Pole assignment of friction-induced vibration for stabilisation through state-feedback control

Friction-induced self-excited linear vibration is often governed by a second-order matrix differential equation of motion with an asymmetric stiffness matrix. The asymmetric terms are product of friction coefficient and the normal stiffness at the contact interface. When the friction coefficient becomes high enough, the resultant vibration becomes unstable as frequencies of two conjugate pairs of complex eigenvalues (poles) coalesce (when viscous damping is low).This short paper presents a receptance-based inverse method for assigning complex poles to second-order asymmetric systems through (active) state-feedback control of a combination of active stiffness, active damping and active mass, which is capable of assigning negative real parts to stabilise an unstable system.     

Passive-active vibration isolation systems to produce zero or infinite dynamic modulus: theoretical and conceptual design strategies

The application of mechanical springs connected in parallel and/or in series with active springs can produce dynamical systems characterised by infinite or zero value stiffness. This mathematical model is extended to more general cases by examining the dynamic modulus associated with damping, stiffness and mass effects. This produces a theoretical basis on which to design an isolation system with infinite or zero dynamic modulus, such that stiffness and damping may have infinite or zero values. Several theoretical designs using a mixture of passive and active systems connected in parallel and/or in series are proposed to overcome limitations of feedback gain experienced in practice to achieve an infinite or zero dynamic modulus. It is shown that such systems can be developed to reduce the weight supported by active actuators as demonstrated, for example, by examining suspension systems of very low natural frequency or with a very large supporting stiffness or with a viscous damper or a self-excited vibration oscillator. A more general system is created by combining these individual systems allowing adjustment of the supporting stiffness and damping using both displacement and velocity feedback controls. Frequency response curves show the effects of active feedback control on the dynamical behaviour of these systems. The theoretical design strategies presented can be applied to design feasible hybrid vibration control systems displaying increased control performance.     

Dynamic stiffness vibration analysis of an elastically connected three-beam system

An exact dynamic stiffness method is developed for predicting the free vibration characteristics of a three-beam system, which is composed of three non-identical uniform beams of equal length connected by innumerable coupling springs and dashpots. The Bernoulli-Euler beam theory is used to define the beams’ dynamic behaviors. The dynamic stiffness matrix is formulated from the general solutions of the basic governing differential equations of a three-beam element in damped free vibration. The derived dynamic stiffness matrix is then used in conjunction with the automated Muller root search algorithm to calculate the free vibration characteristics of the three-beam systems. The numerical results are obtained for two sets of the stiffnesses of springs and a large variety of interesting boundary conditions.     

RESPONSE OF NON-LINEAR DISSIPATIVE SHOCK ISOLATORS

In this paper, a simple technique combining the straightforward perturbation method with Laplace transform has been developed to determine the transient response of a single degree-of-freedom system in the presence of non-linear, dissipative shock isolators. Analytical results are compared with those obtained by numerical integration using the classical Runge–Kutta method. Three types of input base excitations, namely, the rounded step, the rounded pulse and the oscillatory step are considered. The effects of nonlinear damping on the response are discussed in detail. Both the positive and negative coefficients of the nonlinear damping term have been considered. It has been shown that a critical value of the positive coefficient maximizes the peak values of relative and absolute displacements. This is true for any power-law damping force with an index greater than 1. On the other hand, the overall performance of a shock isolator improves if the nonlinear damping term is symmetric and quadratic with a negative coefficient.     

Weakly dissipative dust acoustic solitons in the presence of superthermal particles

An investigation of the linear and non‐linear properties of low‐frequency electrostatic (dust acoustic) waves in a collisional dusty plasma with negative dust grains, Maxwellian electrons, and κ ‐distributed ions is carried out. Low dust–neutral collisions accounting for dissipation (wave damping effect) is considered. The linear properties of dust acoustic excitations are discussed for varying values of relevant plasma parameters. It is shown that large wavelengths (beyond a critical value) are overdamped. In the limit of low dust–neutral collision rate, we have derived a damped Korteweg de Vries (KdV) equation by using the reductive perturbation technique. Supplemented by vanishing boundary conditions, the time‐varying solution of damped KdV equation leads to a weakly dissipative negative potential soliton. The soliton evolution with the damping parameter and other physical plasma parameters (superthermality, dust concentration, ion temperature) is delineated.     

