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1.
非线性动力学常微分方程组高精度数值积分方法   总被引:6,自引:1,他引:5  
郑兆昌  沈松  苏志霄 《力学学报》2003,35(3):284-295
建立了一种求解非线性动力学常微分方程组初值问题的新方法.若非线性函数一阶导数存在,则给出解的积分方程表达式,计算得到按规定误差要求的高精度数值解.引入一般自治或非自治非线性系统的首次近似Jacobi矩阵,不作任何假设重构等价的非线性常微分方程组,简捷而有广泛的适应性,不改变方程的本质,但其主项构成线性化方程组,其它项则代表非线性函数高阶余项而不涉及Taylor级数展开计算,给出该方程组初值问题的Duhamel卷积分解析表达式,在时间步长内进行数值积分选代求解,在指定误差内快速收敛,逐步递推获得非线性常微分方程的瞬态响应和全时域高精度数值解.积分解连续满足微分方程组而不是在离散的步长端点上满足代数方程组,打破了传统用增量法在离散点上建立的代数方程组迭代求解,从而使传统Euler型逐步积分法的各种差分格式算法改变成真正的积分格式算法.数值计算中给出指数矩阵递增展开式,变矩阵乘法为乘积系数的加法,避免了大量矩阵自乘而大大提高计算效率.算法验证为无条件稳定,则保证对线性常微分方程而言,计算中舍入误差的传播不会扩散,不出现计算机字长有限而引起舍入误差导致计算不确定性问题.基于以上理论和数值方法,计算了线性非线性算例并进行了分析,验证了本方法简捷而有广泛的适应性,可以有足够的精确性.  相似文献
2.
超空泡射弹尾拍分析与计算   总被引:6,自引:0,他引:6       下载免费PDF全文
对超空泡射弹进行运动学和动力学分析并数学建模,求解耦合非线性微分方程组,得到水下高速超空泡射弹运动特性。数值模拟结果表明,高速超空泡射弹在航行过程中,由于弹体头部和尾部的阻力作用,水平速度随时间迅速衰减。并且射弹的角速度呈周期性往复变化,即尾拍现象。同时由于空泡尺寸的减小导致尾拍幅度逐渐变小。射弹转动惯量越小,角速度变化幅度越平稳,相同时间内尾拍次数减少。发射深度或发射速度越大,尾拍幅度衰减越快。较大的初始角速度也会使射弹角速度很快衰减。  相似文献
3.
Solution properties of the nonlinear second-order delay-differential equation x(t)=–ax(t)+f[x(t–)] are studied wheref is a piecewise constant function which mimics negative feedback. We show that the solutions can be obtained by a simple geometrical construction which, in principle, can be implemented using a ruler and a compass. Analytical results guarantee the existence and stability properties of limit cycle solutions. Computer-aided constructions reveal a remarkable richness of different types of dynamical behaviors including a variety of unconventional bifurcation schemes.  相似文献
4.
IntroductionOwingtothefactthatthesolutionsoftheDirichletproblemsforaclassofsystemsofdifferentialequationsoftheformu″=v ,εv″ f(x ,u ,u′)v′ -g(x ,u ,u′)v=0   ( 0 <x <1,0 <ε 1) ,u( 0 ;ε) =0 ,u( 1;ε) =0 ,v( 0 ;ε) =α ,v( 1;ε) =β,( 1)havethequalitativepropertiesofsolutionsofamorecomplicatedsyste…  相似文献
5.
We analyze axisymmetric, spatially localized standing wave solutions with periodic time dependence (breathers) of a nonlinear partial differential equation. This equation is derived in the 'continuum approximation' of the equations of motion governing the anti-phase vibrations of a two-dimensional array of weakly coupled nonlinear oscillators. Following an asymptotic analysis, the leading order approximation of the spatial distribution of the breather is shown to be governed by a two-dimensional nonlinear Schrödinger (NLS) equation with cubic nonlinearities. The homoclinic orbit of the NLS equation is analytically approximated by constructing [2N × 2N] Padé approximants, expressing the Padé coefficients in terms of an initial amplitude condition, and imposing a necessary and sufficient condition to ensure decay of the Padé approximations as the independent variable (radius) tends to infinity. In addition, a convergence study is performed to eliminate 'spurious' solutions of the problem. Computation of this homoclinic orbit enables the analytic approximation of the breather solution.  相似文献
6.
In this article, we solve in closed form a system of nonlinear differential equations modelling the elastica in space of a thin, flexible, straight rod, loaded by a constant thrust at its free end. Common linearizations of strength of materials are of course not applicable any way, because we analyze great deformations, even if not so large to go off the linear elasticity range. By passing to cylindrical coordinates ρ, θ, z, we earn a more tractable differential system evaluating ρ as elliptic function of polar anomaly θ and also providing z through elliptic integrals of I and III kind. Deformed rod’s centerline is then completely described under both tensile or compressive load. Finally, the planar case comes out as a degeneracy, where the Bernoulli lemniscatic integral appears.  相似文献
7.
侯宇  沈力行 《上海力学》1999,20(3):291-296
本文研究数学规划加权残值法在非线性微分方程求解中的应用,利用数学规划加权残值法和LP模理论,把非线性微分方程边值问题转化为一个可微分的无约束非线性优化问题,从而运用成熟稳定的寻优方法求得问题的解。文中数字计算例子表明本文方法可以快速有效地求解非线性微分方程。  相似文献
8.
Dynamic systems described by nonlinear differential equations of the second order are studied. It is assumed that certain preliminary information on the dissipative or elastic characteristics of systems is known. A new approach is demonstrated to obtaining full information on unknown or partially known characteristics of a system from measurements of not only displacements but also velocities and accelerations __________ Published in Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 139–143, June 2005.  相似文献
9.
ONTHEBOUNDEDNESSANDTHESTABILITYRESULTSFORTHESOLUTIONOFCERTAINFOURTHORDERDIFFERENTIALEQUATIONSVIATHEINTRINSICMETHODCemilTUNC;A...  相似文献
10.
INSTABILITYOFSOLUTIONFORACLASSOFTHETHIRDORDERNONLINEARDIFFERENTIALEQUATIONLuDeyuan卢德渊(ReceivedNov101994CommunicatedbyZhangShi...  相似文献
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