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Research Papers

Finite Element Model for Hysteretic Friction Damping of Traveling Wave Vibration in Axisymmetric Structures

[+] Author and Article Information
X. W. Tangpong

Department of Mechanical Engineering and Applied Mechanics, North Dakota State University, Fargo, ND 58105

J. A. Wickert1

Department of Mechanical Engineering, Iowa State University, Ames, Iowa 50011wickert@isu.edu

A. Akay

Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213

Symbols in bold denote matrices or vectors.

The proportional damping matrices for the base (B) and damper (D) are taken as CB(D)=(2ζB(D)ΩB(D))KB(D), where ΩB is the base’s first flexible body natural frequency in the absence of contact. The modal damping ratios are taken illustratively as ζB=ζD=0.01%.

The tangential stiffness of the interface is estimated by the shear expression kF=GwDL(hD2), where G is the shear modulus of the damper’s material. The stiffness ratio becomes KFD=12(LhD)4(1+ν), with the Poisson’s ratio of the damper’s material taken as ν=0.3. In parameter studies of M*, only wD is considered to change, so that mD and kD vary with the same proportion.

With EB(D) being the elastic modulus, and ρB(D) being the mass density of the base’s (damper’s) material, the natural frequency ratio becomes γ=(hBhD)EBρD(EDρB). For simplicity, the mass ratio M* is held constant, and only hD and wD are varied to generate different γ values.

1

Corresponding author.

J. Vib. Acoust 130(1), 011005 (Nov 12, 2007) (7 pages) doi:10.1115/1.2775519 History: Received December 29, 2006; Revised May 22, 2007; Published November 12, 2007

A finite element method is developed to treat the steady-state vibration of two axisymmetric structures—a base substructure and an attached damper substructure—that are driven by traveling wave excitation and that couple through a spatially distributed hysteretic friction interface. The base substructure is representative of a rotating brake rotor or gear, and the damper is a ring affixed to the base under preload and intended to control vibration through friction along the interface. In the axisymmetric approximation, the equation of motion of each substructure is reduced in order to the number of nodal degrees of freedom through the use of a propagation constant phase shift. Despite nonlinearity and with contact occurring at an arbitrarily large number of nodal points, the response during sticking, or during a combination of sticking and slipping motions, can be determined from a low-order set of computationally tractable nonlinear algebraic equations. The method is applicable to element types for longitudinal and bending vibration, and to an arbitrary number of nodal degrees of freedom in each substructure. In two examples, friction damping of the coupled base and damper is examined in the context of in-plane circumferential vibration (in which case the system is modeled as two unwrapped rods), and of out-of-plane vibration (alternatively, two unwrapped beams). The damper performs most effectively when its natural frequency is well below the base’s natural frequency (in the absence of contact), and also when its natural frequency is well separated from the excitation frequency.

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Figures

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Figure 3

Nomenclature and illustration of the geometry for adjoining elements in a structure that has periodic boundary conditions and is excited by traveling wave vibration

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Figure 4

Nomenclature and illustration of the geometry for two structures, each having periodic boundary conditions, that couple through a spatially distributed friction interface. The base structure is excited by traveling wave excitation.

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Figure 5

Bilinear hysteresis response at the i-th contact element within the interface

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Figure 6

(a) Base structure and ring damper systems that are subjected to circumferential traveling wave excitation and (b) an idealized model comprising two unwrapped rods that have periodic boundary conditions and that couple through a spatially distributed friction interface

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Figure 8

Response of the base structure’s amplitude for transverse bending vibration. The frequency responses are shown at five levels of preload; M*=5, L∕hD=33.3, γ=4, n=1, N=30, and F=1.

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Figure 9

Phase difference between the base’s and damper’s responses at P=50; other parameters are as specified in Fig. 8

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Figure 10

Relative amplitude along the base/damper interface during transverse bending vibration. The frequency responses are shown at four levels of preload; other parameters are as specified in Fig. 8.

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Figure 1

Side view of an automotive disk brake rotor. The ring damper is affixed to the rotor’s periphery in a groove that is machined across the rotor’s pattern of cooling vanes.

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Figure 2

Measured collocated point frequency response functions for (a) an automotive brake rotor that incorporates a ring damper, as in Fig. 1, and (b) an otherwise identical rotor alone

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Figure 7

(a) Base structure and ring damper systems that are subjected to transverse traveling wave excitation and (b) an idealized model comprising two unwrapped beams that have periodic boundary conditions and that couple through a spatially distributed friction interface. Parameters wB(D) and hB(D) denote the widths and heights of the base’s and damper’s cross sections, respectively.

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Figure 11

Maximum amplitude of the base’s motion when the damper is attached, under constant excitation amplitude, as normalized to the resonant amplitude of the base alone: γ=4, L∕hD=33.3, n=1, and N=30

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Figure 12

Maximum amplitude of the base’s motion when the damper is attached, under constant excitation amplitude, as normalized to the resonant amplitude of the base alone: M*=10, n=1, N=30, and L∕hD=8.3, 12.5, 16.7, 33.3

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