Sec03_SDOF.單自由度系統(tǒng)

1、Copyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.SEC 3單自由度系統(tǒng)的動(dòng)力學(xué)性能單自由度系統(tǒng)的動(dòng)力學(xué)性能Copyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.2l 為什么單自由度系統(tǒng)如此重要？l 再來(lái)回顧一下結(jié)構(gòu)動(dòng)力學(xué)的最重要的一個(gè)概念之一：模態(tài)變換為什么為什么SDOF重要？重要？Copyright 2013 Altair Enginee

2、ring, Inc. Proprietary and Confidential. All rights reserved.3模態(tài)變換模態(tài)變換Modal Approach用模態(tài)來(lái)縮聚質(zhì)量陣、阻尼陣和剛度陣Using Normal Modes to reduce the M,C and K matrices:( )( )( )( )nnnnnnnnnnMxtCxtKxtft模態(tài)縮聚Modal Reduction:變換到模態(tài)坐標(biāo)下方程， 稱為模態(tài)坐標(biāo)Eq in Modal CS():(nn矩陣方程)(mm矩陣方程)( )( )( )( )mmmmmmmmmmMtCtKtft()( )( )nnmmm

3、nxtt Copyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.4解耦方程和半解耦方程解耦方程和半解耦方程u模態(tài)坐標(biāo)下的動(dòng)力學(xué)方程為半解耦方程Semi-uncoupled Eq:( )( )( )( )MtCtKtf t11( )2tttt如直接法般用有限差分法直接計(jì)算u如果用模態(tài)阻尼或者比例阻尼，動(dòng)力學(xué)方程為m個(gè)解耦方程Uncoupled Eq:( )( )( )( ), 1jjjjjjjmtctktf tjm用Duhamel積分求解Copyright 2013 Al

4、tair Engineering, Inc. Proprietary and Confidential. All rights reserved.51.減少自由度！2.用單自由度系統(tǒng)來(lái)代替原結(jié)構(gòu)！模態(tài)變換的本質(zhì)模態(tài)變換的本質(zhì)模態(tài)縮減(t)( )nnmmxt 1,jm由nn階矩陣變換為m個(gè)單自由度方程Copyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.6l 用m個(gè)單自由度(SDOF)系統(tǒng)來(lái)抽象、代替復(fù)雜的原始物理結(jié)構(gòu)l 因此研究SDOF的動(dòng)力學(xué)性能非常重要模態(tài)變換的本質(zhì)

5、模態(tài)變換的本質(zhì)123m-1m具有n個(gè)自由度只需要m個(gè)單自由度彈簧振子(t)( )nnmmxt Copyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.7l 動(dòng)力學(xué)方程l 通常令載荷為正弦周期變化l Why？動(dòng)力學(xué)方程動(dòng)力學(xué)方程( )mxcxkxf t( )sinf tPtCopyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.8l 滿足一定條件的任意

6、周期信號(hào)，可以用無(wú)窮多離散頻率的簡(jiǎn)諧信號(hào)來(lái)表示l 非周期信號(hào)，則用傅里葉積分變換傅里葉級(jí)數(shù)傅里葉級(jí)數(shù)-連接力學(xué)模型與工程的橋梁連接力學(xué)模型與工程的橋梁Fourier series01( )(coscos)nintnnnnnnnf teaantbnt任意載荷可以分解為：定常載荷+諧波載荷定常載荷動(dòng)力學(xué)分析不考慮，通過(guò)靜力學(xué)分析求解諧波載荷是動(dòng)力學(xué)分析的重點(diǎn)Copyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.9任意的信號(hào)(時(shí)域)=傅里葉級(jí)數(shù)傅里葉級(jí)數(shù)-連接力學(xué)模型與工程的

7、橋梁連接力學(xué)模型與工程的橋梁Fourier series確定性的三角周期信號(hào)(時(shí)域)000111222333444,0, ,.,mmma fca fa fa fa fafCopyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.10l 強(qiáng)迫振動(dòng)SDOF的解的解10( )( )( )sinmx tcx tkx tPt2222222122/( )( sincos)sin()(1/)(2/),122(0)/,(0)(1/)2/tan(1/)ntddnndncnnnnnnP kx

8、teAtBttccccmkmuP kABu 通解u1特解u2Copyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.11l 通解的幅值隨時(shí)間迅速呈指數(shù)衰減，衰減周期固定通解通解u1，瞬時(shí)解，瞬時(shí)解11-0.100-0.075-0.050-0.0250.0000.0250.0500.0750.1000.001.002.003.004.005.00DisplacementTime2% Damp5% Damp1( )( sincos)ntddu teAtBt指數(shù)型衰減nteCo

