Modeling and optimization of local constraint elastomer treatments for vibration and noise reduction

This paper presents the vibroacoustic study of a constrained elastomer treatment used in the industry for reducing noise. It can be trimmed and bonded conveniently to vibrating structures for reducing radiated noise. First, an identification of the elastomer viscoelastic characteristics is carried out with a program that models damped vibrations, a conjugate gradient search technique and experimental data extracted rom two contact-free modal analyses. The first modal analysis, adapted to dissipation characterization, is made on a partially covered suspended plate. The second modal analysis, adapted to identifying the elastomer stiffness behavior, concerns a cantilever beam that has almost been covered by a large treatment. The complete dynamic characterization is finally deduced from an iterative procedure that combines information from both experiments. The procedure highlights the influence of the treatment bonding quality on the achieved elastomer damping. Second, practical rules are deduced from a number of parametric studies on beams with baffled radiation conditions. In particular, a design criterion is introduced to help positioning patches where the elastomer damping can be maximized. A threshold, for which an optimal acoustic performance with a minimum of elastomer can be fulfilled, is also identified.     

Detailed ball bearing model for magnetic suspension auxiliary service

Catcher bearings (CBs) provide backup protection for rotating machines with active magnetic bearings (AMBs). The CBs are required in the event of an AMB failure or high transient loads. Numerical simulations of a rotor drop on CBs in flywheel energy storage system are conducted with a detailed CB model which includes a Hertzian load–deflection relationship between mechanical contacts, speed-and-preload-dependent bearing stiffness due to centrifugal force, and a Palmgren's drag friction torque. The transient simulation results show the rotor shaft response variations with the design parameters: shaft/bearing friction coefficients, axial preload, support damping of damper liner, and side loads from magnetic bearings. The results reveal that friction coefficients, support damping, and side loads are critical parameters to satisfy CB design objectives and prevent backward (super) whirl.     

Viscoelastic analysis of multi-body systems using the finite element method

A study of the effect of viscoelastic material damping on the dynamic response of multibody systems, consisting of interconnected rigid, elastic and viscoelastic components, is presented. The motion of each elastic or viscoelastic body is identified by using three sets of modes: rigid body, reference and normal modes. Rigid body modes describe translation and large angular rotation of a body reference. Reference modes are the result of imposing the body-axis conditions. Normal modes define the deformation of the body relative to the body reference. Constraints between different components are formulated by using a set of non-linear algebraic equations that can be introduced to the dynamic formulation by using a Lagrange multiplier technique or can be utilized to eliminate dependent co-ordinates by partitioning the constraint Jacobian matrix. In developing the system equations of motion of the viscoelastic component, an assumption of a linear viscoelastic model is made. A Kelvin-Voigt model is employed, wherein the stress is assumed to be proportional to the strain and its time derivative. The formulation yields a constant damping matrix and the damping forces depend only on the local deformation; thus, no additional coupling between the reference and elastic co-ordinates appears in the formulation when considering the viscoelastic effects. It is demonstrated, by a numerical example, that the viscoelastic material damping can have a significant effect on the dynamic response of multibody systems.     

The steady state out-of-plane response of an internally damped ring supported by springs in some bays

The steady state out-of-plane response of an internally damped ring supported by springs in some bays to a sinusoidally varying point force or moment is determined by use of the transfer matrix technique. For this purpose, the equations of out-of-plane vibration of a uniform circular ring based upon the Timoshenko beam theory are written as a coupled set of first order differential equations by using the transfer matrix of the ring. The matrix is obtained analytically and the steady state response of the ring is determined by the product of the matrices in free bays and those in supported bays. In this case, the elastic moduli of the ring and springs with internal damping are assumed to be complex quantities. The method is applied to rings supported against deflection and torsion in some bays of the same length located at equal angular intervals; the driving point impedance, transfer impedance and the distributions of the deflection, angular rotation, force and moment are calculated numerically, and the effects of the number, the stiffness and the length of supporting springs on them are studied.     

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.     

Quantitative determination of material viscoelasticity using a piezoelectric cantilever bimorph beam

The objective of this paper is to formulate the governing equation of a cantilever bimorph beam associated with a tip mass in contact with a viscoelastic material, which is modeled by a stiffness and a damper in parallel. From the eigenvalue problem, we can obtain the resonant frequencies as functions of the tip mass and material stiffness. The relation between the spectrum and material damping is established by the half-power bandwidth. It is found that the resonant frequencies increase as the material stiffness increases or the tip mass decreases, and the spectrum decreases by increasing the damping. From the analytic results, a cantilever could provide a technique to assess material viscoelasticity by simple measurements of the resonant frequency and the spectrum. Since the cantilever's behavior scales with its geometry, the device can be designed specifically for mechanical measurement of a microscopic system such as living cells and biomaterials.     

