说明讲稿振动

1、无阻尼自由振动k + k2-kém1ù ìüéù ìüì ü0xx01+121=ú íx ýú íx ýí0ýê 0ê-këmû î 2 þëkû î 2 þî þ222ì x1 (t) ü = Asin (wt + f ) = ì A1 üsin (wt +

2、 f )íx(t)ýíA ýîþî 2 þ2广义特征值问题K A = w 2 M AKF= wMF自振频率与振型2iiinX (t) = åai Fi sin(wit + fi )i =1自由振动振动力学1内容回顾自振频率与振型的计算振动力学2m1 x1 = -k1 x1 + k2 (x2 - x1 )m2 x2 = k2 (x1- x2 ) + k3 (x3- x2 )m3 x3 = k3 (x2- x3 ) - k4 x3振动力学3m1 x1 + (k1 + k2 )x1 - k2 x2 = 0m2

3、x2 - k2 x1m3 x3 - k3 x2+ (k2+ (k3+ k3 )x2+ k4 )x3- k3 x3= 0= 0振动力学4m1 x1 = -k1 x1 + k2 (x2 - x1 )m2 x2 = k2 (x1 - x2 ) + k3 (x3 - x2 )m3 x3 = k3 (x2 - x3 ) - k4 x3ìm1 x1 + (k1 + k2 )x1 - k2 x2= 0ïm x - k x + (k+ k )x - k x = 0í222123233ïm x - k x + (k+ k )x = 0îém133323

4、430 ù ì x1 (t) üék1 + k2-k20ù ì x1 (t) üì0ü00 ú ïx (t)ï + êú ïx(t)ï = ïïê 0-k0k + k-kmú íýú íýí0ýêêë 0êëê2222-k3332m3 úû 

5、8;ïx3 (t)þïk3 + k4 ûú îïx3 (t)þïîï0þï0M X (t)+ K X (t) = 0ék1 + k2-k2ù0ém10 ùì x1 (t) ü0ú X (t) = ïx (t)ïK = êM = ê 00 ú-k0k + k-kmíýêêë 0úm3 

6、50;ûêêëú222332ïîx3 (t)ïþk + k úû0-k334振动力学5ì A1 üX (t) = Asin(wt + f) = ï Aïwt + f)íýsin(2ïî A3 ïþX (t) = -w2 Asin(wt + f)-w2 M Asin(wt + f) +K Asin(wt + f) = 0K A = w 2 M A振动力学6M X (t)+ K X (t)

7、= 0Jacobi方法QL（QR）方法逆迭代法（幂法、反幂法）Sturm序列 子空间迭代法Lanczos法等振动力学7K A = w 2 M AGeneralized Eigenproblem(K - w 2 M )A = 0K - w 2 M = 0ék1 + k2-k2ém10 ùùú00M = ê 00 úK = ê-k0k + k-kmêêë 0úm3 úûêêëú222-k3033k3 + k4 ú

8、;û0k + k - m w 2-k1212k + k - m w 2-k0-k= 022323+ k - m w 2-kk3343振动力学8行列式展开法k + k - m w 2-k01212k + k - m w 2-k0-k= 022323- m w 2-kk + k3343k1 = k2= k3= k4= 1= m3= 1= 2m1m22 - w 2-1 0-12 - 2w 2-10-1 2 - w 2= 0振动力学92 - w 2-1 0-12 - 2w 2-10-1 2 - w 2= 0f (w2 ) = w6- 5w4+ 7w2= (w2- 2)(w4- 3w2- 2+

9、1) = 05 -1 = 0.618= 1 +5 = 1.618w= 1.414w =w21322振动力学10w2= 3 -5 = 0.382w 2 = 2.0w2= 3 +5 = 2.61812232(K - w 2 M )A = 0é2 - w2êù ì A ü-12 - 2w2-10-1 2 - w1 ïú ï-10í A ý = 0êú2ïïêú2Aû î3 þëì(2 - w

10、2 ) A - A = 0ìïï1 +512ïA1 - A2= 0w ) A - A- A + (2 - 2= 02í2123ï- A + (2 - w 2 ) A = 0îï23í- A1ï+ (-1 +5) A2 - A3= 0ï- A+ 1 +5 A5 -1= 0ïw = 0.618232î12振动力学11ì1 +5A1 - A2= 0ïï2ïí- A1ï+ (-1 +5) A2 - A3 = 0&#

11、239;- A+ 1 +5 A= 0ï232î= 1 +5 = 1.618A = 1.0A = 1.0A1322ì A1 üì1.0üïïïïí A2 ý = í1.618ýF = 1.01.0T1.618ïî A ïþïî1.0ïþ13振动力学12ì(2 - w 2 ) A - A = 01+ (2 - 22ïw ) A - A- A= 02í1

12、2= 03ï- A+ (2 - w 2 ) Aî23F2 = 1.0-1.0T0.0w= 1.41421 +F = 1.01.0T5-0.618w3 = 1.61832振动力学13K A = w2 M AM -1M -1 K A = w 2 AS = M -1 K S A(i)= w 2 A(i+1)振动力学14幂法S = M -1 K S A(i)= w 2 A(i+1)ék1 + k2-k2ém10 ùùú00M = ê 00 úK = ê-k0k + k-kmêê

13、35; 0úm3 úûêêë= m3ú222-k333k3 + k4 úû0k1 = k2= k3= k4 = 1= 1m2 = 2m1-1 2-1é10ùé 20 ù020M = ê00úK = ê-1-1úêêë0ú1úûêêë 0ú2 úû振动力学15S = M -1 K S A(i)= w 2 A(i

