A simplified model to explore the root cause of stick-slip vibrations in drilling systems with drag bits

In this paper a study of the self-excited stick-slip oscillations of a rotary drilling system with a drag bit, using a discrete model that takes into consideration the axial and torsional vibration modes of the system, is described. Coupling between these two vibration modes takes place through a bit-rock interaction law, which accounts for both the frictional contact and the cutting processes. The cutting process introduces a delay in the equations of motion that is responsible for the existence of self-excited vibrations, which can degenerate into stick-slip oscillations and/or bit bouncing under certain conditions. From analysis of this new model it is concluded that the experimentally observed decrease of the reacting torque with the angular velocity is actually an expression of the system response, rather than an intrinsic rate dependence of the interface laws between the rock and the drill bit, as is commonly assumed.     

Audio signal modulation caused by self-excited vibrations of magnetic tape

The longitudinal vibrations of a magnetic tape can cause distortion of a reproduced signal. The dependence of the distortion on the various system parameters has been investigated. The self-excited vibrations of a tape, caused by friction against the heads and guides, have been analyzed. It can be stated on the basis of the relationships obtained both when the self-excited vibrations can be generated and what parameters the amplitude of these vibrations depends on. The tape path can be optimized on this basis. The amplitude of vibration is inversely proportional to the natural frequency. The fundamental natural mode is the most important because of the frequency range of a tape recorder. Stability of multi-frequency cycles has been tested. The internal loss limits the amplitude of the vibrations and prevents generation of higher natural modes. The tests have been made on a high fidelity reel tape recorder. Sidebands caused by vibrations of the tape have been observed. An additional scrape flutter idler does not eliminate these vibrations. The self-excited vibrations are still generated but with lower frequency and higher amplitude. The sidebands become closer to the carrier signal and because of this are better masked by the carrier frequency.     

Stability and transient dynamics of a propeller–shaft system as induced by nonlinear friction acting on bearing–shaft contact interface

This paper investigates the friction-induced instability and the resulting self-excited vibration of a propeller–shaft system supported by water-lubricated rubber bearing. The system under consideration is modeled with an analytical approach by involving the nonlinear interaction among torsional vibrations of the continuous shaft, tangential vibrations of the rubber bearing and the nonlinear friction acting on the bearing–shaft contact interface. A degenerative two-degree-of-freedom analytical model is also reasonably developed to characterize system dynamics. The stability and vibrational characteristics are then determined by the complex eigenvalues analysis together with the quantitative analysis based on the method of multiple scales. A parametric study is conducted to clarify the roles of friction parameters and different vibration modes on instabilities; both the graphic and analytical expressions of instability boundaries are obtained. To capture the nature of self-excited vibrations and validate the stability analysis, the nonlinear formulations are numerically solved to calculate the transient dynamics in time and frequency domains. Analytical and numerical results reveal that the nonlinear coupling significantly affects the system responses and the bearing vibration plays a dominant role in the dynamic behavior of the present system.     

Squeal and chatter phenomena generated in a mountain bike disc brake

This paper examines squeal and chatter phenomena generated experimentally in mountain bike disc brakes. There are two kinds of frictional self-excited vibrations in the bike disc brakes, called squeal with frequency of 1 kHz and chatter with frequency of 500 Hz. In order to reproduce the squeal and chatter, a bench test apparatus using an actual bike was set up to determine the associated frequency characteristics experimentally. The results show the frequencies to be independent of pad temperature and disc rotating speed. Squeal is shown to be in-plane vibration in the direction of the disc surface which is caused by the frictional characteristics having negative slope with respect to the relative velocity in the vibrating system, which includes brake unit, spokes and hub. Chatter is generated within a limited high temperature region. Again, it is frictional vibration in which the squeal and out-of-plane vibration of the disc due to Coulomb friction combine through the internal resonance relation between in-plane and out-of-plane nonlinear vibration caused by the temperature increase of the disc during braking.     

On the limiting of vibration amplitudes by a sequential friction-spring element

The possibility of using an additional sequentially connected friction spring element in order to reduce vibration amplitudes both for the self-excited oscillations and for the forced vibrations is discussed in the paper. The analysis is based on the averaging technique for systems with “slave variables” and demonstrates two main effects: damping during slipping in the additional element and fast switching between different natural frequencies due to alternating sticking/slipping phases. Analytic predictions for the oscillations’ amplitudes are obtained as steady state solutions of the equations governing slow motions of the system. The obtained analytic results enable optimal choice of friction in order to achieve maximal damping effect in case of the forced vibrations. The reasonable choice of the friction by the self-excited vibrations is a compromise between the acceptable amplitude and the robustness of the corresponding limit cycle. The asymptotic results are confirmed by numeric simulations.     

