華洪良 廖振強(qiáng) 張相炎

(南京理工大學(xué)機(jī)械工程學(xué)院，南京210094)

軸向移動(dòng)懸臂梁高效動(dòng)力學(xué)建模及頻率響應(yīng)分析1)

華洪良 廖振強(qiáng)2)張相炎

(南京理工大學(xué)機(jī)械工程學(xué)院，南京210094)

軸向移動(dòng)梁動(dòng)力學(xué)問題具有廣泛的工程應(yīng)用背景，如：機(jī)械手、機(jī)床主軸、武器身管等.計(jì)算軸向移動(dòng)梁動(dòng)力學(xué)響應(yīng)是評(píng)估結(jié)構(gòu)動(dòng)力學(xué)性能以及最終指導(dǎo)結(jié)構(gòu)設(shè)計(jì)的一個(gè)重要手段.采用Rayleigh-Ritz法、拉格朗日方程推導(dǎo)了軸向移動(dòng)懸臂梁時(shí)變動(dòng)力學(xué)方程.選取冪級(jí)數(shù)函數(shù)構(gòu)造試函數(shù)對(duì)軸向移動(dòng)系統(tǒng)動(dòng)力問題進(jìn)行求解.冪級(jí)數(shù)函數(shù)良好的積分與微分性能，使得推導(dǎo)容易以矩陣的形式快速進(jìn)行，便于符號(hào)運(yùn)算軟件直接生成MATLAB程序.由于MATLAB基本數(shù)據(jù)單位為矩陣，符號(hào)軟件生成的程序只需經(jīng)過簡單修改便可進(jìn)行動(dòng)力學(xué)計(jì)算.大大縮短了軸向移動(dòng)梁從建模到動(dòng)力學(xué)分析的時(shí)間，過程十分高效.通過四組算例，將本文方法計(jì)算得到的動(dòng)力學(xué)響應(yīng)與文獻(xiàn)數(shù)據(jù)進(jìn)行對(duì)比，對(duì)該方法準(zhǔn)確性進(jìn)行了驗(yàn)證，并給出了冪級(jí)數(shù)函數(shù)擬合階數(shù)的選取原則.以此為基礎(chǔ)，研究了軸向移動(dòng)梁的頻率響應(yīng)特性.分為考慮重力與忽略重力兩種情況，討論了軸向振動(dòng)幅度對(duì)其頻率響應(yīng)特性的影響.

計(jì)算動(dòng)力學(xué)，冪級(jí)數(shù)基函數(shù)，頻響分析,軸向移動(dòng)梁

引言

軸向移動(dòng)動(dòng)力系統(tǒng)具有廣泛的工程應(yīng)用背景，常見的有：機(jī)械手、機(jī)床主軸、武器身管[1-3]、傳送帶[4-6]、衛(wèi)星結(jié)構(gòu)等.近年來，軸向移動(dòng)系統(tǒng)動(dòng)力學(xué)是一研究熱點(diǎn).由于動(dòng)力響應(yīng)在評(píng)估結(jié)構(gòu)動(dòng)力學(xué)性能以及最終結(jié)構(gòu)設(shè)計(jì)中不可或缺，因此，一種快速、準(zhǔn)確的建模與響應(yīng)分析方法在軸向移動(dòng)系統(tǒng)動(dòng)力學(xué)研究過程中至關(guān)重要.

在軸向移動(dòng)系統(tǒng)動(dòng)力學(xué)研究中，通常將軸向移動(dòng)系統(tǒng)簡化為軸向移動(dòng)弦或軸向移動(dòng)梁.根據(jù)已有文獻(xiàn)，目前軸向移動(dòng)系統(tǒng)動(dòng)力學(xué)研究主要集中于如下幾類問題：軸向移動(dòng)彈性梁與旋轉(zhuǎn)剛性支撐的耦合動(dòng)力學(xué)[7-9]，軸向移動(dòng)黏彈性梁 (或弦)[10-14]、外流體軸向移動(dòng)梁[19-22]、含有摩擦邊界的軸向移動(dòng)梁[23-24]、軸向移動(dòng)梁的參數(shù)振動(dòng)[25-26]、軸向移動(dòng)梁橫向振動(dòng)抑制技術(shù)研究[27-28]、軸向與橫向耦合振動(dòng)研究[29-31]等.