Simplified models of the vibration of mannequins in car seats

A simplified two-dimensional modelling approach to predict the vibration response of mannequin occupied car seats about a static settling point is demonstrated to be feasible. The goal of the research is to develop tools for car seat designers. The two-dimensional model, consisting of interconnected masses, springs and dampers is non-linear due to geometric effects but, under the excitations considered, the model behaviour is linear. In this approach to modelling, the full system is initially broken down into subsystems, and experiments are conducted with subsystems to determine approximate values for the stiffness and damping parameters. This approach is necessary because of the highly non-linear behaviour of foam where stiffness changes with compression level, and because the simplified model contains more structure than is necessary to model the relatively simple measured frequency response behaviour, thus requiring a good initial starting point from which to vary parameters. A detailed study of the effects of changing model parameters on the natural frequencies, the mode shapes and resonance locations in frequency response functions is given, highlighting the influence of particular model parameters on features in the seat-mannequin system's vibration response. Reasonable qualitative as well as good quantitative agreement between experimental and simulation frequency response estimates is obtained. In particular, the two-dimensional motions at the peaks in the frequency response, a combination of up and down and rotational behaviour is predicted well by the model. Currently research is underway to develop a similar model with non-linear springs, surface friction effects and viscoelastic elements, that predicts the static settling point, a necessary step to aid in the subsystem modelling stage in this dynamic modelling approach.     

Negative stiffness-induced extreme viscoelastic mechanical properties: stability and dynamics

Use of negative stiffness inclusions allows one to exceed the classic bounds upon overall mechanical properties of composite materials. We here analyse discrete viscoelastic ‘spring’ systems with negative stiffness elements to demonstrate the origin of extreme properties, and analyse the stability and dynamics of the systems. Two different models are analysed: one requires geometrical nonlinear analysis with pre-load as a negative stiffness source and the other is a linearized model with a direct application of negative stiffness. Material linearity is assumed for both models. The metastability is controlled by a viscous element. In the stable regime, extreme high mechanical damping tan?δ can be obtained at low frequency. In the metastable regime, singular resonance-like responses occur in tan?δ. The pre-stressed viscoelastic system is stable at the equilibrium point with maximal overall compliance and is metastable when tuned for maximal overall stiffness. A reversal in the relationship between the magnitude of complex modulus and frequency is also observed. The experimental observability of the singularities in tan?δ is discussed in the context of designed composites and polycrystalline solids with metastable grain boundaries.     

Structure of resonances and formation of stationary points in symmetrical chains of bilinear oscillators

Dynamics of strongly nonlinear systems can in many cases be modelled by bilinear oscillators, which are the oscillators whose springs have different stiffnesses in compression and tension. This underpins the analysis of a wide range of phenomena, from oscillations of fragmented structures, connections and mooring lines to deformation of geological media. Single bilinear oscillators were studied previously and the presence of multiple resonances both super- and sub-harmonic was found. Less attention was paid to systems of multiple bilinear oscillators that describe many natural and engineering processes such as for example the behaviour of fragmented solids. Here we fill this gap concentrating on the simplest case – 1D symmetrical chains of bilinear oscillators. We show that the presence and structure of resonances in a symmetric chain of bilinear oscillators with fixed ends depends upon the number of oscillating masses. Two elementary chains act as the basic ones: a single mass bilinear chain (a mass connected to the fixed points by two bilinear springs) that behaves as a linear oscillator with a single resonance and a two mass chain that is a coupled bilinear oscillator (two masses connected by three bilinear springs). The latter has multiple resonances. We demonstrate that longer chains either do not have resonances or get decomposed, in the resonance, into either the single mass or two mass elementary chains with stationary masses in between. The resonance frequencies are inherited from the basic chains of decomposition. We show that if the number of masses is odd the chain can be decomposed into the single mass bilinear chains separated by stationary masses. It then inherits the resonances of the single mass bilinear chain. The chains with the number of masses minus 2 divisible by 3 can be decomposed into the two mass bilinear chains separated by stationary masses and inherit the resonances of the two mass chains. The chains whose lengths satisfy both criteria (such as chains with 5, 11, 17 … masses) allow both types of resonances.     

Exploring the performance of a nonlinear tuned mass damper

We explore the performance of a nonlinear tuned mass damper (NTMD), which is modeled as a two degree of freedom system with a cubic nonlinearity. This nonlinearity is physically derived from a geometric configuration of two pairs of springs. The springs in one pair rotate as they extend, which results in a hardening spring stiffness. The other pair provides a linear stiffness term. We perform an extensive numerical study of periodic responses of the NTMD using the numerical continuation software AUTO. In our search for optimal design parameters we mainly employ two techniques, the optimization of periodic solutions and parameter sweeps. During our investigation we discovered a family of detached resonance curves for vanishing linear spring stiffness, a feature that was missed in an earlier study. These detached resonance response curves seem to be a weakness of the NTMD when used as a passive device, because they essentially restore a main resonance peak. However, since this family is detached from the low-amplitude responses there is an opportunity for designing a semi-active device.     

