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Optimum FG Rotating Disk of Constant Mass: Lightweight and Economical alternatives Based on Limit Angular Speed

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Abstract

The operating range of permissible angular speeds defines the performance of rotating disks. The upper limit of elastic operating range is the maximum angular speed at which a disk can be rotated before material yielding sets and is called limit elastic speed. Limit elastic speed can be enhanced in two ways: first, by changing the geometry of the disks, and second, by changing the material. In mathematical formulation, the limit elastic speed is ascertained by reaching the von Mises stress, and as a result, in the present work, assuming radial displacement field, the formulation is carried out using the minimum potential energy principle. The approach used is the variational principle, and the solution has been achieved using Galerkin's error minimization theory. The von Mises stress is computed at various angular speeds, and the speed at which the equivalent stress coincides with the material's yield strength is the limit speed of such disks, and the corresponding location is the position of yield initiation in the disk. In the current study, the effect of different grading indices, aspect ratios, and material gradations (i.e., power law, exponential, and sigmoid gradation) on disk performance is investigated. For uniform and variable disk geometries of constant mass, limit angular speed analysis is performed, and the optimum material and geometry parameters are determined. Therefore, the results reveal solutions toward optimal performance, lightweight and economical functionally graded disks of various profiles, demonstrating a substantial improvement in the range of operating speeds.

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Abbreviations

a, b, r :

Inner, outer radii and radius.

c i :

Vector of unknown coefficients

h 0 :

Root thickness of the disk

n :

Grading parameter or grading index

V m, V c :

Volume fraction of metal and ceramic

ρ m, ρ c :

Density of metal and ceramic

E m, E c :

Elasticity modulus of metal and ceramic

E 0, ρ 0, σ y 0 :

Elasticity, density, and yield strength of the material at the base/root of the disk

σ yf :

Yield stress at any radius r

ρ f :

Density at any radius r

E f :

Elasticity modulus at any radius r

u :

Displacement field of the disk

U :

Strain energy of the disk

W :

Work potential of the disk

ε r , ε θ :

Radial and tangential strain, respectively

σ r, σ θ :

Radial and tangential stress, respectively

δ :

The variational operator

μ :

Poisson’s ratio

φ i :

Set of orthogonal polynomials

ξ :

Normalized radial coordinate

ξ y :

Location of yielding in the disk

ω :

Angular speed of the disk

Ω l :

Normalized limit elastic speed of the disk

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Authors and Affiliations

  1. Department of Mechanical Engineering, Chandigarh University, Mohali, Punjab, 140413, India

    Royal Madan

  2. Department of Mechanical Engineering, National Institute of Technology Raipur (C.G), 492010, Raipur, India

    Shubhankar Bhowmick

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  1. Royal Madan
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  2. Shubhankar Bhowmick
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Correspondence to Royal Madan.

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Madan, R., Bhowmick, S. Optimum FG Rotating Disk of Constant Mass: Lightweight and Economical alternatives Based on Limit Angular Speed. Iran J Sci Technol Trans Mech Eng 47, 1019–1033 (2023). https://doi.org/10.1007/s40997-022-00553-6

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