在軸向移動(dòng)系統(tǒng)動(dòng)力學(xué)建模過程中，主要有兩種方法.第一種為分析力學(xué)方法[7-10,29-30,32-35]，并且應(yīng)用最為廣泛的分析力學(xué)方法為第二類拉格朗日方程.在該方法中，首先采用能量法對(duì)系統(tǒng)進(jìn)行能量分析，得到相應(yīng)的拉格朗日函數(shù)，經(jīng)離散后采用拉格朗日方程推導(dǎo)得到其運(yùn)動(dòng)方程.第二種為彈性力學(xué)方法[12,14-24,26-28,31,34,37].采用該方法時(shí)，首先對(duì)研究對(duì)象取分離體進(jìn)行受力分析，得到其控制方程，經(jīng)離散后直接得到其運(yùn)動(dòng)方程.

在離散過程中，兩種主要的離散技術(shù)為有限單元法[8-9,32-33,38]與假設(shè)模態(tài)法[34-37].在假設(shè)模態(tài)法中，需要一系列滿足所有(或者部分)邊界條件的基函數(shù)來構(gòu)造一個(gè)試函數(shù)對(duì)原函數(shù)進(jìn)行逼近.在軸向移動(dòng)梁控制方程離散過程中，通常選取其振型函數(shù)來構(gòu)造試函數(shù).梁的振型函數(shù)由一系列三角函數(shù)與雙曲函數(shù)構(gòu)成，形式復(fù)雜，其復(fù)雜程度取決于其具體邊界條件.在常見的邊界條件中，只有簡支邊界梁的振型函數(shù)相對(duì)簡單，而固支邊界、自由邊界、懸臂邊界等梁的振型函數(shù)較復(fù)雜.在建模過程中，通常需要采用振型函數(shù)之間的正交性對(duì)方程進(jìn)行解耦.然而，由于梁的軸向運(yùn)動(dòng)，在其動(dòng)力學(xué)建模過程中會(huì)產(chǎn)生諸多不同于定常結(jié)構(gòu)的陀螺項(xiàng)，利用目前已知的模態(tài)函數(shù)正交性條件尚無法使這些陀螺項(xiàng)完全解耦[39]，因此軸向移動(dòng)梁動(dòng)力學(xué)建模尚無統(tǒng)一方法.

基于上述因素，本文嘗試采用冪級(jí)數(shù)作為基函數(shù)構(gòu)造試函數(shù)對(duì)軸向移動(dòng)系統(tǒng)動(dòng)力問題進(jìn)行求解.由于冪級(jí)數(shù)基函數(shù)具有較好的積分與微分性能，更重要的是對(duì)任何兩個(gè)基函數(shù)之間的正交性并不敏感.這一特點(diǎn)，使得積分演算變得非常迅速，并且對(duì)于不同問題均很容易以矩陣的形式進(jìn)行推導(dǎo)，推導(dǎo)過程更為簡潔.采用矩陣形式的最主要優(yōu)勢在于：能夠采用符號(hào)運(yùn)算軟件推導(dǎo)并直接生成MATLAB程序.由于MATLAB基本數(shù)據(jù)單位為矩陣，符號(hào)軟件生成的程序只需經(jīng)過簡單修改便可進(jìn)行動(dòng)力學(xué)計(jì)算.這樣大大縮短了軸向移動(dòng)梁從建模到動(dòng)力學(xué)分析的時(shí)間，提高了效率.

本文主要工作：

(1)基于歐拉梁理論，首先推導(dǎo)了軸向移動(dòng)梁動(dòng)能與勢能函數(shù)，考慮時(shí)變軸向移動(dòng)速度.隨后，采用Rayleigh-Ritz法與冪級(jí)數(shù)基函數(shù)對(duì)系統(tǒng)動(dòng)能與勢能函數(shù)進(jìn)行離散，并通過拉格朗日方程推導(dǎo)得到系統(tǒng)運(yùn)動(dòng)方程.

