Abstract
In-plane dynamics of rotating rings on elastic foundation is a topic of continuous research, especially in the field of tire dynamics. When the inner surface of a ring is connected to a stiff foundation, the through-thickness variation of radial and shear stress needs to be accounted for. This effect is often overlooked in the ring models proposed in the literature. In this paper, a new high order theory is developed for the in-plane vibration of rotating rings whose inner surface is connected to an immovable hub by distributed springs while the outer surface is stress-free. The high-order terms are chosen such that the boundary conditions at the inner and outer surfaces are satisfied at all times. Instability, which is usually overlooked in the literature, is predicted using the present model. Resonant speeds are investigated, at which modes appear as a stationary displacement pattern to a space-fixed observer. The exact satisfaction of boundary conditions at the inner and outer ring surfaces together with the through-thickness variation of the radial and shear stresses are shown to be of significant importance when the ring rotates at high speeds or is supported by relatively stiff foundation.
Highlights
Ρ is the mass density of the ring, E is the Young’s modulus, A is the cross-sectional area and I is the cross-sectional moment of inertia, and b is the out-of-plane width of the ring
The in-plane motions of a ring can either be considered as a plane strain or plane stress problem depending on the ratio b/h
The inner surface of the ring is connected by means of distributed radial and circumferential springs to an immovable axis
Summary
The radial and circumferential displacements of the ring are designated by w(z, θ, t) and u(z, θ, t), respectively. A rotating ring on an elastic foundation in which w0(θ, t) and u0(θ, t) are the radial and circumferential displacements of the middle surface, respectively. According to [1], the nonlinear strain-displacement relation for the circumferential strain εθ, the radial strain εr and the shear strain γθr of a differential element in the ring are given by εθ. The inner surface of the ring is connected by means of distributed radial and circumferential springs to an immovable axis. To obtain to characteristic equation from the linearised governing equations, it is assumed that the dynamic displacements are w0(θ, t) = R W einθ+iωt, u0(θ, t) = R U einθ+iωt, φ1(θ, t) = Φ einθ+iωt (14).
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