Erratum: ‘‘Transverse electromagnetic standing waves with E∥B’’ [Am. J. Phys. 56, 801–806 (1988)]

Abstract Natural frequencies and vibrating motions are determined in terms of the material and geometric properties of a radial tire modeled as a thin ring on an elastic foundation. Experimental checks of resonant frequencies show good agreement. Forced vibration solutions obtained are shown to consist of a superposition of resonant vibrations, each rotating around the tire at a rate depending on the mode number and the tire rotational speed. Theoretical rolling speeds that are upper bounds at which standing waves occur are determined and checked experimentally. Digital Fourier transform, transfer function, and modal analysis techniques used to determine the resonant mode shapes of a radial tire reveal that antiresonances are the primary transmitters of vibration to the tire axle.

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A Finite Element Analysis of the General Rolling Contact Problem for a Viscoelastic Rubber Cylinder

Tire Science and Technology ◽

10.2346/1.2148795 ◽

1988 ◽

Vol 16 (1) ◽

pp. 18-43 ◽

Cited By ~ 10

Author(s):

J. T. Oden ◽

T. L. Lin ◽

J. M. Bass

Keyword(s):

Finite Element ◽

Variational Principles ◽

Standing Waves ◽

Rolling Contact ◽

Finite Element Models ◽

Viscoelastic Cylinder ◽

Element Analysis ◽

Elastic Rubber ◽

Frictional Effects ◽

Rough Foundation

Abstract Mathematical models of finite deformation of a rolling viscoelastic cylinder in contact with a rough foundation are developed in preparation for a general model for rolling tires. Variational principles and finite element models are derived. Numerical results are obtained for a variety of cases, including that of a pure elastic rubber cylinder, a viscoelastic cylinder, the development of standing waves, and frictional effects.

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Formation of Standing Waves in Radial Tires

Tire Science and Technology ◽

10.2346/1.2150982 ◽

1984 ◽

Vol 12 (1) ◽

pp. 44-63 ◽

Cited By ~ 8

Author(s):

Y. D. Kwon ◽

D. C. Prevorsek

Keyword(s):

High Speed ◽

Critical Speed ◽

Unit Length ◽

Standing Waves ◽

Rolling Resistance ◽

Damping Coefficient ◽

Circular Membrane ◽

Critical Speeds ◽

Steel Cord ◽

The Individual

Abstract Radial tires for automobiles were subjected to high speed rolling under load on a testing wheel to determine the critical speeds at which standing waves started to form. Tires of different makes had significantly different critical speeds. The damping coefficient and mass per unit length of the tire wall were measured and a correlation between these properties and the observed critical speed of standing wave formation was sought through use of a circular membrane model. As expected from the model, desirably high critical speed calls for a high damping coefficient and a low mass per unit length of the tire wall. The damping coefficient is particularly important. Surprisingly, those tire walls that were reinforced with steel cord had higher damping coefficients than did those reinforced with polymeric cord. Although the individual steel filaments are elastic, the interfilament friction is higher in the steel cords than in the polymeric cords. A steel-reinforced tire wall also has a higher density per unit length. The damping coefficient is directly related to the mechanical loss in cyclic deformation and, hence, to the rolling resistance of a tire. The study shows that, in principle, it is more difficult to design a tire that is both fuel-efficient and free from standing waves when steel cord is used than when polymeric cords are used.

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