(2)將本文計(jì)算得到的動(dòng)力學(xué)響應(yīng)與相關(guān)文獻(xiàn)數(shù)據(jù)進(jìn)行比較，對(duì)本文方法與模型的準(zhǔn)確性進(jìn)行驗(yàn)證.并給出冪級(jí)數(shù)函數(shù)擬合階數(shù)的選取原則.

(3)研究軸向往復(fù)運(yùn)動(dòng)頻率對(duì)橫向振動(dòng)的影響，即頻率響應(yīng)分析(身管武器應(yīng)用背景).分為考慮重力與忽略重力兩種情況進(jìn)行討論.

1 動(dòng)力學(xué)建模

考慮一軸向移動(dòng)懸臂梁模型，如圖1.梁總長LB，懸臂長L(t).楊氏模量、慣性矩、密度、截面積分別為E,I,ρ,A.梁橫向位移采用w(x(t),t)進(jìn)行描述.

梁上任意一點(diǎn)發(fā)生變形后，其位移矢量為

圖1 軸向移動(dòng)懸臂梁Fig.1 Axially moving cantilever beam

將式(1)對(duì)時(shí)間求導(dǎo)得到速度矢量

式中，點(diǎn)和撇分別表示對(duì)時(shí)間、空間求偏導(dǎo).根據(jù)式(2)，系統(tǒng)動(dòng)能為

梁彎曲變形產(chǎn)生的應(yīng)變能為

考慮梁軸向運(yùn)動(dòng)存在加速度，其軸向慣性力產(chǎn)生的拉伸應(yīng)變能為

根據(jù)式(3)～式(5)，得到系統(tǒng)拉格朗日函數(shù)

2 離散與運(yùn)動(dòng)方程

首先構(gòu)造試函數(shù)

對(duì)梁的橫向位移進(jìn)行離散.式中

在式(8)中

為滿足懸臂梁邊界條件的冪級(jí)數(shù)函數(shù).式中，n1為冪級(jí)數(shù)函數(shù)階數(shù)；αji為冪級(jí)數(shù)函數(shù)系數(shù)，確定αji的一個(gè)簡易方法就是對(duì)懸臂梁振型函數(shù)進(jìn)行擬合.在擬合前，首先將懸臂梁振型函數(shù)

映射至 1-1空間.式中，cosλi+chλi/2shλisinλi為一系列幅值映射系數(shù)；λi為特征方程cosλichλi+1=0的根.經(jīng)過該映射后，梁長度與自由端橫向位移均被無量綱化為1.將梁長度映射為1后，有利于后續(xù)縮放為梁實(shí)際懸臂長度進(jìn)行動(dòng)力學(xué)計(jì)算.將梁自由端橫向位移映射為1后，結(jié)果后處理將變得十分簡單，梁自由端真實(shí)的橫向位移即為各基函數(shù)坐標(biāo)的和.

式(10)中冪級(jí)數(shù)系數(shù)可根據(jù)式(11)擬合得到.對(duì)于自適應(yīng)任意懸臂長度的基函數(shù)可以將式(10)沿X軸縮放得到

將式(7)代入式(6)，并將結(jié)果代入拉格朗日方程，把重力作為廣義力考慮，得到系統(tǒng)運(yùn)動(dòng)方程

值得一提的是，式(13)中C為梁彎曲變形與軸向運(yùn)動(dòng)產(chǎn)生的陀螺項(xiàng)，而非結(jié)構(gòu)阻尼.并且，Vx=與Ax=分別為梁軸向運(yùn)動(dòng)速度與加速度.Vx=V0+Axt，V0為初始速度.由于式(17)為一系列冪級(jí)數(shù)函數(shù)乘積的積分運(yùn)算，很容易采用符號(hào)運(yùn)算軟件進(jìn)行推導(dǎo)得到其代數(shù)格式，并自動(dòng)生成MATLAB程序.

3 模型驗(yàn)證

本節(jié)采用4個(gè)算例對(duì)本文方法準(zhǔn)確性進(jìn)行驗(yàn)證.前兩個(gè)算例源于文獻(xiàn)[36]，在該算例中，梁作軸向勻速運(yùn)動(dòng).后兩個(gè)算例源于文獻(xiàn)[29]，算例中梁作軸向勻加速運(yùn)動(dòng).4個(gè)算例中，梁截面積、楊氏模型、慣性矩、密度均一致，分別為1.4661×10?3m2,6.8335N/m2,1.1073×10?8m4,2738.6kg/m3.其余參數(shù)設(shè)置見表1.

