Dynamic instability of supercritical driveshafts mounted on dissipative supports—Effects of viscous and hysteretic internal damping

The case of a rotating shaft with internal damping mounted either on elastic dissipative bearings or on infinitely rigid bearings with viscoelastic suspensions is investigated in order to obtain the stability region. A Euler-Bernoulli shaft model is adopted, in which the transverse shear effects are neglected and the effects of translational and rotatory inertia, gyroscopic moments, and internal viscous or hysteretic damping are taken into account. The hysteretic damping is incorporated with an equivalent viscous damping coefficient. Free motion analysis yields critical speeds and threshold speeds for each damping model in analytical form. In the case of elastic dissipative bearings, the present results are compared with the results of previous studies on finite element models. In the case of infinitely rigid bearings with viscoelastic suspensions, it is established that viscoelastic supports increase the stability of long shafts, thus compensating for the loss of efficiency which occurs with classical bearings. The instability criteria also show that the effect of the coupling which occured between rigid modes introducing external damping and shaft modes are almost more important than damping factor. Lastly, comparisons between viscous and hysteretic damping conditions lead to the conclusion that an appropriate material damping model is essential to be able to assess these instabilities.     

A damping mechanics model and a beam finite element for the free-vibration of laminated composite strips under in-plane loading

A theoretical framework is presented for predicting the nonlinear damping and damped vibration of laminated composite strips due to large in-plane forces. Nonlinear Green-Lagrange axial strains are introduced in the governing equations of a viscoelastic composite and new nonlinear damping and stiffness matrices are formulated including initial stress effects. Building upon the nonlinear laminate mechanics, a damped beam finite element is developed. Finite element stiffness and damping matrices are synthesized and the static equilibrium is predicted using a Newton-Raphson solver. The corresponding linearized damped free-vibration response is predicted and modal frequencies and damping of the in-plane deflected strip are calculated. Numerical results quantify the nonlinear effect of in-plane loads on structural modal damping of various laminated composite strips. The modal loss-factors and natural frequencies of cross-ply Glass/Epoxy beams subject to in-plane loading are measured and correlated with numerical results.     

Galerkin method for steady-state response of nonlinear forced vibration of axially moving beams at supercritical speeds

The present paper investigates the steady-state periodic response of an axially moving viscoelastic beam in the supercritical speed range. The straight equilibrium configuration bifurcates in multiple equilibrium positions in the supercritical regime. It is assumed that the excitation of the forced vibration is spatially uniform and temporally harmonic. Under the quasi-static stretch assumption, a nonlinear integro-partial-differential equation governs the transverse motion of the axially moving beam. The equation is cast in the standard form of continuous gyroscopic systems via introducing a coordinate transform for non-trivial equilibrium configuration. For a beam constituted by the Kelvin model, the primary resonance is analyzed via the Galerkin method under the simply supported boundary conditions. Based on the Galerkin truncation, the finite difference schemes are developed to verify the results via the method of multiple scales. Numerical simulations demonstrate that the steady-state periodic responses exist in the transverse vibration and a resonance with a softening-type behavior occurs if the external load frequency approaches the linear natural frequency in the supercritical regime. The effects of the viscoelastic damping, external excitation amplitude, and nonlinearity on the steady-state response amplitude for the first mode are illustrated.     

Transverse hysteretic damping characteristics of a serpentine belt: Modeling and experimental investigation

In this paper, the transverse dynamic hysteretic damping characteristics (HDC) of a serpentine belt are investigated. The variable stiffness and variable damping model (VSDM) constituted of a variable-stiffness spring and a variable-damping damper is developed to estimate the HDC of the belt. A test rig is designed to test the force–displacement hysteresis damping curve and resonance frequencies of serpentine belts with different lengths under diverse loading conditions. The force–displacement hysteresis damping curve getting from the experiment is then used to determine the transverse stiffness and damping coefficients needed for the VSDM. The experiment particularly shows that the orientation of the hysteresis curve swings left and right around each natural frequency as it is a symmetrical point. This interesting phenomenon is explicated in detail with the loss angle which is calculated by two methods. Moreover, two sub-analytical models included in the VSDM are proposed to model the dependence of transverse dynamic stiffness and damping coefficient of a belt on belt length, pretension and excitation frequency. A comparison of the hysteresis curves obtained from the VSDM and experiment indicates that they are in good agreement.     