Micro/macro-slip damping in beams with frictional contact interface

Friction in contact interfaces of assembled structures is the prime source of nonlinearity and energy dissipation. Determination of the dissipated energy in an assembled structure requires accurate modeling of joint interfaces in stick, micro-slip and macro-slip states. The present paper proposes an analytical model to evaluate frictional energy loss in surface-to-surface contacts. The goal is to develop a continuous contact model capable of predicting the dynamics of friction interface and dissipation energy due to partial slips. To achieve this goal, the governing equations of a frictional contact interface are derived for two distinct contact states of stick and partial slip. A solution procedure to determine stick–slip transition under single-harmonic excitations is derived. The analytical model is verified using experimental vibration test responses performed on a free-frictionally supported beam under lateral loading. The theoretical and experimental responses are compared and the results show good agreements between the two sets of responses.     

Experimental and numerical investigation of friction-induced vibration of a beam-on-beam in contact with friction

The vibrations generated by friction are responsible for various noises such as squealing, squeaking and chatter. Although these phenomena have been studied for a long time, it is not well-understood. In this study, an experimental and numerical study of friction-induced vibrations of a system composed of two beams in contact is proposed. The experimental system exhibits periodic steady state vibrations of different types. To model and understand this experimental vibratory phenomenon, complex eigenvalue and dynamic transient analyses are performed. In the linear complex eigenvalue analysis, flutter instability occurs via the coalescence of two eigenmodes of the system. This linear study provides an accurate value of the experimental frequency of vibration. To understand what happens physically during friction-induced instability, a dynamic transient analysis that takes account of the non-linear aspect of a frictional contact is performed. In this analysis, friction-induced instability is characterized by self-sustained vibrations and by stick, slip and separation zones occurring at the surface of the contact. The results stemming from this analysis show that good correlation between numerical and experimental vibrations can be obtained (in time and frequency domains). Moreover, time domain simulations permit understanding the physical phenomena involved in two different vibratory behaviours observed experimentally.     

Comments on “finite element model for dynamic analysis of Timoshenko beam”

An analysis is presented of the effect of dry friction on vibrations of two self-excited systems. Attention is directed to cases where practical quenching of self-excited vibrations cannot be achieved merely by the action of linear additional damping. It is shown that at correct tuning of the systems discussed, a combination of linear additional damping with dry friction is so effective in limiting their amplitudes that self-excited vibrations may be regarded as practically suppressed.     

A note on safety-relevant vibrations induced by brake squeal

Brake squeal is mostly considered as a comfort problem only but there are cases in which self-excited vibrations of the brake system not only cause an audible noise but also result in safety-relevant failures of the system. In particular this can occur if lightweight design rims having very low damping are used. Considering the special conditions of lightweight design rims, a minimal model for safety-relevant self-excited vibrations of brake systems is presented. It is shown that most of the knowledge emanated from investigations of the comfort problem can be used to understand and avoid safety-relevant failures of the brake system.     

Method of dimensionality reduction in contact mechanics and tribology. Heterogeneous media

The method of dimensionality reduction in contact mechanics is based on the mapping of three-dimensional contact problems onto one-dimensional systems. The method was developed and verified for homogeneous media. The present paper discusses the possibilities of generalizing the method to heterogeneous media with an illustration of basic ideas on the example of an elastic medium with a coating. The proposed method is easy to extend to media with lateral heterogeneity or to arbitrary heterogeneous media; this is illustrated by considering linear non-dissipative media. The proposed method can be used for calculation of contact and frictional interactions and for interpretation of experiments on indentation of heterogeneous media.     

The coupled transverse vibrations of a spinning membrane disk with a central hub

The results of an investigation of the effect of hub coupling on the transverse free vibrations of a spinning membrane disk are presented. The equations of motion are formulated by an energy method and Rayleigh-Ritz procedure. It is found by using assumed modal distributions consisting of the normal modes for the spinning membrane with a rigidly clamped hub that the problem partitions into sub-problems associated with (i) symmetric vibrations involving only the generalized coordinates from the symmetric input modes, (ii) unsymmetric vibrations involving only the input modes with one nodal diameter, and (iii) unsymmetric vibrations involving input modes with more than one nodal diameter. The degree of hub coupling is found to depend primarily on the ratio of the mass of the membrane to the mass of the disk and on the ratio of the radius of the hub to the radius of the disk.     