計(jì)算得到的梁自由端動(dòng)力學(xué)響應(yīng)與文獻(xiàn)[36]和文獻(xiàn)[29]對(duì)比見圖2和圖3.本文計(jì)算結(jié)果與文獻(xiàn)結(jié)果吻合較好，說明本文采取的方法以及建立的模型準(zhǔn)確可靠.

圖2 勻速收縮、伸展運(yùn)動(dòng)Fig.2 The retracting and extruding motion with constant velocity

圖3 勻加速收縮、伸展運(yùn)動(dòng)Fig.3 The retracting and extruding motion with constant acceleration

表1 驗(yàn)證算例物理參數(shù)Table 1 Physical parameters of the veri fi cation examples

4 冪級(jí)數(shù)基函數(shù)擬合階數(shù)的影響

第2節(jié)中，冪級(jí)數(shù)函數(shù)擬合階數(shù)n1的大小將直接決定冪級(jí)數(shù)基函數(shù)的擬合精度，并最終影響到動(dòng)力學(xué)計(jì)算精度.本節(jié)主要討論冪級(jí)數(shù)擬合階數(shù)對(duì)動(dòng)力學(xué)計(jì)算精度的影響.

根據(jù)式(11)，采用MATLAB軟件編程，在計(jì)算域內(nèi)對(duì)各振型函數(shù)均勻采樣，本文采用500個(gè)樣本點(diǎn).然后根據(jù)得到的樣本點(diǎn)以及給定的擬合階數(shù)，采用poly fi t函數(shù)便可擬合得到冪級(jí)數(shù)函數(shù)系數(shù)αji.為了給出擬合階數(shù)的選取原則，采用冪級(jí)數(shù)函數(shù)的最大相對(duì)擬合誤差作為量化指標(biāo)

需要說明的是，筆者在研究過程中發(fā)現(xiàn)在多數(shù)情況下，細(xì)長懸臂梁結(jié)構(gòu)動(dòng)力響應(yīng)主要取決于其一階模態(tài).因此，本節(jié)只討論第一階模態(tài)擬合階數(shù)對(duì)動(dòng)力學(xué)計(jì)算精度的影響.通過計(jì)算，在表2中給出了擬合階數(shù)為1至10范圍內(nèi)的最大相對(duì)擬合誤差.對(duì)比發(fā)現(xiàn)，隨著擬合階數(shù)的增大，冪級(jí)數(shù)函數(shù)的最大相對(duì)擬合誤差逐漸減小.

表2 最大相對(duì)擬合誤差Table 2 The maximum relative fi tting error

圖4 冪級(jí)數(shù)函數(shù)擬合階數(shù)對(duì)動(dòng)力學(xué)計(jì)算精度的影響Fig.4 The e ff ect of fi tting order of the power series function on computational accuracy

隨后，以第3節(jié)算例2為例，分別采用不同擬合階數(shù)的冪級(jí)數(shù)函數(shù)進(jìn)行動(dòng)力學(xué)計(jì)算，得到梁自由端位移響應(yīng)曲線對(duì)比如圖4.以10階擬合精度的冪級(jí)數(shù)函數(shù)計(jì)算得到的動(dòng)力學(xué)響應(yīng)曲線為參考進(jìn)行對(duì)比研究.圖4(a)表明，當(dāng)擬合階數(shù)在1至4階范圍內(nèi)時(shí)，動(dòng)力學(xué)計(jì)算產(chǎn)生了較大的誤差.尤其是當(dāng)擬合階數(shù)為1時(shí)，計(jì)算得到的響應(yīng)曲線根本無法反應(yīng)梁的真實(shí)振動(dòng)情況.這是因?yàn)椴捎靡浑A冪級(jí)數(shù)函數(shù)時(shí)，式(17)中N00對(duì)空間X求導(dǎo)兩次后為零，相當(dāng)于結(jié)構(gòu)剛度消失.當(dāng)擬合階數(shù)增加至5階及其以上后，響應(yīng)曲線與10階響應(yīng)曲線基本吻合，差異很小.因此，為了保證動(dòng)力學(xué)計(jì)算精度，應(yīng)采用至少5階冪級(jí)數(shù)函數(shù)進(jìn)行擬合.由表2可知，5階冪級(jí)數(shù)函數(shù)對(duì)應(yīng)的最大相對(duì)擬合誤差為0.0034%.因此，筆者建議在選取冪級(jí)數(shù)函數(shù)擬合階數(shù)時(shí)，應(yīng)將其最大相對(duì)擬合誤差控制在千分之一以內(nèi).

