本文選題：圓薄板大撓度 + 非線性振動�。� 參考：《蘭州大學(xué)》2017年碩士論文

【摘要】：圓薄板被應(yīng)用于各類工程結(jié)構(gòu)之中,尤其在航天航空器、儲存罐、船舶、以及傳感器中得到廣泛使用,如飛機(jī)蒙皮、儲存罐底、壓力儀表中的彈性膜片等。這類結(jié)構(gòu)由于剛度較小,在外界激勵下極易產(chǎn)生大振幅的振動,嚴(yán)重影響著整個系統(tǒng)的有效性、服役安全、使用壽命和舒適性等,必須加以研究。然而由于其違背了線性理論的小變形假設(shè),呈現(xiàn)出明顯的非線性特征,即幾何非線性,導(dǎo)致研究起來非常困難。典型的如圓薄板的大擾度彎曲問題,從基本方程的建立到給出其收斂解中間跨域了近一個世紀(jì)。而對于圓薄板的非線性振動問題,尤其是強(qiáng)非線性振動問題,目前依然缺乏非常有效的求解方法。針對圓薄板的非線性振動問題,目前最常使用的是有限元方法。然而在其求解過程中,由于有限元方法無法實(shí)現(xiàn)時空完全解耦,即其剛度矩陣顯式依賴于時間離散格式。這一方面增大了計算量,因?yàn)槠鋭偠染仃囋诿恳粫r刻步均需更新。同時,由于時間積分過程中累積的誤差,有可能導(dǎo)致結(jié)構(gòu)剛度矩陣存在較大的偏差,進(jìn)而致使長時間追蹤結(jié)果失蹤,甚至獲得錯誤的近似解。有鑒于此,本課題擬在本小組原有研究的基礎(chǔ)之上,探索提出一套分析圓薄板結(jié)構(gòu)非線性行為的高精度小波算法。本文主要內(nèi)容有:(1)推導(dǎo)了任意平方可積函數(shù)在有限區(qū)間上(邊界Lagrange延拓)基于廣義Coiflets小波的逼近公式,對逼近公式在有限區(qū)間上的誤差給予了證明,并給出了幾類在利用小波伽遼金方法求解微分方程的過程中經(jīng)常遇到的連接系數(shù)的推導(dǎo)過程及計算結(jié)果;(2)建立了針對中心彈性約束圓薄板大撓度問題的小波求解格式,通過和以往結(jié)果對比發(fā)現(xiàn):用多項(xiàng)相乘連接系數(shù)離散微分方程所得結(jié)果的精度更高;(3)建立了針對圓薄板軸對稱非線性振動問題的小波求解格式,并結(jié)合Newmark方法對其展開了定量研究,得到了諸如:中心撓度達(dá)到板厚2倍時自由振動周期減至線性振動周期65%;薄板中心響應(yīng)振幅隨激勵力頻率增大而減小等結(jié)論。
[Abstract]:Circular thin plates are widely used in various engineering structures, especially in aerospace aircraft, storage tanks, ships and sensors, such as aircraft skin, storage tank bottom, elastic diaphragm in pressure meters, etc. Due to the small stiffness, it is easy to produce large amplitude vibration under external excitation, which seriously affects the effectiveness, service safety, service life and comfort of the whole system, and must be studied. However, due to its violation of the hypothesis of small deformation in linear theory, it presents obvious nonlinear characteristics, that is, geometric nonlinearity, which makes it very difficult to study. Typical problems such as the large perturbed bending of circular thin plates have been used for nearly a century from the establishment of the basic equation to the solution of its convergence. However, for the nonlinear vibration problems of circular thin plates, especially for the strong nonlinear vibration problems, there is still a lack of very effective methods to solve them. Finite element method (FEM) is the most commonly used method for nonlinear vibration of circular thin plates. However, in the process of its solution, the finite element method can not realize the complete decoupling of time and space, that is, its stiffness matrix is explicitly dependent on the time discrete scheme. This increases the computational complexity because the stiffness matrix needs to be updated at every step. At the same time, due to the accumulated errors in the process of time integration, it is possible that there is a large deviation in the stiffness matrix of the structure, which leads to the disappearance of the long time tracking results and even the obtaining of the wrong approximate solution. In view of this, this paper proposes a set of high-precision wavelet algorithms to analyze the nonlinear behavior of circular thin plate structures on the basis of the original research of this group. In this paper, the approximation formula of arbitrary square integrable function on finite interval (boundary Lagrange extension) based on generalized Coiflets wavelet is derived, and the error of approximation formula on finite interval is proved. The derivation process and calculation results of several connection coefficients often encountered in the process of solving differential equations by wavelet Galerkin method are given. A wavelet solution scheme for the large deflection problem of circular thin plates with central elastic constraints is established. By comparing with the previous results, it is found that the accuracy of the results obtained from the discrete differential equations of multiplying the connecting coefficients is higher than that of the previous ones) and a wavelet scheme for solving the axisymmetric nonlinear vibration problems of circular thin plates is established. Based on the Newmark method, some conclusions are obtained, such as the reduction of the free vibration period to the linear vibration period of 65 when the central deflection reaches 2 times the thickness of the plate, and the decrease of the central response amplitude of the thin plate with the increase of the excitation frequency.
【學(xué)位授予單位】：蘭州大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：TB12

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本文編號：2007965