Control of hysteretic instability in rotating machinery by elastic suspension systems subject to dry and viscous friction

Most of the undesired whirling motions of rotating machines can be efficiently reduced by supporting journal boxes elastically and controlling their movement by viscous dampers or by dry friction surfaces normal to the shaft axis, which rub against the frame. In the case of dry dampers, resonance ranges of the floating support configuration can be easily cut off by planning a motionless adhesive state of the friction surfaces. On the contrary, the dry friction contact must change automatically into sliding conditions when the fixed support resonances are to be feared. Moreover, the whirl amplitude can be restrained throughout the speed range by a proper choice of the suspension-to-shaft stiffness ratio and of the support-to-rotor mass ratio.This theoretical research deals firstly with the natural precession speeds and looks for Campbell plots in dependence on the shaft angular speed, for several rotor-suspension systems. Then, the steady response to unbalance is investigated, in terms of rotor and support orbits and of conical path of the rotor axis. In this search, the ranges of adhesive or sliding contact are identified in particular for system with dry friction damping. At last, the destabilizing influence of the shaft hysteresis in the supercritical regime is focalized and the counterbalancing effect of the other dissipative sources is verified. In the nonlinear case of dry friction dampers, the control of linear stability is fulfilled by a perturbation procedure, checking the magnitude of Floquet characteristic multipliers on the complex plane. Moreover, the nonlinear stability far from steady motion is tested by the direct numerical solution of the full motion equations. The comparison configuration of suspension systems with viscous dampers and no dry friction is examined through an analytical first approximation approach and closed-form results for stability thresholds are derived in particular for the symmetric case.     

Galerkin methods for natural frequencies of high-speed axially moving beams

In this paper, natural frequencies of planar vibration of axially moving beams are numerically investigated in the supercritical ranges. In the supercritical transport speed regime, the straight equilibrium configuration becomes unstable and bifurcate in multiple equilibrium positions. The governing equations of coupled planar is reduced to two nonlinear models of transverse vibration. For motion about each bifurcated solution, those nonlinear equations are cast in the standard form of continuous gyroscopic systems by introducing a coordinate transform. The natural frequencies are investigated for the beams via the Galerkin method to truncate the corresponding governing equations without nonlinear parts into an infinite set of ordinary-differential equations under the simple support boundary. Numerical results indicate that the nonlinear coefficient has little effects on the natural frequency, and the three models predict qualitatively the same tendencies of the natural frequencies with the changing parameters and the integro-partial-differential equation yields results quantitatively closer to those of the coupled equations.     

Whirl and whip instabilities in rotor-bearing system considering a nonlinear force model

Linear models and synchronous response are generally adequate to describe and analyze rotors supported by hydrodynamic bearings. Hence, stiffness and damping coefficients can provide a good model for a wide range of situations. However, in some cases, this approach does not suffice to describe the dynamic behavior of the rotor-bearing system. Moreover, unstable motion occurs due to precessional orbits in the rotor-bearing system. This instability is called “oil whirl” or “oil whip”. The oil whirl phenomenon occurs when the journal bearings are lightly loaded and the shaft is whirling at a frequency close to one-half of rotor angular speed. When the angular speed of the rotor reaches approximately twice the natural frequency (first critical speed), the oil whip phenomenon occurs and remains even if the rotor angular speed increases. Its frequency and vibration mode correspond to the first critical speed. The main purpose of this paper is to validate a complete nonlinear solution to simulate the fluid-induced instability during run-up and run-down. A flexible rotor with a central disk under unbalanced excitation is modeled. A nonlinear hydrodynamic model is considered for short bearing and laminar flow. The effects of unbalance, journal-bearing parameters and rotor arrangement (vertical or horizontal) on the instability threshold are verified. The model simulations are compared with measurements at a real vertical power plant and a horizontal test rig.     

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.     

Extreme damping in composite materials with a negative stiffness phase

Composites with negative stiffness inclusions in a viscoelastic matrix are shown to have higher stiffness and mechanical damping tandelta than that of either constituent and exceeding conventional bounds. The causal mechanism is a greater deformation in and near the inclusions than the composite as a whole. Though a block of negative stiffness is unstable, negative stiffness inclusions in a composite can be stabilized by the surrounding matrix. Such inclusions may be made from single domains of ferroelastic material below its phase transition temperature or from prebuckled lumped elements.     

Hysteretic damping and causality

It is proven that linear oscillatory systems with hysteretic damping in the form of complex stiffness and/or complex elastic moduli satisfy the causality principle: the response of such a system to an arbitrary external force cannot appear earlier than the onset of the force. The proof, based on a rigorous solution to the problem of forced oscillations, is presented in detail for an oscillator with a complex stiffness, as well as in a brief explanation for a system with N mass. It is also shown that these systems are Lyapunov-unstable. A comparison is made to other linear hysteretic damping models.     