5 頻響分析

本節(jié)討論軸向振動(dòng)頻率對(duì)梁橫向振動(dòng)的影響.其工程應(yīng)用背景為武器浮動(dòng)身管.很多身管武器在設(shè)計(jì)過程中，為了減小后坐力，經(jīng)常將其身管設(shè)計(jì)為浮動(dòng)形式，使得身管在發(fā)射過程中能夠沿著槍(炮)身做前后相對(duì)運(yùn)動(dòng)，吸收一部分后坐動(dòng)能.武器連續(xù)發(fā)射過程中，身管不停地做軸向往復(fù)運(yùn)動(dòng)，使得膛口橫向振動(dòng)受到擾動(dòng).由于身管的射擊精度直接于彈頭出膛口瞬時(shí)膛口的振動(dòng)狀態(tài).因此，研究此類振動(dòng)系統(tǒng)的頻響特性對(duì)匹配身管武器系統(tǒng)動(dòng)力學(xué)特性、避免結(jié)構(gòu)共振具有重要意義.

在研究頻響特性時(shí)，使軸向移動(dòng)梁做軸向簡諧運(yùn)動(dòng)，其懸臂長度、軸向速度、加速度由下式給出

式中，L0表示梁平均懸臂長度；Amp和ω分別表示簡諧運(yùn)動(dòng)幅值與頻率.算例所用參數(shù)設(shè)置如表3.

表3 算例參數(shù)Table 3 Example parameters

5.1 考慮重力影響

圖5給出了不同軸向振動(dòng)幅值下的梁自由端頻率響應(yīng)特性曲線.從圖5可以發(fā)現(xiàn)，軸向振動(dòng)幅值越大，幅頻響應(yīng)曲線中特殊頻率點(diǎn)越多.當(dāng)軸向運(yùn)動(dòng)幅值為10mm時(shí)(圖5(a))，幅頻響應(yīng)曲線中只有c和d兩個(gè)特殊頻率點(diǎn).當(dāng)軸向運(yùn)動(dòng)幅值增加至100mm時(shí)，幅頻響應(yīng)曲線中共出現(xiàn)了5個(gè)特殊頻率響應(yīng)點(diǎn)，如圖5(d)，各頻率值見表4.

圖5 軸向移動(dòng)梁頻率響應(yīng)特性(考慮重力)Fig.5 Frequency response characteristic of the axially moving beam(considering gravity)

表4 各頻率響應(yīng)點(diǎn)頻率值(Hz)Table 4 Frequency of each response point(Hz)

當(dāng)梁軸向振動(dòng)頻率由頻率點(diǎn)c向頻率點(diǎn)d靠近過程中，梁橫向振動(dòng)幅度逐漸減小，并在頻率點(diǎn)d達(dá)到最小值.因此，在往復(fù)軸向運(yùn)動(dòng)系統(tǒng)動(dòng)力學(xué)設(shè)計(jì)過程中，匹配其軸向振動(dòng)頻率使其盡可能接近d頻率點(diǎn)，能夠有效減小橫向振動(dòng)，改善系統(tǒng)動(dòng)力性能.

然而，需要注意的是d頻率點(diǎn)會(huì)因梁軸向振動(dòng)幅度而變得不穩(wěn)定，見圖5(b)～圖5(d).隨著梁軸向振動(dòng)幅度的增加，在頻率曲線中d頻率點(diǎn)附近會(huì)出現(xiàn)e頻率點(diǎn)，該頻率點(diǎn)為不穩(wěn)定頻率點(diǎn).當(dāng)梁在e頻率點(diǎn)做軸向振動(dòng)時(shí)，梁橫向振動(dòng)幅度迅速增加并失穩(wěn)，如圖5(d).并且，e頻率點(diǎn)頻帶寬度會(huì)隨著梁橫向振動(dòng)幅度的增大而增大，對(duì)梁的橫向振動(dòng)十分不利.避免e頻率點(diǎn)出現(xiàn)的一個(gè)有效方法就是調(diào)整梁軸向振動(dòng)幅度.