Interface solution for writing-induced nano-deformation of slider body

Writing-induced nano-deformation of slider body becomes a big concern when the mechanical spacing between the head and disk is continuously reduced to achieve higher magnetic recording areal density. Reduced head–disk spacing increases the risk of head/disk contact and causes the thermal instability in head–disk interface (HDI). This paper reports authors’ efforts towards exploration of interface solutions for writing-induced instability in ultra-low head–disk spacing magnetic system. Multi-shallow step structure with optimized rail position is analyzed and a new femto slider with such structure is explored. The results of numerical simulation indicate that the multi-shallow step structure is an effective approach in reducing the flying height change caused by the writing-induced nano-deformation of the slider body.     

Transverse vibration of a rotor system driven by a Cardan joint

The transverse vibration of a rotor system driven by a Cardan joint is analyzed and the effect of the transmitted torque on the dynamic stability of the system evaluated. As a result of the analysis, the following facts are proved: when the driving shaft and driven shaft (rotor shaft) are included, both parametric and self-excited vibrations arise due to transmitted torque; asymmetrical stiffness of the rotor supports has the effect of stabilizing this self-excited vibration.     

Polymer cover induced self-excited vibrations of nipped rolls

As a result of increased speeds, the dynamic instability of rotatory machines including polymer-covered nipped rolls has grown. The instability originates from the viscoelastic behavior of the covers and leads to strong barring vibrations, which limit the operating speed of many machines. In this work, the self-excited vibrations of a nipped two-roll system with a polymer cover on the other roll are investigated using an analytical model developed for the roll system. The viscoelastic properties of the cover are accounted for by the standard linear solid (SLS) model. The numerical results display wave-like roll cover deformation patterns, separate instability regions of the system and moving wave patterns near the resonances. The roll system is unstable when the excitation frequency of the polygonal cover deformation lies in the vicinity of the higher eigenfrequency of the system. By using a speed-up ramp, it is shown that at high speeds the instability regions may become too wide and unstable to be crossed in industrial machines. An experiment was carried out, and a good agreement is found between the numerical and experimental results.     

Modal Amplitude Stability Analysis and its application to brake squeal

In the present study, a new approach is proposed to predict the occurrence of squeal in brake systems. This strategy, called Modal Amplitude Stability Analysis (MASA), is based on the calculation of the first harmonic state-space system of nonlinear original equations using a specific linearization of the nonlinear contact forces at the frictional interfaces. An estimation of the occurrence and generation of increasing self-excited vibration is proposed on the basis of monitoring and the evolution of the real parts of the dynamic system considered as a function of modal amplitudes.     

Vibration of a circular disk as a test method for damping characteristics of constrained layer material

The theoretical analysis of free vibrations of a disk of constrained layer sandwich material is considered. The results enable the real and imaginary parts of the shear parameter to be determined from experimental data. A correction for a small mass attached to the centre of a disk is introduced. An example of test results illustrates the method. An exact method involving numerical solution of the governing differential equation has been checked by approximate formulae based on potential energy of deformation.     

Tuneable vibration absorber design to suppress vibrations: An application in boring manufacturing process

Dynamic vibration absorbers are used to reduce the undesirable vibrations in many applications such as electrical transmission lines, helicopters, gas turbines, engines, bridges, etc. Tuneable vibration absorbers (TVA) are also used as semi-active controllers. In this paper, the application of a TVA for suppression of chatter vibrations in the boring manufacturing process is presented. The boring bar is modeled as a cantilever Euler–Bernoulli beam and the TVA is composed of mass, spring and dashpot elements. In addition, the effect of spring mass is considered in this analysis. After formulation of the problem, the optimum specifications of the absorber such as spring stiffness, absorber mass and its position are determined using an algorithm based on the mode summation method. The analog-simulated block diagram of the system is developed and the effects of various excitations such as step, ramp, etc. on the absorbed system are simulated. In addition, chatter stability is analyzed in dominant modes of boring bar. Results show that at higher modes, larger critical widths of cut and consequently more material removal rate (MRR) can be achieved. In the case of self-excited vibration, which is associated with a delay differential equation, the optimum absorber suppresses the chatter and increases the limit of stability.