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An elastic bar of length L spins with angular velocity w about an axis, as shown in the figure below. The radial acceleration at a generic point x along the bar is a(x) = w²x. Due to this radial acceleration, the bar stretches along x with displacement function u(x). The displacement u(x) is governed by the following equations: d ₫ (σ(x)) + pa(x) = 0 PDE dx |σ(x) = Edu dx Hooke's law (1) where σ(x) is the axial stress in the rod, p is the mass density, and E is the (constant) Young's modulus. The bar is pinned on the rotation axis at x = 0, and it is free at x = L. == ω → X L

An elastic bar of length L spins with angular velocity w about an axis, as shown in the figure below. The radial acceleration at a generic point x along the bar is a(x) = w²x. Due to this radial acceleration, the bar stretches along x with displacement function u(x). The displacement u(x) is governed by the following equations: d ₫ (σ(x)) + pa(x) = 0 PDE dx |σ(x) = Edu dx Hooke's law (1) where σ(x) is the axial stress in the rod, p is the mass density, and E is the (constant) Young's modulus. The bar is pinned on the rotation axis at x = 0, and it is free at x = L. == ω → X L

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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An elastic bar of length L spins with angular velocity ω about an axis, as shown in the figure below. The radial acceleration at a generic point x along the bar is a(x) = ω 2 x. Due to this radial acceleration, the bar stretches along x with displacement function u(x). The displacement u(x) is governed by the following equations: ( d dx (σ(x)) + ρa(x) = 0 PDE σ(x) = E du dx Hooke’s law (1) where σ(x) is the axial stress in the rod, ρ is the mass density, and E is the (constant) Young’s modulus. The bar is pinned on the rotation axis at x = 0, and it is free at x = L.

 

Determine:
1. Appropriate BCs for this physical problem.
2. The displacement function u(x).
3. The stress function σ(x).

An elastic bar of length L spins with angular velocity w about an axis, as shown in the figure below. The radial acceleration at a generic point x along the bar is a(x) = w²x. Due to this radial acceleration, the bar stretches along x with displacement function u(x). The displacement u(x) is governed by the following equations: d ₫ (σ(x)) + pa(x) = 0 PDE dx |σ(x) = Edu dx Hooke's law (1) where σ(x) is the axial stress in the rod, p is the mass density, and E is the (constant) Young's modulus. The bar is pinned on the rotation axis at x = 0, and it is free at x = L. == ω → X L
Transcribed Image Text:An elastic bar of length L spins with angular velocity w about an axis, as shown in the figure below. The radial acceleration at a generic point x along the bar is a(x) = w²x. Due to this radial acceleration, the bar stretches along x with displacement function u(x). The displacement u(x) is governed by the following equations: d ₫ (σ(x)) + pa(x) = 0 PDE dx |σ(x) = Edu dx Hooke's law (1) where σ(x) is the axial stress in the rod, p is the mass density, and E is the (constant) Young's modulus. The bar is pinned on the rotation axis at x = 0, and it is free at x = L. == ω → X L
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