筆者發(fā)現(xiàn)，在重力作用下，各頻率點(diǎn)頻率值分布有一定規(guī)律性：梁平均懸臂長度對(duì)應(yīng)的第一階模態(tài)頻率為14.07Hz，a,b,e頻率點(diǎn)頻率值分別分布在梁第一階模態(tài)頻率的1/3,1/2,2倍處附近.因此，在軸向運(yùn)動(dòng)系統(tǒng)動(dòng)力學(xué)設(shè)計(jì)過程中，不僅要使軸向振動(dòng)頻率避開一階模態(tài)頻率.在軸向振動(dòng)幅值較大的情況下，避開一階模態(tài)頻率的1/3,1/2,2倍頻率值也是必要的.

5.2 忽略重力影響

將重力加速度g設(shè)為0，則可忽略重力對(duì)梁橫向振動(dòng)的影響.同樣分析不同軸向振動(dòng)幅值下的梁頻率響應(yīng)特性，結(jié)果見圖6.

對(duì)比圖5與圖6可以發(fā)現(xiàn)，忽略重力影響后，梁的頻率響應(yīng)特性變得相對(duì)簡單.當(dāng)梁軸向振動(dòng)幅度較小時(shí)，軸向運(yùn)動(dòng)頻率對(duì)梁的橫向振動(dòng)影響很小，如圖6(a).隨著梁軸向振動(dòng)幅度的增大，頻響曲線在28.36Hz附近出現(xiàn)共振峰(即5.1節(jié)中e頻率點(diǎn))，使系統(tǒng)失穩(wěn).梁軸向振動(dòng)幅度越大，e頻率點(diǎn)共振帶寬越大，這一現(xiàn)象與文獻(xiàn)[33]得出的結(jié)論一致.

關(guān)于軸向運(yùn)動(dòng)梁的穩(wěn)定性，主要是由于式(15)中的C為一陀螺項(xiàng)，其符號(hào)受到梁運(yùn)動(dòng)方向的影響，即：梁做伸展運(yùn)動(dòng)時(shí)，C符號(hào)為正，具有阻尼效應(yīng)，能夠耗散系統(tǒng)能量使系統(tǒng)獲得穩(wěn)定.當(dāng)梁做收縮運(yùn)動(dòng)時(shí)，C符號(hào)為負(fù)，收縮運(yùn)動(dòng)過程中梁會(huì)從固定邊界處吸收能量，使系統(tǒng)處于不穩(wěn)定運(yùn)動(dòng)狀態(tài)，甚至失穩(wěn).關(guān)于軸向運(yùn)動(dòng)梁的穩(wěn)定性的詳細(xì)討論，可參考文獻(xiàn)[32].

圖6 軸向移動(dòng)梁頻率響應(yīng)特性(忽略重力)Fig.6 Frequency response characteristic of the axially moving Beam(neglecting gravity)

6 結(jié)論

本文采用Rayleigh-Ritz法、拉格朗日方程推導(dǎo)了軸向移動(dòng)懸臂梁時(shí)變動(dòng)力學(xué)方程.與以往離散方法不同的是，本文選取冪級(jí)數(shù)函數(shù)構(gòu)造試函數(shù)對(duì)軸向移動(dòng)系統(tǒng)動(dòng)力問題進(jìn)行求解.由于冪級(jí)數(shù)基函數(shù)具有較好的積分與微分性能，這一特點(diǎn)使積分運(yùn)算變得非常迅速，并且很容易以矩陣的形式進(jìn)行推導(dǎo)，并可以采用符號(hào)運(yùn)算軟件直接生成MATLAB程序.由于MATLAB基本數(shù)據(jù)單位為矩陣，符號(hào)軟件生成的程序只需經(jīng)過簡單修改便可進(jìn)行動(dòng)力學(xué)計(jì)算.這樣大大縮短了軸向移動(dòng)梁從建模到動(dòng)力學(xué)分析的時(shí)間，過程十分高效.相關(guān)算例對(duì)比表明，本文采取的方法與建立的模型準(zhǔn)確可靠.