Nonlinear analysis of cylindrical and conical hysteretic whirl motions in rotor-dynamics

The internal friction of a rotor–shaft-support system is mainly due to the shaft structural hysteresis and to some possible shrink-fit release of the assembly. The experimentation points out the destabilizing effect of the internal friction in the over-critical rotor running. Nevertheless, this detrimental influence may be efficiently counterbalanced by other external dissipative sources located in the supports or by a proper anisotropic configuration of the support stiffness. The present analysis considers a rotor–shaft system which is symmetric with respect to the mid-span and is constrained by viscous-flexible supports with different stiffness on two orthogonal planes. The cylindrical and conical whirling modes are easily uncoupled and separately analysed. The internal dissipation is modelled by nonlinear Coulombian forces and moments, which counteract the translational and rotational motion of the rotor relative to a frame rotating with the shaft ends. The nonlinear equations of motion are solved by averaging approaches of the Krylov–Bogoliubov type. In both the over-critical whirling motions, cylindrical and conical, stable limit cycles may be attained whose amplitude is as large as the external dissipation applied by the supports is low. The stiffness anisotropy of the supports may be recognised as quite beneficial for the cylindrical whirl.     

Implication of conservative and gyroscopic forces on vibration and stability of an elastically tailored rotating shaft modeled as a composite thin-walled beam

Problems related with the implications of conservative and gyroscopic forces on vibration and the stability of a circular cylindrical shaft modeled as a thin-walled composite beam and spinning with constant angular speed about its longitudinal axis are addressed. Taking into account the directionality property of fiber reinforced composite materials, it is shown that for a shaft featuring flapwise-chordwise-bending coupling, a dramatic enhancement of both the vibrational and stability behavior can be reached. In addition, the effects played in the same context by transverse shear, rotatory inertias as well as by the various boundary conditions are discussed and pertinent conclusions are outlined.     

ANALYSIS OF NON-LINEAR DYNAMIC STABILITY FOR A ROTATING SHAFT-DISK WITH A TRANSVERSE CRACK

In this paper, the nonlinear dynamic stability of a rotating shaft-disk with a transverse crack is studied. The crack and the disk are located in arbitrary positions of the shaft respectively. Using the equivalent line-spring model, the deflections of the system with a crack are constructed by adding a deflection to the deflections of the uncracked system. The unstable regions are confirmed by Runge-Kutta method and the Floquet theory. The effects of crack depth, crack position, disk position, disk thickness and rotating speed on the principal unstable regions are discussed. The numerical results are compared with available data.     

Effects of internal viscous damping on the stability of a rotating shaft driven through a universal joint

A rotating flexible shaft, with both external and internal viscous damping, driven through a universal joint is considered. The mathematical model consists of a set of coupled, linear partial differential equations with time-dependent coefficients. Use of Galerkin's technique leads to a set of coupled linear differential equations with time-dependent coefficients. Using these differential equations some effects of internal viscous damping on parametric and flutter instability zones are investigated by the monodromy matrix technique. The flutter zones are also obtained on discarding the time-dependent coefficients in the differential equations which leads to an eigenvalue analysis. A one-term Galerkin approximation aided this analysis. Two different shafts (“automotive” and “lab”) were considered. Increasing internal damping is always stabilizing as regards to parametric instabilities. For flutter type instabilities it was found that increasing internal damping is always stabilizing for rotational speeds v below the first critical speed, v₁. For v>v₁, there is a value of the internal viscous damping coefficient, C_iv, which depends on the rotational speed and torque, above which destabilization occurs.The value of C_iv (“critical value”) at which the unstable zone first enters the practical range of operation was determined. The dependence of C_iv critical on the external damping was investigated. It was found for the automotive case that a four-fold increase in external damping led to an increase of about 20% of the critical value. For the lab model an increase of two orders of magnitude of the external damping led to an increase of critical value of only 10%.For the automotive shaft it was found that this critical value also removed the parametric instabilities out of the practical range. For the lab model it is not always possible to completely stabilize the system by increasing the internal damping. For this model using C_iv critical, parametric instabilities are still found in the practical range of operation.     

Mechanics and finite elements for the damped dynamic characteristics of curvilinear laminates and composite shell structures

Integrated mechanics and a finite element method are presented for predicting the damping of doubly curved laminates and laminated shell composite structures. Damping mechanics are formulated in curvilinear co-ordinates from ply to structural level and the structural modal loss factors are calculated using the energy dissipation method. The modelling of damping at the laminate level is based on first order shear shell theory. An eight-node shell damping finite element is developed. Comparisons of the present model with classical and discrete layer laminate damping theory predictions are shown. Modal damping and natural frequencies of composite plates and open cylindrical shells were measured and correlated with predicted results. Parametric studies illustrate the effect of curvature and lamination on modal damping and natural frequency.     

