f2,k2) r=solve(g1,g2,g3,g4,g5,g6,g7,g8,g9,h10,h11,h12,h13,h14,h15,h16,h17,h18,h19,h20,'a1','a2','a3','a4','b0','b1','b2','b3','b4','c0','c1','c2','c3','c4','lamda1','lamda2','lamda3','lamda4','lamda5','lamda6') a1=r.a1 a2=r.a2 a3=r.a3 a4=r.a4 b0=r.b0 b1=r.b1 b2=r.b2 b3=r.b3 b4=r.b4 c0=r.c0 c1=r.c1 c2=r.c2 c3=r.c3 c4=r.c4 lamda1=r.lamda1 lamda2=r.lamda2 lamda3=r.lamda3 lamda4=r.lamda4 lamda5=r.lamda5 lamda6=r.lamda6 x=solve(lamda6,'omega')