Modeling of automotive drum brakes for squeal and parameter sensitivity analysis

Many fundamental studies have been conducted to explain the occurrence of squeal in disc and drum brake systems. The elimination of brake squeal, however, still remains a challenging area of research. Here, a numerical modeling approach is developed for investigating the onset of squeal in a drum brake system. The brake system model is based on the modal information extracted from finite element models for individual brake components. The component models of drum and shoes are coupled by the shoe lining material which is modeled as springs located at the centroids of discretized drum and shoe interface elements. The developed multi degree of freedom coupled brake system model is a linear non-self-adjoint system. Its vibrational characteristics are determined by a complex eigenvalue analysis. The study shows that both the frequency separation between two system modes due to static coupling and their associated mode shapes play an important role in mode merging. Mode merging and veering are identified as two important features of modes exhibiting strong interactions, and those modes are likely candidates that lead to coupled-mode instability. Techniques are developed for a parameter sensitivity analysis with respect to lining stiffness and the stiffness of the brake actuation system. The influence of lining friction coefficient on the propensity to squeal is also discussed.     

The use of asymptotic modelling in vibration and stability analysis of structures

The use of springs with very large stiffness to model constraints in vibratory systems has been a popular approach to overcome the limitations on the choice of admissible functions in the Rayleigh-Ritz method. The maximum possible error resulting from this asymptotic modelling can be determined by using positive and negative stiffness values, or in general terms using positive and negative penalty functions. This paper illustrates how this method could be used to determine the critical loads of structures.     

Note on the Stability of a Slowly Rotating Timoshenko Beam with Damping

This paper continues the senior author's previous investigation of the slowly rotating Timoshenko beam in a horizontal plane whose movement is controlled by the angular acceleration of the disk of the driving motor into which the beam is rigidly clamped. It was shown before that this system preserves the total energy. We consider the problem of stability of the system after introducing a particular type of damping. We show that the energy of only part of the system vanishes. We illustrate obtained solution with the critical case of the infinite value of the damping coefficient.     

Physical mechanism of the (tri)critical point generation

We discuss some ideas resulting from a phenomenological relation recently declared between the tension of string connecting the static quark-antiquark pair and surface tension of corresponding cylindrical bag. This relation analysis leads to the temperature of vanishing surface tension coefficient of the QGP bags at zero baryonic charge density as T _?? = 152.9 ± 4.5 MeV. We develop the view point that this temperature value is not a fortuitous coincidence with the temperature of (partial) chiral symmetry restoration as seen in the lattice QCD simulations. Besides, we argue that T _?? defines the QCD (tri)critical endpoint temperature and claim that a negative value of surface tension coefficient recently discovered is not a sole result but is quite familiar for ordinary liquids at the supercritical temperatures.     

Monte Carlo analysis of critical properties of the two-dimensional randomly site-diluted Ising model via Wang-Landau algorithm

The influence of random site dilution on the critical properties of the two-dimensional Ising model on a square lattice was explored by Monte Carlo simulations with the Wang-Landau sampling. The lattice linear size was L=20-120 and the concentration of diluted sites q=0.1,0.2,0.3. Its pure version displays a second-order phase transition with a vanishing specific heat critical exponent α, thus, the Harris criterion is inconclusive, in that disorder is a relevant or irrelevant perturbation for the critical behaviour of the pure system. The main effort was focused on the specific heat and magnetic susceptibility. We have also looked at the probability distribution of susceptibility, pseudocritical temperatures and specific heat for assessing self-averaging. The study was carried out in appropriate restricted but dominant energy subspaces. By applying the finite-size scaling analysis, the correlation length exponent ν was found to be greater than one, whereas the ratio of the critical exponents (α/ν) is negative and (γ/ν) retains its pure Ising model value supporting weak universality.     

The static conductivity of disordered systems at zero temperature and the localization of states

The static conductivity of a disordered system at T = 0 is calculated by means of a gauge-invariant approach proposed by us earlier.We obtain localization of eigenstates in a strict sense, i.e. in the sense of vanishing of the conductivity, both in 1-D systems where an exponential vanishing with the length of the chain follows and in 3-D systems where localized states are shown to correspond to the vanishing of the imaginary part of the self-energy.     

Viscoelastic properties of confined molecular layers

We study the viscoelastic properties of a film of n layers of spherical molecules confined between two walls. We find that the dynamic response arises from two competing contributions: the effective stiffness of n + 1 springs in series and softening due to strain fluctuations. In particular, the latter are the origin of the oscillatory behavior of the stiffness and the damping coefficient. The dissipation is strongest at the minima of the stiffness; the inverse behavior may occur for a modulated relaxation time. As a corollary we show that confined molecular layers cannot be described as Maxwell fluids.     

