【摘要】：多體系統(tǒng)動力學(xué)建模時,可以將系統(tǒng)中的運動副和物體簡化為剛體或柔性體。其中,梁是多體系統(tǒng)中廣泛應(yīng)用的柔性構(gòu)件,其建模是多體領(lǐng)域的一個研究熱點,剛?cè)狁詈隙囿w系統(tǒng)中梁的幾何非線性問題顯著。高速旋轉(zhuǎn)柔性梁呈現(xiàn)動力剛化現(xiàn)象,基于小變形假設(shè)的梁單元能夠在附加動力剛化項的條件下解決動力剛化問題,卻不適用于大變形分析。幾何精確梁除剛截面外不對模型做其他假設(shè),它的位移-應(yīng)變關(guān)系適用于任意大小的位移和轉(zhuǎn)動,適合梁的幾何非線性分析。傳統(tǒng)幾何精確梁單元大多對位移和轉(zhuǎn)動獨立插值,應(yīng)用在細長梁上將面臨剪切閉鎖的問題,而多體系統(tǒng)中的柔性梁多為細長梁,使用歐拉-伯努利梁符合其受力特征。歐拉-伯努利梁中截面與形心線垂直,這一約束為形心位移與截面轉(zhuǎn)動的插值增加了難度。對節(jié)點轉(zhuǎn)動矢量插值的梁單元引起了許多數(shù)值求解方面的困難,容易丟失單元應(yīng)變對剛體位移的客觀性。一類幾何精確梁單元直接對應(yīng)變離散,由此能夠避開轉(zhuǎn)動插值引起的麻煩,但對于空間梁不易由應(yīng)變導(dǎo)出節(jié)點位移和轉(zhuǎn)動。針對以上幾何非線性梁的建模問題,本文基于幾何精確歐拉-伯努利梁彈性虛功率和形心線與曲率的關(guān)系,先通過單元形心曲線插值得到端面曲率及其對弧長的變化率,進而對單元域內(nèi)的曲率進行擬合,提出了一種以整體節(jié)點位移和轉(zhuǎn)動矢量為基本變量,既可避免轉(zhuǎn)動矢量插值,同時又不增加節(jié)點參數(shù)的幾何非線性空間梁單元。單元對任意剛體位移保有應(yīng)變客觀性,具備可積的顯式節(jié)點力表達式,在小變形情況下能夠退化到線性單元。單元考慮了軸向變形與彎、扭變形的耦合效應(yīng),能夠完全計及“動力剛化”項,數(shù)值算例驗證了該單元處理幾何大變形,以及動力剛化問題的能力,證實了單元的收斂性與精確性。以上位移-應(yīng)變混合插值梁單元具有可積的彈性節(jié)點力表達式,這對于幾何非線性問題是個難得的優(yōu)點。這一優(yōu)點得益于單元內(nèi)做了幾何假設(shè)。同樣基于幾何精確梁模型,本文構(gòu)造了一種嚴格滿足截面與形心線約束和保持應(yīng)變客觀性的幾何非線性歐拉-伯努利梁單元。歐拉-伯努利梁截面在運動中始終與形心線垂直,可認為截面轉(zhuǎn)動四元數(shù)令初始直梁的截面法線轉(zhuǎn)到變形后形心線切線上,它是一組4變量的3個四元數(shù)方程。插值形心線曲線,使其滿足邊界連續(xù)性和位移非線性要求。由形心線切線和四元數(shù)方程的通解輔助確定截面轉(zhuǎn)動場,它能夠自動滿足截面與形心線約束,有效避免轉(zhuǎn)動參數(shù)的奇異性。根據(jù)該建模策略構(gòu)造的幾何非線性梁單元形式簡潔,能夠保證單元應(yīng)變由位移場和轉(zhuǎn)動場一致地導(dǎo)出。此外,多體系統(tǒng)動力學(xué)建模要求提供柔性梁截面以及剛性物體的質(zhì)心、質(zhì)量、轉(zhuǎn)動慣量等參數(shù),為自動化建模帶來不便。對于復(fù)雜形狀和不規(guī)則幾何體來說,確定這些參數(shù)需要很大的工作量,而且不易計算準確。隨著多體系統(tǒng)動力學(xué)的推廣,多體理論和商業(yè)軟件的發(fā)展趨向于減少用戶的人工操作,以便降低輸入?yún)?shù)有誤的幾率。為此,本文基于虛功率原理推導(dǎo)了一種自動集成多體系統(tǒng)中剛體動力學(xué)方程的新方法。建模流程是先對剛體采用網(wǎng)格剖分,以組成物體的剛體單元為基本元素,該建模方法與有限元法一樣自動集成系統(tǒng)的動力方程。根據(jù)需要具體構(gòu)造了剛性四面體單元和剛性梁單元,對復(fù)雜形狀的梁截面,構(gòu)造了對平面形狀逼近性較好的三角形單元。以剛體單元為基礎(chǔ)并內(nèi)嵌網(wǎng)格剖分模塊的分析軟件能夠自動獲得剛體的幾何參數(shù)和慣性參數(shù),從而具備了獨立處理任意復(fù)雜形狀系統(tǒng)的能力。針對多體系統(tǒng)中大變形細長梁的動力學(xué)建模,本文基于幾何精確梁理論,提出了兩種大變形空間歐拉-伯努利梁單元。針對系統(tǒng)中剛體的動力學(xué)建模存在的問題,本文構(gòu)造了剛體單元。
[Abstract]:In multi-body system dynamics modeling, the motion pairs and objects in the system can be simplified as rigid or flexible bodies. Beams are widely used flexible components in multi-body systems, and their modeling is a research hotspot in multi-body field. The beam element based on the hypothesis of small deformation can solve the dynamic stiffness problem under the condition of additional dynamic stiffness term, but it is not suitable for large deformation analysis. Most of the traditional geometrically accurate beam elements are interpolated independently for displacement and rotation, so they will face the problem of shear locking when applied to slender beams, while the flexible beams in multi-body systems are mostly slender beams. The Euler-Bernoulli beams are used to fit the mechanical characteristics of the beams. A class of geometrically exact beam elements are directly discrete to the strain, thus avoiding the trouble caused by the rotation