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This document discusses the analysis of twist in a closed thin-walled shell subjected to torque. It defines key terms like shear stress, shear flow, and torque. The document presents equations to calculate the torque on a shell section based on the shear flow and area. It also provides Castigliano's theorem to calculate the angle of twist based on the applied torque and the material and geometric properties of the shell.
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This document defines and provides examples of curves and parametrized curves. It discusses regular and unit-speed curves. The key points are:
i) A parametrized curve is a continuous function from an interval to Rn. Examples of parametrized curves include ellipses, parabolas, and helices.
ii) A regular curve is one where the derivative of the parametrization is never zero. A unit-speed curve has a derivative of constant length 1.
iii) The arc-length of a curve is defined as the integral of the derivative of the parametrization. Any reparametrization of a regular curve is also regular. A curve has a unit-
QMC: Operator Splitting Workshop, Thresholdings, Robustness, and Generalized ...
QMC: Operator Splitting Workshop, Thresholdings, Robustness, and Generalized ...The Statistical and Applied Mathematical Sciences Institute
The document discusses statistical estimation methods that use joint regularization. Specifically, it discusses using thresholding rules and nonconvex penalties within an additive robust framework for statistical regression. Key points:
- Thresholding rules Θ can induce nonconvex penalty functions P and allow reformulating regression as a proximity operator problem.
- An additive robust framework combines thresholding Θ with a ψ function to perform M-estimation, as long as Θ + ψ = identity.
- Generalized group sparsity pursuit extends this to multiple nonconvex penalties and response variables. An algorithm is developed using linearization and scaled thresholding rules.
- Challenges include analyzing convergence of nonconvex algorithms, understanding statistical performance, and acceleratingR. Jimenez - Fundamental Physics from Astronomical Observations
The SEENET-MTP Workshop BW2011
Particle Physics from TeV to Plank Scale
28 August – 1 September 2011, Donji Milanovac, Serbia
Similar to 簡報1 (20)
QMC: Operator Splitting Workshop, Thresholdings, Robustness, and Generalized ...
QMC: Operator Splitting Workshop, Thresholdings, Robustness, and Generalized ...
R. Jimenez - Fundamental Physics from Astronomical Observations
R. Jimenez - Fundamental Physics from Astronomical Observations
簡報1
- 2. 封閉薄殼的扭轉剪應力 RB RA 由靜力平衡 R A = τA ⋅ t A = τB ⋅ t B = R B 定義剪力流 (Shear Flow) q q = τavg ⋅ t 封閉薄殼軸受一扭力 T 薄殼厚度小， 定義剪力流為薄殼斷面單位 長度的剪力 假設剪應力在徑向均勻分佈 沿著薄殼圓週剪力流相同
- 4. ds h t dF dA dT = h ⋅ dF = hq ⋅ ds dA = 1 h ⋅ ds 2 T = ∫ dT = ∫ hq ⋅ ds = 2 ∫ q ⋅ dA T = 2qAm dF T τ= 2tAm 作用在斷面的合力 dF Am 中線所圍成的面積 dF = τ avg ⋅ tds = q ⋅ ds
- 5. 扭轉角 Angle of Twist 以應變能分析，運用 Castigliano's Theorem L TL ds 2 AmG ∫ t φ= 2 TL s = 2 2 AmG ∑t
- 6. π 4 J = ( Ro − Ri4 ) 2 πDmt 3 ≅ 2πRmt = 3 4 TR τ= J T T = = 2πRmt 2 Amt 2 TL TL φ= = GJ 2πGRmt 3 TL πRm TL ds 2GAm ∫ t = ⋅ = ( 2G π Rm t 2 4 ) 2