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1、38null1吉林大学学报 ( 工学版 ) null Vol. 38 null No. 12008M1Journal o f Jilin U niv ersity ( Engineering and T echnolo gy Edition) null Jan. 2008l: 2006- 12- 27.:npVSS( 200402133022) .Te:( 1941- ) ,o,q,pV3=.Z_:+.,ddL.E- mail: lig x hit . edu. cnpddLsZZE李瑰贤,于广滨,温建民,管null 竹(哈尔滨工业大学机电工程学院,哈尔滨 150001)Knull1:首先建立
2、了描述齿轮系统扭转振动的动力学分析模型, 并推导出综合考虑时变啮合刚度a齿侧间隙a动态传递误差等非线性因素的齿轮系统非线性动力学的统一微分方程b介绍了用于求解齿轮系统非线性动力学微分方程的多尺度方法的原理,并推导了频率响应方程b利用多尺度方法获得的近似解析解与直接进行数值积分所获得的精确解吻合得较好, 表明多尺度方法是求解复杂非线性微分方程最有效的方法之一b1oM:机械设计;齿轮系统; 非线性振动; 多尺度方法ms|: T H 132. 4 null nullDSM: A null nullcI|: 1671-5497( 2008) 01-0075-05Method of multiplesc
3、ales in solving nonlinear dynamicdifferential equations of gear systemsLi Gu-i x ian, Yu Guang-bin, Wen Jian-m in, Guan Zhu( School of Mechanical and Electrical Engineering, H arbin Institute of Technology , H arbin, 150001, China)Abstract: The dynamic mo del w hich describes the torsional vibr atio
4、n behav iors o f gear sy stem w asintroduced accurately in this paper. T he differential equation of gear system nonlinear dynamicsex hibiting combined no nlinearity influence such as time-vary ing stiffness, tooth backlash and dy nam ictr ansm issio n er ror ( DTE) w as proposed. The theory of mult
5、iple scales m ethod w as used to solveno nlinear differential equatio ns of gear systems and the frequency response equation w as obtained.The fact that the appr oxim ate analy tical solution by using the method of multiple scales is in g oodag reem ent w ith the ex act so lutions by numerically int
6、egr ating differ ential equations and it proved thatthe metho d of multiple scales is one of the most frequently used m ethods in solving differentialequations, especially for lar ge and complicated differential equations.Keywords: mechanical desig n; gear system s; no nlinear oscillatio ns; m ethod
7、 of m ultiple scalesnull nulld+dLsZd,pV?,yNdLZ?pbEi?|EbZEdLsZEK1b-ZE1Z7EaLindstedt-Poincar eEa?ZEaZEaMEa(E 1-3b b0, null null null - b null null null bnull+ b, null null 1- 1 x (t) 1x(t) - 1VE7QT,f ( x ) = a1 x + a3 x3 +a5x5 + a7 x7b,T(9)dKsZxnullnull(t) + 2nullx ( t) + 1+ null5j= 1Bj cos(j nullet +
8、 nullj )( a1x + a3x3 + a5x5 + a7x7 ) =FaT cos( nulleht + nullT ) + Fahnull2eh cos( nulleht+ nullh)(10)2 nullpdLsZZEnull nullT(10)dsZd,pV?,yN?ZEbT-KWZEnull null nullZEpVbn5|T( 10)Z7,T( 10)dLlnull,7a1 = null20 , null1 = a3 , null2 =a5, null3= a7 ,5xnullnull(t) + 2nullnullxnull(t) + null20x + null null
9、5j= 1Bj cos(j nullet+ nullj) null20x +1+ null5j= 1Bj cos(j nullet+ nullj) ( null3x3 + null5x5 + null7x7) =Fm + nullFahnull2eh cos( nulleht+ nullh) (11)null nullZE,!T(11)Tx(T,null) = x0 (T0 , T1 ) + nullx 1( T0, T1 ) + null(12)T: T0HWyM, T 0= t; T 1HWM, T1 = nulltb|T( 12)T(11),7null0anullM,x(2, 0)0 (
10、T 0, T1) + null20x0( T0, T1) - Fm = 0(13)x(2, 0)1 (T0, T1) + 2x(1,1)0 (T 0, T1) + 2nullx (1,0)0 (T0, T1) +null20x1(T0, T1) + null4k= 2nullk- 1 x0(T0 ,T 1) 2k- 1 +null5j = 1Bjnull20x0(T0, T1)cos(jnullet + nullj ) +null4k= 2nullk- 1 null5j= 1Bj x0 (T 0, T1) 2k- 1cos(jnulleT 0 + nullj ) -Fahnull2eh cos
11、(null2eht+ nullh) = 0 (14)V!T(13)Yx0 (T0 , T1 ) = A( T1 )einull0 T0 + A(T 1) e- inull0 T0 + Fmnull20(15)T: A(T1 )aA(T 1)sY1HWT 1fb|T(15)T(14),Vx(2, 0)1 (T 0, T 1) + null20 x1(T 0, T 1) =- einull0 T0 3null1F2mA (T 1)null40+2inull0nullA (T 1) + 3null1 A(T1 ) 2A(T 1) +2inull0A (T 1) + 10null2A(T 1) 3 A
12、(T 1) 2 +35null3 A( T1) 4A(T 1) 3 +30null2F2mA(T 1) 2 A(T1 )null40+210null3 F2m A( T1 ) 3A(T 1) 2null40+5null2F4mA( T1)null80 +105null3F4mA(T 1) 2 A(T 1)null80 +7null3F6mA (T 1)null120+ NST( 762) (16)T: NST(762)VUcy0einull0 T0762b3 null=T/dYsnull nullT=T/ddYsb=T/,?3H,nulleh nullnull0bBnull= O( 1)9,=qnull0, null=qnullehnull0bnullV1sZZLst(l),yN,7nulleh = null0 + nullnull (17)|T(15)a( 17)T(16),3,APT( 16)ceinull0T 0,bnull77null吉林大学学报( 工学版) 第38卷NV3null1A(T 1) F2mnull40+ A(T 1)A(T 1) +2inull0nu