interpolation, but it is not easy to derive the node from the strain for the spatial beam. In this paper, based on the geometrically accurate Euler-Bernoulli beam's elastic virtual power and the relationship between the center line and the curvature, the end-to-end curvature and its rate of change of the arc length are interpolated by the element center curve, and then the curvature in the element domain is fitted, and a global method is proposed. The nodal displacement and rotation vectors are the basic variables, which can avoid the interpolation of rotation vectors without adding the nodal parameters. The element retains the strain objectivity for any rigid body displacement, has an integrable explicit nodal force expression, and can degenerate to a linear element under small deformation. The coupling effect of deformation, bending and torsion can fully take into account the term of "dynamic stiffness". Numerical examples show that the element is capable of dealing with large geometric deformation and dynamic stiffness problems, and the convergence and accuracy of the element are verified. The above displacement-strain hybrid interpolated beam element has an integrable elastic nodal force expression, which is suitable for geometry. This advantage is due to the geometric assumptions made in the element. Also, based on the geometric exact beam model, a geometrically nonlinear Euler-Bernoulli beam element is constructed which strictly satisfies the constraints of section and centroid and maintains the objectivity of strain. The section of Euler-Bernoulli beam is always in motion with the center of shape. Line perpendicular, it can be considered that the section rotation quaternion makes the section normal of the initial straight beam turn to the tangent of the centroid after deformation. It is a set of three quaternion equations with four variables. The geometrically nonlinear beam element based on this modeling strategy is simple and can ensure that the element strain is uniformly derived from the displacement field and the rotation field. With the development of multi-body dynamics, the development of multi-body theory and commercial software tends to reduce the user's manual operation in order to reduce the transmission. Based on the principle of virtual power, a new method for automatically integrating the dynamic equations of rigid bodies in multibody systems is presented in this paper. Rigid tetrahedron element and rigid beam element are constructed according to the requirement, and triangular element with good approximation to plane shape is constructed for beam section with complex shape. In this paper, based on the geometric exact beam theory, two large deformation spatial Euler-Bernoulli beam elements are proposed for the dynamic modeling of slender beams with large deformation in multibody systems.
【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】：2016
【分類號】：O313.7

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