Reflection of lamb waves in a solid layer by a grating formed by mechanical resonators

The scattering of the mth Lamb mode by a grating formed by P chains of closely spaced identical mechanical resonators (springs with loads) attached to the boundaries of a solid layer is considered. The distance between the chains is identical to the half-wavelength of the given mode at a frequency ω identical to or close to the natural frequency of the resonator chain with allowance for the interaction of neighboring chains through inhomogeneous modes. The coefficient of reflection of the mth Lamb mode from the aforementioned diffraction grating is calculated.     

Transient analysis of nonlinear Euler–Bernoulli micro-beam with thermoelastic damping,via nonlinear normal modes

In this paper an Euler–Bernoulli model has been used for vibration analysis of micro-beams with large transverse deflection. Thermoelastic damping is considered to be the dominant damping mechanism and introduced as imaginary stiffness into the equation of motion by evaluating temperature profile as a function of lateral displacement. The obtained equation of motion is analyzed in the case of pure single mode motion by two methods; nonlinear normal mode theory and the Galerkin procedure. In contrast with the Galerkin procedure, nonlinear normal mode analysis introduces a nonconventional nonlinear damping term in modal oscillator which results in strong damping in case of large amplitude vibrations. Evaluated modal oscillators are solved using harmonic balance method and tackling damping terms introduced as an imaginary stiffness is discussed. It has been shown also that nonlinear modal analysis of micro-beam with thermoelastic damping predicts parameters such as inverse quality factor, and frequency shift, to have an extrema point at certain amplitude during transient response due to the mentioned nonlinear damping term; and the effect of system?s characteristics on this critical amplitude has also been discussed.     

EXISTENCE OF NATURAL FREQUENCIES OF SYSTEMS WITH ARTIFICIAL RESTRAINTS AND THEIR CONVERGENCE IN ASYMPTOTIC MODELLING

A major limitation of the Rayleigh-Ritz method for determining the natural frequencies of a system is the need to choose admissible functions that do not violate the geometric constraints of that system (Courant 1943 Bulletin of the American Mathematical Society49, 1-23). Several researchers have attempted to overcome this problem by asymptotically modelling the rigid constraints with artificial (imaginary) restraints of very large stiffness (Courant 1943Bulletin of the American Mathematical Society49 , 1-23; Warburton and Edney 1984 Journal of Sound and Vibration95, 537-552; Gorman 1989 Journal of Applied Mechanics56, 893-899; Kim et al. 1990 Journal of Sound and Vibration143, 379-394; Yuan and Dickinson 1992 Journal of Sound and Vibration153, 203-216; Yuan and Dickinson 1992 Journal of Sound and Vibration159, 39-55; Cheng and Nicolas 1992 Journal of Sound and Vibration155, 231-247; Yuan and Dickinson 1994Computers and Structures53 , 327-334; Lee and Ng 1994 Applied Acoustics42, 151-163; Amabili and Garziera 1999 Journal of Sound and Vibration224, 519-539; Amabili and Garziera 2000 Journal of Fluids and Structures14, 669-690). While the numerical results thus obtained for the systems considered in the literature were in close agreement with exact values for the natural frequencies corresponding to the first few modes, sample calculations show that the error introduced by the asymptotic modelling increases with mode number and therefore to obtain accurate results for higher modes the magnitude of stiffness should also be increased. In any event, the error due to the asymptotic modelling would remain uncertain, except when the correct frequency values are known. However, the use of artificial restraints with negative stiffness, a new concept which was introduced in a recent publication (Ilanko and Dickinson 1999 Journal of Sound and Vibration219, 370-378) paves the way for estimating the error due to asymptotic modelling. This is possible since in this work, the Rayleigh-Ritz frequencies of the constrained system were found to be bracketed by the frequencies of the asymptotic models with positive and negative restraints. However, the use of artificial restraints with negative stiffness has raised some important questions: would a system with a large negative restraint become unstable, and if so what is the guarantee that the frequencies of the asymptotic model would converge to that of the constrained system? This paper is the result of the author's attempt to answer these questions and gives a proof of existence of natural frequencies for systems with artificial restraints (springs) having positive or negative stiffness coefficients, and their convergence towards constrained systems. Based on Rayleigh's theorem of separation, it has been shown that a vibratory system obtained by the addition of h restraints to an n -degree-of-freedom (d.o.f.) system, where h<n, will have at least (n÷h) natural frequencies and modes and that as the magnitude of the stiffness of the added restraints becomes very large, these (n÷h) natural frequencies will converge to the (n÷h) natural frequencies of a constrained system in which the displacements restrained by the springs are effectively constrained.