9、pyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.12特解特解u2，穩(wěn)態(tài)解，穩(wěn)態(tài)解12222 22/()sin()(1/)(2/)nnP ku tt 靜力載荷 = 0時(shí)的解隨著 n 振幅逐漸增加 當(dāng) = n 時(shí)，該項(xiàng)等于4z2 , 控制振幅相對(duì)于輸入的相位角響應(yīng)22221(1/)(2/)nnDMFQ 定義振幅放大因子(Dynamic Magnification Factor)：QQ是動(dòng)力學(xué)幅值和靜力學(xué)幅值之間的比值。Copyright 2013 Altair Engin

10、eering, Inc. Proprietary and Confidential. All rights reserved.13振幅放大因子隨頻率比的曲線振幅放大因子隨頻率比的曲線13振幅放大因子振幅放大因子Q=Copyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.14振幅放大因子Q=特解，穩(wěn)態(tài)解特解，穩(wěn)態(tài)解A.當(dāng) 放大因子 1 (穩(wěn)定解) 相位角 00 (響應(yīng)與外力同相)B.當(dāng) 放大因子 (共振) 相位角 900C. 當(dāng) 放大因子 0 (無(wú)響應(yīng)) 相位角 1800

11、(響應(yīng)與外力反相）nnn22221(1/)(2/)nnQ 12ABC2( )sin()Pu tQtkCopyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.15振幅放大因子，振幅放大因子，A類類準(zhǔn)準(zhǔn)靜態(tài)解靜態(tài)解靜力學(xué)和動(dòng)力學(xué)的誤差范圍靜力學(xué)和動(dòng)力學(xué)的誤差范圍15frequency放大因子Q固有頻率 n最高激勵(lì)頻率 HLHL 1/31/5nHLHL/ /n nCritical Critical damping(damping( ) )阻尼影響00.010.030.050.1

12、0.2%0.20 0.20 1.041667 1.041667 1.041658 1.041585 1.041585 1.041441 1.040764 1.038068 -0.35%0.25 1.066667 1.066651 1.066530 1.066530 1.066288 1.065153 1.060650 -0.56%0.33 0.33 1.125000 1.125000 1.124968 1.124715 1.124715 1.124210 1.121849 1.112553 -1.11%0.50 1.333333 1.333215 1.332268 1.332268 1.330

13、380 1.321637 1.288313 -3.38%0.60 1.562500 1.562225 1.560034 1.560034 1.555678 1.535738 1.463014 -6.37%0.70 1.960784 1.960046 1.954169 1.954169 1.942572 1.890836 1.718781 -12.34%-12.34%振幅放大因子在1/5時(shí)，準(zhǔn)靜態(tài)和動(dòng)態(tài)分析的相對(duì)誤差為4.2%振幅放大因子在1/3時(shí)，準(zhǔn)靜態(tài)和動(dòng)態(tài)分析的相對(duì)誤差為12.5%同時(shí)發(fā)現(xiàn)，振幅放大因子在這一范圍內(nèi)對(duì)阻尼也不敏感。0,80,91,01,11,21,31,40,000,100

14、,200,300,400,50/nAmplitude Anx0.02x0.2x1.0Copyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.16振幅放大因子，振幅放大因子，A類準(zhǔn)靜態(tài)解類準(zhǔn)靜態(tài)解靜力學(xué)的重新定義：靜力學(xué)的重新定義：靜力時(shí)序響應(yīng)分析靜力時(shí)序響應(yīng)分析16線性靜力線性靜力Linear Static理想的靜態(tài)載荷 時(shí)序響應(yīng)時(shí)序響應(yīng)Static Response analysis當(dāng)載荷頻率比結(jié)構(gòu)基頻小得多時(shí)的近似Load-1FEMLoad-2時(shí)序結(jié)果是多體動(dòng)力學(xué)和非

15、線性靜力學(xué)常見(jiàn)的結(jié)果Copyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.17振幅放大因子，振幅放大因子，A類準(zhǔn)靜態(tài)解類準(zhǔn)靜態(tài)解動(dòng)力學(xué)動(dòng)力學(xué)精度精度模態(tài)縮減(t)( )nnmmxt 1,jm由nn階矩陣變換為m個(gè)單自由度方程模態(tài)縮聚Modal Reduction1求解Solving2( )mt物理數(shù)據(jù)恢復(fù)Physical Data Recovery3Copyright 2013 Altair Engineering, Inc. Proprietary and Confi

16、dential. All rights reserved.18l 這里 n = n階保留主模態(tài) c = 不保留的截?cái)嗄B(tài)l c 通常是 的一個(gè)大子集 l 模態(tài)解的品質(zhì)依賴于線性組合 c表示實(shí)際的作用載荷導(dǎo)致的實(shí)際解的程度. l 也就是，模態(tài)截?cái)鄷?huì)導(dǎo)致少量的結(jié)果誤差。模態(tài)截?cái)嗟哪B(tài)截?cái)嗟母拍罡拍?( )nnccx t Copyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.19l 從前面的分析可以知道，當(dāng)固有頻率大于激勵(lì)的最高頻率的5倍以上時(shí)，動(dòng)力學(xué)響應(yīng)的結(jié)果和靜力學(xué)計(jì)算的

