Brian F. Feeny

A Historical Review on Dry Friction and Stick-Slip Phenomena

This article gives a historical overview of structural and mechanical systems with friction. Friction forces between sliding surfaces arise due to complex mechanisms and lead to mathematical models which are highly nonlinear, discontinuous and nonsmooth. Humankind has a long history of magnificent usage of friction in machines, buildings and transportation. Regardless, our state of knowledge of the friction-influenced dynamics occurring in such systems as well as in our daily lives was, until recently, rather primitive. To represent our understanding of friction in nonlinear dynamics, we first trace examples from the earliest prehistoric technologies and the formulation of dissipation laws in mechanics. The work culminates with examples of friction oscillators and stick-slip. This review article contains 304 references.

Interpreting proper orthogonal modes of randomly excited vibration systems

Proper orthogonal modes (POMs) of displacements are interpreted for linear vibration systems under random excitation. Excitations are considered for which the Fourier transform is convergent, meaning that the input must have zero mean, and no sustained sinusoidal component. In such a case, the POMs in undamped discrete linear symmetric systems can represent linear natural modes if the mass distribution is known. POMs in one-dimensional distributed-parameter self-adjoint systems can approximately represent the linear normal modes if the mass distribution is known. Simulation examples are presented. Simulations show that these ideas are also applicable under light modal damping.

Identifying Coulomb and Viscous Friction from Free-Vibration Decrements

This study focuses on an algorithm for the simultaneous identification of Coulomb and viscous damping effects from free-vibration decrements in a damped linear single degree-of-freedom (DOF) mass-spring system. Analysis shows that both damping effects can indeed be separated. Numerical study of a combined-damping system demonstrates a perfect match between the simulation parameters and the estimated values. Experimental study includes two types of real systems. The method is applied to an experimental industrial bearing. Experimental results are compared with numerical simulations to illustrate the reliability of this method. An analysis provides conservative bounds on error estimates. An example of the effect of quantization error on the estimations is included.

Dynamical friction behavior in a forced oscillator with a compliant contact

Contact compliance, which may arise from elastic deformation near the contact point or in the surrounding structure, affects the dynamical friction behaviors in mechanical oscillators. An idealized model consisting ofa mass sliding harmonically on a massless compliant contact produces hysteresis in friction-velocity plots. Dynamical friction features, depending on the contact stillness, friction level, and the frequency and amplitude of oscillation, are predicted and quantified. Contact compliance can also lead to oscillations at the transition from slip to stick. Experiments and simulations verify the model and tie together phenomena of both continuous sliding and stick-slip.

Enhanced Proper Orthogonal Decomposition for the Modal Analysis of Homogeneous Structures

Proper orthogonal decomposition (POD) is studied in an effort to increase its applicability as a modal analysis tool. A modification is proposed to make better use of spatial resolution and to accommodate arbitrary spacing in the discretization. The theory for this modification is rooted in the discrete approximation of the integral orthogonality condition for continuous normal modes. The modified POD is applied to a finite element beam and an experimental beam sensed with accelerometers, and the resulting proper orthogonal modes (POMs) are compared to the theoretical modes of the beam. The POMs are used as a basis for decomposing the signal ensemble into proper modal coordinates. The proper modal coordinates are used to evaluate the POMs and to match modes with modal frequencies and damping.

A nonsmooth Coulomb friction oscillator

Abstract A forced Coulomb friction oscillator, whose frictional force is allowed to vary with displacement, is analyzed geometrically. The equation of motion for the oscillator is piecewise linear. We geometrically observe the nature of the flow in each region of solvability, and then see how these solutions interact at the boundary of the regions. The dynamics of the flow is viewed in terms of a map on the boundary between the regions. For chaotic motion, we geometrically construct the strange attractor, and show that its exact behavior is that of a one-dimensional map. The following dynamical properties arise from the nonsmooth nature of the Coulomb friction law: the flow may not be invertible; the flow may reach its attractor in finite time; the dimension of the attractor may be less than or equal to two; embeddings of an observable may not be diffeomorphic to the full phase flow.

Part 1: Dynamical Characterization of a Frictionally Excited Beam

The dynamics of an experimental frictionally excited beam areinvestigated. The friction is characterized and shown to involve contactcompliance. Beam displacements are approximated from strain gagesignals. The system dynamics are rich, including a variety of periodic,quasi-periodic and chaotic responses. Proper orthogonal decomposition isapplied to chaotic data to obtain information about the spatialcoherence of the beam dynamics. Responses for different parameter valuesresult in a different set of proper orthogonal modes. The number ofproper orthogonal modes that account for 99.99% of the signalpower is compared to the corresponding number of linear normal modes,and it is verified that the proper orthogonal modes are more efficientin capturing the dynamics.

Parametric Identification of Chaotic Systems

Parameters are identified in chaotic systems. Periodic orbits are first extracted from a chaotic set. The harmonic-balance method is applied to these periodic orbits, resulting in a linear equation in the unknown parameters, which can then be solved in the least squares sense. The idea is applied numerically to forced and autonomous systems. The effects of noise and errors in the periodic orbit extraction are outlined. The benefit of extracting several periodic orbits from the chaotic set is revealed.

Autocorrelation on Symbol Dynamics for a Chaotic Dry-Friction Oscillator

Abstract An autocorrelation function based on symbol dynamics is applied to a chaotic dry-friction oscillator to estimate the largest Lyapunov exponent. The friction problem is well suited for symbol dynamics since two distinct states of motion can be identified: sticking and slipping. In addition, the dynamics of the oscillator can be reduced to a non-invertible one-dimensional map, which has been studied in terms of binary symbol sequences. The study is done for an experimental oscillator and for a numerical model. The numerical result is compared to the Lyapunov exponent estimated from the continuous flow.

Identifying Coulomb and Viscous Friction in Forced Dual-Damped Oscillators

This paper presents a method for estimating Coulomb and viscous friction coefficients from responses of a harmonically excited dual-damped oscillator with linear stiffness. The identification method is based on existing analytical solutions of non-sticking responses excited near resonance. The method is applicable if the damping ratio of viscous component can be considered small. The Coulomb and viscous friction parameters can be extracted from two or more input-output amplitude pairs at resonance. The method is tested numerically and experimentally. Experimental results are cross checked with estimations from free-vibration decrements and also from friction measurements.