本文選題：約束系統(tǒng) + 質(zhì)量矩陣��； 參考：《大連理工大學(xué)》2017年博士論文

【摘要】：約束多體系統(tǒng)的運動學(xué)和動力學(xué)是CAD(計算機輔助設(shè)計)和CAE(計算機輔助工程)的重要組成部分。由于約束條件的存在,約束系統(tǒng)的動力學(xué)方程呈現(xiàn)出多種表達形式。這些不同的數(shù)學(xué)描述在理論和數(shù)值上都體現(xiàn)出不同的特征。過往幾十年,已經(jīng)有大量的研究工作致力于約束系統(tǒng)的各種不同的理論表述以及相關(guān)的數(shù)值仿真研究。盡管如此,在目前已有的數(shù)值仿真方面的著作中,還沒有不同表述對精度的影響這方面較為透徹的研究工作。本文對約束多體動力學(xué)方程慣量(質(zhì)量)矩陣的各種表達式以及數(shù)值仿真中不同慣量表示對于精度的影響進行了詳細的研究,主要的研究內(nèi)容可以概括為如下三個部分:(1)以Schur分解為基礎(chǔ),對約束多體系統(tǒng)質(zhì)量矩陣的增廣表達進行了詳細分析,然后通過定義廣義角速度,提出了質(zhì)量矩陣的一種特殊表述形式,被稱為主慣量表示。在標(biāo)準(zhǔn)形式中,主慣量表示將質(zhì)量矩陣分為兩部分:一個單位矩陣和一個位移相關(guān)矩陣,兩矩陣相互之間的比例由伸縮參數(shù)σ控制,其中伸縮參數(shù)σ可以為任意常數(shù)。通過建立質(zhì)量矩陣的增廣表達式與主慣量表示在數(shù)學(xué)上的等價性,確保了主慣量表示對于一般約束動力學(xué)系統(tǒng)的普遍存在性。(2)在主慣量表示的框架下,詳細推導(dǎo)了約束動力學(xué)系統(tǒng)的離散動能誤差估計。誤差估計表明:在Lagrange框架下,動能的離散誤差是伸縮參數(shù)的線性函數(shù);在Hamilton框架下,動能的離散誤差是伸縮參數(shù)倒數(shù)的線性函數(shù)。并且當(dāng)質(zhì)量矩陣為廣義位移的函數(shù)時,誤差函數(shù)的斜率與離散誤差同階。與之相反,動能的離散誤差與質(zhì)量的增廣表達式的具體形式無關(guān)。因此,主慣量表示與質(zhì)量的增廣表達式在數(shù)值上是不等價的。根據(jù)誤差分析結(jié)果,本文進一步建議慣量主值的算術(shù)平均以及調(diào)和平均可以作為伸縮參數(shù)σ的合理的預(yù)條件值,以得到較小的離散誤差。(3)以主慣量表示為基礎(chǔ),針對約束系統(tǒng)提出了一種通過確定伸縮參數(shù)的(最佳)預(yù)條件值來改進數(shù)值積分精度的新方法,并且將其應(yīng)用于二維和三維剛體的數(shù)值仿真。數(shù)值結(jié)果驗證了誤差分析結(jié)論的正確性,在主慣量表示下數(shù)值積分的精度得到了大幅度的改善。在基于對流基矢量或單位四元數(shù)的三維剛體旋轉(zhuǎn)的算例中,采用慣量主值的算術(shù)平均作為伸縮參數(shù)預(yù)條件值的數(shù)值積分,其精度提高了一個量級以上。根據(jù)上述研究結(jié)果,主慣量表示作為約束動力系統(tǒng)質(zhì)量矩陣的一種具體的表示形式,為改善數(shù)值積分的精度提供了一種新方式,其相關(guān)的理論和數(shù)值研究亟待進一步展開。
[Abstract]:Kinematics and dynamics of constrained multibody systems are important components of CAD and CAE. Because of the existence of constraint conditions, the dynamic equations of constrained systems take on many forms of expression. These different mathematical descriptions show different characteristics both theoretically and numerically. In the past few decades, a great deal of research work has been devoted to the various theoretical representations of constrained systems and related numerical simulation. However, in the existing works on numerical simulation, there is no more thorough research on the effect of different expressions on accuracy. In this paper, the effects of various expressions of inertia (mass) matrix of constrained multi-body dynamics equations and different inertia representations in numerical simulation on the accuracy are studied in detail. The main research contents can be summarized as follows: 1) based on the Schur decomposition, the augmented expression of the mass matrix of constrained multibody systems is analyzed in detail, and then the generalized angular velocity is defined. A special representation of mass matrix is presented, which is called principal inertia representation. In the standard form, the principal inertia representation divides the mass matrix into two parts: a unit matrix and a displacement correlation matrix. The proportion between the two matrices is controlled by the scaling parameter 蟽, where the scaling parameter 蟽 can be an arbitrary constant. By establishing the mathematical equivalence between the augmented expression of the mass matrix and the representation of the principal inertia, the universal existence of the representation of the principal inertia for the general constrained dynamical system is ensured under the framework of the representation of the principal inertia. The estimation of discrete kinetic energy error for constrained dynamical systems is derived in detail. The error estimates show that the discrete error of kinetic energy is a linear function of the scaling parameter under Lagrange framework, and the discrete error of kinetic energy is a linear function of the inverse of the scaling parameter in the Hamilton framework. When the mass matrix is a function of generalized displacement, the slope of the error function is the same as the discrete error. On the contrary, the discrete error of kinetic energy is independent of the specific form of the mass augmentation expression. Therefore, the expression of principal inertia and the augmented expression of mass are not numerically equivalent. According to the results of error analysis, it is further suggested that the arithmetic average and harmonic average of the principal value of inertia can be regarded as the reasonable preconditioned value of the extensional parameter 蟽, so as to obtain a smaller discrete error, which is based on the representation of the principal inertia. A new method to improve the numerical integration accuracy by determining the (optimal) preconditioned values of the stretching parameters for constrained systems is proposed and applied to the numerical simulation of 2D and 3D rigid bodies. Numerical results verify the correctness of the error analysis, and the accuracy of numerical integration is greatly improved under the representation of principal inertia. In the example of 3D rigid body rotation based on convection basis vector or unit quaternion, the arithmetic average of the principal value of inertia is used as the numerical integral of the preconditioned value of the telescopic parameter, and its precision is improved by more than one order of magnitude. Based on the above results, as a concrete representation of the mass matrix of constrained dynamic systems, the principal inertia representation provides a new way to improve the accuracy of numerical integration. The related theory and numerical research need to be further developed.
【學(xué)位授予單位】：大連理工大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O313.7

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