17、結(jié)果誤差不超過(guò)5%。l 因此模態(tài)截?cái)嗟恼`差可以通過(guò)靜力學(xué)計(jì)算結(jié)果再補(bǔ)充回來(lái)，這就是殘余矢量法。l 殘余矢量法是動(dòng)力學(xué)的默認(rèn)方法，即在Optistruct中RESVEC是默認(rèn)使用的。l 殘余矢量法RESVEC可以有效提高求解精度。振幅放大因子，振幅放大因子，A類類準(zhǔn)準(zhǔn)靜態(tài)解靜態(tài)解動(dòng)力學(xué)精度準(zhǔn)則動(dòng)力學(xué)精度準(zhǔn)則-殘余矢量法殘余矢量法RESVECfrequency放大因子Q固有頻率 n最高激勵(lì)頻率 HLHL 35 HL最高激勵(lì)頻率HL模態(tài)截?cái)囝l率HLTruncate frequency模態(tài)數(shù)據(jù)恢復(fù)方法：殘余矢量法殘余矢量法(default)模態(tài)截?cái)嗟念l率標(biāo)準(zhǔn)：模態(tài)截?cái)囝l率是最高激勵(lì)頻率的3-5倍。在使

18、用殘余矢量法的情況下，模態(tài)截?cái)囝l率可以放寬到最高激勵(lì)頻率的1.5-2倍。Copyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.21質(zhì)量M剛度K正弦位移，幅值 X阻尼CtimeA=1正弦力，幅值F，頻率1.0123450frequency傳遞函數(shù)正弦力振幅放大因子，振幅放大因子，B類共振解類共振解共振時(shí)的最大峰值的頻域表現(xiàn)共振時(shí)的最大峰值的頻域表現(xiàn)在0.1Hz到5.0Hz的頻響區(qū)域上施加單位載荷阻尼是臨界阻尼的1%.12PkCopyright 2013 Altair En

19、gineering, Inc. Proprietary and Confidential. All rights reserved.22振幅放大因子，振幅放大因子，B類共振解類共振解共振時(shí)的最大峰值的時(shí)域表現(xiàn)共振時(shí)的最大峰值的時(shí)域表現(xiàn)DisplacementTime放大因子Q = 1/2z = 1/G = 50靜態(tài)響應(yīng) = P/k = 0.01注: 可以使用log函數(shù)幫助定義低階響應(yīng) (1/2zz (-1/2z z )穩(wěn)態(tài)振幅Q P/kCopyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights

20、 reserved.23l 實(shí)際模型非常復(fù)雜，不可能通過(guò)理論求解得到。l 一般需要疊加幾百階幾千階的模態(tài)+殘余矢量法才能得到精確的響應(yīng)結(jié)果。l 峰值可以有有限元計(jì)算得到。l 接下來(lái)討論瞬時(shí)沖擊情況下什么時(shí)候系統(tǒng)達(dá)到峰值？振幅放大因子，振幅放大因子，B類共振解類共振解達(dá)到最大達(dá)到最大峰值峰值的時(shí)間的時(shí)間Copyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.24l 先考察一個(gè)單自由度彈簧振子。質(zhì)量固定為1kg。l 瞬時(shí)沖擊激勵(lì)可以采用半正弦波、三角波、方波等。這里采用最經(jīng)典

21、的半正弦波，使用如下周期為1秒的半正弦波載荷去激勵(lì)質(zhì)量塊。l 通過(guò)調(diào)整剛度K可以得到任意固有頻率，查看不同固有頻率下的響應(yīng)情況。單自由度系統(tǒng)模型單自由度系統(tǒng)模型M=1kgK 00.20.40.60.8100.511.522.53Copyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.250.05Hz/0.1HzCopyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights r

22、eserved.260.25Hz/0.5HzCopyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.271Hz/2HzCopyright 2013 Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.28l 在各類設(shè)計(jì)規(guī)范中，常用的半正弦時(shí)長(zhǎng)為20ms，即“等效頻率”為50Hz。l 從圖表可以看出，結(jié)構(gòu)基頻小于載荷“等效頻率”的1/2時(shí)，峰值將在激勵(lì)時(shí)間外得到。單自由度系統(tǒng)單自由度系統(tǒng)模型結(jié)果模型結(jié)果00.20.40.60.8100.010.020.030.040.050.06等效頻率(hz)scale結(jié)構(gòu)基頻(hz)放大因子Q時(shí)間倍數(shù)500.052.50.205.47500.150.002.98500.2512.50.481.74500.5251.551.00501501.710.675021001.260.40時(shí)間倍數(shù)=峰值時(shí)間/載荷