Jon Juel Thomsen

Analytical approximations for stick-slip vibration amplitudes

Abstract The classical “mass-on-moving-belt” model for describing friction-induced vibrations is considered, with a friction law describing friction forces that first decreases and then increases smoothly with relative interface speed. Approximate analytical expressions are derived for the conditions, the amplitudes, and the base frequencies of friction-induced stick–slip and pure-slip oscillations. For stick–slip oscillations, this is accomplished by using perturbation analysis for the finite time interval of the stick phase, which is linked to the subsequent slip phase through conditions of continuity and periodicity. The results are illustrated and tested by time-series, phase plots and amplitude response diagrams, which compare very favorably with results obtained by numerical simulation of the equation of motion, as long as the difference in static and kinetic friction is not too large.

Slow Effects of Fast Harmonic Excitation for Elastic Structures

High-frequency excitation may affect the ‘slow’ behavior of a dynamical system. For example, equilibria may move, disappear, or gain or loose stability. We consider such slow effects of fast excitation for a simple mechanical system that incorporates features of many engineering structures. The study is intended to contribute to the general understanding of periodically excited linear and nonlinear systems, as well as to the current attempts to utilize high-frequency excitation for altering the low-frequency properties of structures.

Stiffening Effects of High-Frequency Excitation: Experiments for an Axially Loaded Beam

According to theoretical predictions one can change the effective stiffness or natural frequency of an elastic structure by employing harmonic excitation of very high frequency. Here we examine this effect for a hinged-hinged beam subjected to longitudinal harmonic excitation. A simple analytical expression is presented, that relates the effective natural frequencies of the beam to the intensity of harmonic excitation. Experiments performed with a laboratory beam confirm the general tendency of this prediction, though there are discrepancies that cannot be explained in the framework of the linear Galerkin-discretized beam model.

Predicting vibration-induced displacement for a resonant friction slider

Abstract A mathematical model is set up to quantify vibration-induced motions of a slider, sandwiched between friction layers with different coefficients of friction, and equipped with an imbedded resonator that oscillates at high frequency and small amplitude. This model is highly nonlinear, involving non-smooth functions with strong harmonic excitation terms. The method of averaging is extended to hold for systems of this class, and used to derive approximate expressions for predicting average velocities of the slider. These expressions are shown to produce results that agree very well with numerical integration of the full equations of motion. The expressions are used to estimate and explain the influence of system parameters.

Chaotic vibrations of non-shallow arches

Non-shallow arches inherently possess a nearly two-to-one internal resonance between the lowest modes of symmetric and antisymmetric vibration. This implies that non-linear modal interaction may entirely dominate the dynamic response, even at extremely small excitation levels. In this paper the effects of such non-linearities are studied by perturbation analysis and numerical simulations. Special emphasis is laid on chaotic vibrations, which are shown to occur for excitation levels and frequencies occupying significant areas of the primary region of dynamic instability. Thus, this is a case of a structure in widespread practical use, which may display unpredictable chaotic behaviour not very far from normal operating conditions. Evidence for chaotic motion is given through Poincare sections, frequency spectres and Lyapunov exponents. The routes to chaos are shown to include quasi-periodic break-up, intermittency and long transients.

Modelling human tibia structural vibrations.

Mode shapes and natural frequencies of human long bones play an important role in the interpretation, prediction and control of their dynamic response to external mechanical loads. This paper describes an experimental and theoretical study of free vibrations in an excised human tibia. Experimentally, seven tibial natural frequencies in the range 0-3 kHz were identified through measured structural transfer functions. Theoretically, a beam type Finite Element model of a human tibia is suggested. Unknown parameters in this model are determined by a Bayesian parameter estimation approach, by which very fine model/observation-accordance was achieved with realistic parameter estimates. A sensitivity analysis of the model confirms that the human tibia in a vibrational sense is more uniform than its complicated geometry would immediately suggest. Accordingly, two simple tibia models are identified, based on uniform beam theory with inclusion of shear deformations.

Theories and experiments on the stiffening effect of high-frequency excitation for continuous elastic systems

Abstract One effect of strong mechanical high-frequency excitation may be to apparently “stiffen” a structure, a well-described phenomenon for discrete systems. The present study provides theoretical and experimental results on this effect for continuous elastic structures. A laboratory experiment is set up for demonstrating and measuring the stiffening effect in a simple setting, in the form of a horizontal piano string subjected to longitudinal high-frequency excitation at the clamped base and free at the other end. A simplest possible theoretical model is set up and analyzed using a hierarchy of three approximating theories, each providing valuable insight. One of these is capable of predicting the vertical string lift due to stiffening in terms of simple expressions, with results that agree very well with experimental measurements for a wide range of conditions. It appears that resonance effects cannot be ignored, as was done in a few related studies—unless the system has very low modal density or heavy damping; thus first-order consideration to resonance effects is included. Using the specific example with experimental support to put confidence on the proposed theory, expressions for predicting the stiffening effect for a more general class of continuous systems in differential operator form are also provided.

Chelomei's pendulum explained

Chelomeis pendulum has a sliding disc on its rod, and is mounted on a support that vibrates vertically with small amplitude and high frequency. In 1982, V. N. Chelomei demonstrated experimentally that a configuration can be stable where the pendulum points against gravity, and the disc ‘floats’ on the rod. This phenomenon has never been satisfactorily explained. The present work considers why and where the disc floats. It suggests that the phenomenon is caused by resonant flexural rod vibrations, which are excited through small symmetry-breaking imperfections, such as a small deviation from perfectly vertical excitation. This hypothesis is supported by laboratory experiments, and by perturbation analysis and numerical analysis of a new mathematical three-degree-of-freedom model of the system. Simple analytical expressions for the prediction of stable states of the system are set up, providing frequency responses that agree closely with numerical simulation, and agree qualitatively with experimental observations.

Vibration-Induced Displacement Using High-Frequency Resonators and Friction Layers

A mathematical model is set up to quantify vibration-induced motions of a slider with an imbedded resonator. A simple approximate expression is presented for predicting average velocities of the slider, agreeing fairly well with numerical integration of the full equations of motion. The simple expression can be used to the estimate influence of system parameters, and to plan and interpret laboratory experiments.

Post-critical behavior of Beck's column with a tip mass

This study examines how a tip mass with rotary inertia affects the stability of a follower-loaded cantilevered column. Using nonlinear modeling and perturbation analysis, expressions are set up for determining the stability of the straight column and the amplitude of post-critical flutter oscillations. Bifurcation diagrams are given, showing how the vibration amplitude changes with follower load and other parameters. These results agree closely with numerical simulation. It is found that sufficiently large values of tip mass rotary inertia can change the primary bifurcation from supercritical into subcritical. This can imply very large motions for follower loads just beyond critical, contrasting the finite amplitude motions accompanying supercritical bifurcations. Also, the straight column may be destabilized by a sufficiently strong disturbance at loads far below the value of critical load predicted by linear theory. A similar change in bifurcation is found to occur with increased external (as compared to internal) damping, and with a shortening in column length. These effects are not revealed by linear modeling and analysis, which may consequently fail to predict even qualitatively the real critical load for a column with tip mass.