14、+1)-1 2-1é10ùé 20 ù020M = ê00úK = ê-1-1úêêë0é1ú1úû00.50êêë 0ú2 úû0ù= ê00úM -1êêë0ú1úû00.50-1 2-1-1 1-1é10ù é 20 ùéù20S =

15、 M -1 K = ê00ú ê-1-1ú = ê-0.5-0.5úêêë0ú ê1úû êë 0ú2 úûêêëúúû02振动力学16-1 1-1éù20S = ê-0.5-0.5úS A(i)= w 2 A(i+1)êêëúúû02ì1ü

16、;= ï ï( 0)í1ýïî1ïþA-1 1-1éù ì1üì1ü20ï ïï ï= ê-0.5úS A( 0) (1)-=0=A0.5ú í1ýí ýêêëûú îï1þïîï1þï02振动力学17-1 1-1

17、33;ù20S = ê-0.5-0.5úS A(i)= w 2 A(i+1)êêëú2úû0ì1ü= ï0ïA(1)í ýïî1ïþ-1 1-1éù ì1üì 2 üìü201-0.5ú ï0ï = ï-ï = 2 ï-0.5ï= ê-0.5S A(

18、1)ú í ýí1ýíýêêëûú îï1þïîï 2 þïîïþï021ìü1= ï-0.5ïA( 2)íýïîïþ1振动力学18ìüìüìüìü11112.5 ï-0

19、.6ï ® 2.6 ï-0.615ï ® 2.615 ï-0.618ï ® 2.618 ï-0.618ïíýíýíýíýïîïþïîïþïîïþïîïþ1111w 2= 2.6183F3 = 11T-0.618振动力学19K A = w2 M AK A = 1

20、 M Al = 1 w 2lM A = l K AK -1 M A = l AK -1T = K -1 M T A(i )= l A(i +1)振动力学20反幂法T A(i )é 2T = K -1 M = l A(i +1)-1 2-1é10ù0 ù020M = ê00úK = ê-1-1úêêë0ú1úû0.51.00.5êêë 0ú2 úûé0.750.25ù= 

21、4; 0.50.5 úK -1êêë0.25ú0.75úû0.51.00.5é0.750.25ù é10ùé0.750.25ù0201.02.01.0T = K -1 M = ê 0.50.5 ú ê00ú = ê 0.50.5 úêêë0.25ú ê0.75úû êë0ú1úûê

22、;êë0.25ú0.75úû振动力学21ì1üT A(i )é0.750.25ù= l A(i +1)1.02.01.0= ï ïT = ê 0.50.5 úA( 0)í1ýïî1ïþêêë0.25ú0.75úûé0.750.25ù ì1üì2üì 1 ü1.02.01

23、.00.5 ú ï ï = ï3ï =ïï= ê 0.5T A( 0)ú í1ýí ý2 í1.5ýîï 1 þïêêë0.250.75ûú îï1þïîï2þïì 1 üìüìüìü111ï

24、;ïïïïïïï2.5 í1.6ý ® 2.6 í1.615ý ® 2.615 í1.618ý ® 2.618 í1.618ýïî 1 ïþïîïþïîïþïîïþ111振动力学22ì 1 üìüìü

25、ìü111ïïïïïïïï2.5 í1.6ý ® 2.6 í1.615ý ® 2.615 í1.618ý ® 2.618 í1.618ýïî 1 ïþïîïþïîïþïîïþ111l = 2.618= 1 = 0.382w 21l

26、F = 11T1.6181振动力学23k1 = k2= k3 = k4 = 1m1 = m3 = 1m2 = 2ì1üx = ïr ïí ýïî1ïþ对称振型振动力学24综合法ì1üx = ïr ïí ýïî1ïþK A = w 2 M A-1 2-1é 20 ù ì1üé10ù ì1ü020ï ï

27、0ú ïr ïê-1úê0= w-2ú í ýú í ý1rêêë 0êëê02 úû îï1þï1ûú îï1þï振动力学25-1 2-1é 20 ù ì1üé10ù ì1ü020ï ï0ú

28、; ïr ïê-1úê0= w-2ú í ýú í ý1rêêë 0êëê02 úû îï1þï1ûú îï1þï2 - r = w 22r - 2 = 2w 2 r- r -1 = 0r2= 1 += 1 -5 = 1.6185 = -0.618rrab22= 3 -= 3 +5 = 0.3825 = 2.

29、618w2w2ab22振动力学26ì 1 üx = ï 0 ïíý称振型ïî-1ïþK A = w 2 M A-1 2-1é 20 ù ì 1 üé10ù ì 1 ü020ïï0ú ï 0 ïê-1úê0w-=2ú íýú íý10êêë 0

30、4;êë02 úû îï-1þï1ûú îï-1þïw 2= 2c振动力学27= 3 -= 3 +5 = 0.3825 = 2.618w 2= 2w2w2cab22= 3 -= 3 +5 = 0.3825 = 2.618w 2= 2w2w221322ìüì 1 üìü11= ïïïïïï1F2 = í 0 ýï

31、;î-1ïþF3 = í-0.618ýFí1.618ý1ïîïþïîïþ1振动力学28K A = w 2 M AK -1= F A = w 2 F M A振动力学29均质简支梁长为L，不计自重在距二支点各L/6处及中点处分别有集中质量m 梁截面的抗弯刚度为EI各阶自振频率及振型振动力学30算例25L3L3d= d33d 22=113888EI48EI13L317L3d= d21= d23 = d32d13= d31 =121296EI3888EI

32、é25ê17 ùúé1ê0ùú398139010L3M = m ê0êë0=F3939ú25úû0ú1úûê3888EIêë17振动力学31A = w 2 F M Aé25ê17 ùú398139w2 mL339úA25úûA=39ê3888EIêë17mw 2 L33888EI ll =w =mL33888EIé2517 ù398139A = l ê3939úAêê