通過建立的時(shí)變動(dòng)力學(xué)方程對(duì)梁的軸向共振問題進(jìn)行了研究，得到如下結(jié)論：

(1)在選取冪級(jí)數(shù)函數(shù)擬合階數(shù)時(shí)，應(yīng)將其最大相對(duì)擬合誤差控制在千分之一以內(nèi)，以保證較好的動(dòng)力學(xué)計(jì)算精度.

(2)軸向振動(dòng)幅值越大，幅頻響應(yīng)曲線中特殊頻率點(diǎn)越多，并且容易導(dǎo)致系統(tǒng)失穩(wěn).同時(shí)，失穩(wěn)頻率寬度越大.

(3)考慮重力影響時(shí)，會(huì)在梁一階模態(tài)頻率值的1/3、1/2、2倍處附近產(chǎn)生共振峰.共振峰峰值隨著軸向振動(dòng)幅度的增加而增加.忽略重力影響，即梁在水平面內(nèi)的振動(dòng)，梁的頻率響應(yīng)特性相對(duì)簡單，只在梁一階模態(tài)頻率值的2倍處附近出現(xiàn)系統(tǒng)失穩(wěn).

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AN EFFICIENT DYNAMIC MODELING METHOD OF AN AXIALLY MOVING CANTILEVER BEAM AND FREQUENCY RESPONSE ANALYSIS1)

Hua Hongliang Liao Zhenqiang2)Zhang Xiangyan
(School of Mechanical Engineering，Nanjing University of Science and Technology，Nanjing210094，China)

The dynamics of the axially moving beam has wide application in engineering,such as robot manipulators,machine tools and gun barrel,et al.Computing the dynamic response of axially moving beam is an important method to evaluate the dynamic performance and fi nally the structure design.The time-varying motion equations of the axially movingcantileverbeamarederivedusingtheRayleigh-RitzmethodandLagrange’sequation.Thepowerseriesfunctionis used to construct the trial function to solve the dynamic problem.Due to the good integral and di ff erential performance of power series function,the derivation is easy to be carried out in the form of matrix.In this way,the symbolic computation software can generate the MATLAB program directly.And the generated MATLAB program can be used to conduct the dynamic computation with few modi fi cations,because the basic data unit of MATLAB is matrix.The overall process is efficiency and the time from dynamic modeling to computation is greatly reduced.Through four sets of numerical examples,the computational accuracy of the presented method is validated by comparing the dynamic responses with those from previous literatures.Then,the e ff ects of fi tting order of the power series function on computational accuracy are discussed.And the principle to select the fi tting order of the power series function to achieve good convergence and computational accuracy is given.Based on the dynamic model,the e ff ects of axial motion frequency on transverse vibration are studied.The e ff ects of axial vibration amplitude on the frequency response characteristic are explored.And the di ff erence between considering gravity and neglecting gravity e ff ect are compared.

computational dynamics，power series basis function，frequency response analysis，axially moving beam

O313

A doi：10.6052/0459-1879-17-182

2017–05–16 收稿，2017–08–07 錄用,2017–08–10 網(wǎng)絡(luò)版發(fā)表.

1)國家自然科學(xué)基金資助項(xiàng)目(51375241).

2)廖振強(qiáng)，教授，主要研究方向：自動(dòng)武器發(fā)射動(dòng)力學(xué).E-mail:huahl123@126.com

華洪良，廖振強(qiáng)，張相炎.軸向移動(dòng)懸臂梁高效動(dòng)力學(xué)建模及頻率響應(yīng)分析.力學(xué)學(xué)報(bào),2017,49(6):1390-1398

Hua Hongliang，Liao Zhenqiang，Zhang Xiangyan.An efficient dynamic modeling method of an axially moving cantilever beam and frequency response analysis.Chinese Journal of Theoretical and Applied Mechanics,2017,49(6):1390-1398