VIBRATION ATTENUATION VIA CONSTRAINED LAYER DAMPING AND DASHPOT ON A SPRING-MASS-PLATE SYSTEM

The vibrations and damping characteristics of an annular plate with constrained layer damping (CLD) treatment subject to a traveling spring-mass-damper (SMD) are investigated. The equations of the CLD-treated plate are first derived from the energy principle. These equations are simplified via the Donnell-Mushtari-Vlasov assumptions. The response equations are eventually uncoupled for each mode and are in terms of a single-degree-of-freedom (s.d.o.f.) linear oscillator with hysteretic damping. The receptance method follows to joint the plate and the SMD, and the resulting change of natural frequencies and damping ratios are investigated. Individual effects due to the inertia and the stiffness are illustrated as well. The results shows that the damping ratios resulted from the viscoelastic core are more significant than that from the viscous damper. In addition, there exists a best design on the thickness of the viscoelastic material core to have the maximum damping ratios. The results also show that the attachment of SMD bifurcated the plate's natural frequencies for every mode but n = 0. The bifurcation becomes more obvious with the rotational speed. These results provide useful information for vibration suppression in engineering design.     

Diffuse field transmission into infinite sandwich composite and laminate composite cylinders

This paper is concerned with the modelling of diffuse field transmission into composite laminate and sandwich composite infinite cylinders. Two models are presented and compared: Symmetrical Laminate composite and discrete thick laminate composite. The latter is shown to handle accurately, as a particular case, the first model, and the important case of sandwich composite shells. In both models, membrane, bending, transverse shearing as well as rotational inertia effects and orthotropic ply angle of the layers are considered. Starting from the dynamic equilibrium relations and stress–strain–displacement relations, a dispersion system is given in a wave approach context. Next, expressions for the matrix systems governing the structural impedance, critical frequencies and ring frequency are given. The developed equations are applied to the calculation of the diffuse field transmission of an infinite cylinder. Predictions with the presented models are compared to results presented in the literature for both laminate composite and sandwich composite configurations. They confirm the accuracy of both models and the general nature of the presented discrete thick laminate composite model.     

Single-mode fibre optic Bragg grating sensing on the base of birefringence in surface-mounting and embedding applications

Birefringence effects in the two typical installation techniques of fibre Bragg grating(FBG) sensor are investigated: surface-mounting and embedding configurations. When the FBG is bonded on a host material, the sensitivity loss in ultrasonic measurement caused by glue-induced low-birefringence is first reported. Next, the transverse stress-induced high birefringence when the FBG is embedded into a fabric composite laminate is measured as 3.6×10⁻⁴. Such induced-birefringence effects are experimentally analysed in mechanical applications. Simple and effective solutions with respect to the respective installation configurations for removing the birefringence effect are proposed and the obtained zero-birefringence cases are compared with the birefringent cases.     

Damping characteristics of elastic-viscoelastic composite curved bars and helical springs

The paper deals with the analysts of damping characteristics of a curved composite shaft. The composite shaft consists of an outer elastic curved tube with a similarly curved rod placed inside. The annular space between the tube and the rod is filled with viscoelastic material. The composite shaft is clamped at one end and a concentrated load (varying harmonically) acts on a radial arm fixed to the outer shell at the free end. It is shown that optimum design values exist for maximizing the total damping capacity of the system. It also is indicated that direct use of this method for increasing the damping capacity is not effective, as the damping factor reduces sharply after reaching a maximum value. It is shown how this difficulty can be overcome and how helical springs, possessing considerable damping capacity, can be designed. Sometimes this may help in simplifying the design of vibration isolators which will take the shape of a simple spring only.     

Universal crossover of liquid dynamics in supercritical region

We demonstrate that all liquids in supercritical region may exist in two qualitatively different states: solid-like and gas-like. Solid-like to gas-like crossover corresponds to the condition τ ≈ τ₀, where τ is liquid relaxation time and τ₀ is the minimum period of transverse waves. This condition corresponds to the loss of shear stiffness of a liquid at all frequencies and defines a new narrow crossover zone on the phase diagram. We show that the intersection of this zone corresponds to the disappearance of high-frequency sound, qualitative changes of diffusion and viscous flow, increase in particle thermal speed to half of the speed of sound and reduction of the specific heat at constant volume to 2k _B per particle. The new crossover is universal: it separates two liquid states at arbitrarily high pressure and temperature, and even exists in systems where liquid-gas transition and the critical point are absent overall.