Finite element analysis and experimental study on dynamic properties of a composite beam with viscoelastic damping

This paper investigates the frequency dependent viscoelastic dynamics of a multifunctional composite structure from finite element analysis and experimental validation. The frequency-dependent behavior of the stiffness and damping of a viscoelastic material directly affects the system's modal frequencies and damping, and results in complex vibration modes and differences in the relative phase of vibration. A second order three parameter Golla–Hughes–McTavish (GHM) method and a second order three fields Anelastic Displacement Fields (ADF) approach are used to implement the viscoelastic material model, enabling the straightforward development of time domain and frequency domain finite elements, and describing the frequency dependent viscoelastic behavior. Considering the parameter identification a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Agreement between the curve fits using both the GHM and ADF and experiment is within 0.001 percent error. Continuing efforts are addressing the material modulus comparison of the GHM and the ADF model. There may be a theoretical difference between viscoelastic degrees of freedom at nodes and elements, but their numerical results are very close to each other in the specific frequency range of interest. With identified model parameters, numerical simulation is carried out to predict the damping behavior in its first two vibration modes. The experimental testing on the layered composite beam validates the numerical predication. Experimental results also show that elastic modulus measured from dynamic response yields more accurate results than static measurement, such as tensile testing, especially for elastomers.     

Dynamics of a supercritical composite shaft mounted on viscoelastic supports

The damping in a carbon fiber reinforced plastic (CFRP) laminate is greater than that which occurs in most metallic materials. In the supercritical regime, the damping can trigger unstable whirl oscillations, which can have catastrophic effects. The vibrations occurring in a supercritical composite drive shaft are investigated here in order to predict instabilities of this kind. A simply supported carbon/epoxy composite tube mounted on viscoelastic supports is studied, using an approximation of the Rayleigh–Timoshenko equation. The damping process is assumed to be hysteretic. The composite behavior is described in terms of modulus and loss factor, taking homogenized values. The critical speeds are obtained in several analytical forms in order to determine the effects of factors such as the rotatory inertia, the gyroscopic forces, the transverse shear and the supports stiffness. Assuming that the hysteretic damping can be expressed in terms of the equivalent viscous model, the threshold speed is obtained in the form of an analytical criterion. The influence of the various factors involved is quantified at the first critical speed of a subcritical composite shaft previously described in the literature. The influence of the coupling mechanisms on the unsymmetrical composite laminate and the end fittings is also investigated using a finite element model. None of these parameters were found to have a decisive influence in this case. Those having the greatest effects were the transverse shear and the supports stiffness. The effects of the composite stacking sequence, the shaft length and the supports stiffness on the threshold speed were then investigated. In particular, drive shafts consisting only of ±45° or ±30° plies can be said to be generally unstable in the supercritical regime due to their very high loss factors.     

Wave attenuation in nonlinear periodic structures using harmonic balance and multiple scales

We study the attenuation, caused by weak damping, of harmonic waves through a discrete, periodic structure with frequency nominally within the Propagation Zone (i.e., propagation occurs in the absence of the damping). The period of the structure consists of a linear stiffness and a weak linear/nonlinear damping. Adapting the transfer matrix method and using harmonic balance for the nonlinear terms, a four-dimensional linear/nonlinear map governing the dynamics is obtained. We analyze this map by applying the method of multiple scales upto first order. The resulting slow evolution equations give the amplitude decay rate in the structure. The approximations are validated by comparing with other analytical solutions for the linear case and full numerics for the nonlinear case. Good agreement is obtained. The method of analysis presented here can be extended to more complex structures.     

Parametric response of viscoelastically supported beams

The stability of viscoelastic beams with an attached mass and viscoelastic end supports under axial and tangential periodic loads is investigated. Viscoelastic end supports are substituted for translational and rotational springs with viscoelastic damping. The regions of instability for simple and combination resonances are obtained from the ordinary Mathieu equation which is obtained from the equation of motion by application of Galerkin's method. In numerical computations, the influences of the direction of loading, the attached mass, the support stiffness, and the damping on the regions of instability for simple and combination resonances are clarified.     

More on finite element modeling of damped composite systems

Finite element procedures are developed and verified for layered beams and rings having either continuously or discontinuously constrained viscoelastic damping layers. The two configurations considered are (1) a three-layered sandwich beam or ring (closed curved beam) consisting of two thin elastic layers with a viscoelastic core in between, and (2) a damped composite made of a thin-walled elastic structure having a finite number of mass segments or elastic segments adhered to it by a viscoelastic material. Viscoelastic material dependence on frequency and temperature is accounted for. Numerical predictions of transverse driving point impedances agree very well with available experimental data.     

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.     

Rotor drop and following thermal growth simulations using detailed auxiliary bearing and damper models

Catcher bearings (CBs) or auxiliary bearings provide mechanical backup protection in the events of magnetic bearing failure. This paper presents numerical analysis for a rotor drop on CBs and following thermal growths due to their mechanical rub using detailed CB and damper models. The detailed CB model is determined based on its material, geometry, speed and preload using the nonlinear Hertzian load–deflection formula, and the thermal growths of bearing components during the rotor drop are estimated using a 1D thermal model. A finite-element squeeze film damper provides the pressure profile of an annular oil film and the resulting viscous damping force. Numerical simulations of an energy storage flywheel with magnetic suspensions failed reveal that an optimal CB design using the detailed simulation models stabilizes the rotor drop dynamics and lowers the thermal growths while preventing the high-speed backward whirl. Furthermore, CB design guides based on the simulation results are presented.     

Analytical and empirical modeling of multilayered elastomeric isolators from damping experiments

Experimental studies have been performed on elastomeric layered composites to characterize the nonlinearity in dynamic stiffness and specific damping energy, so that their performance can be enhanced as isolators. The present study is divided into two parts: (a) analytical modeling of isolator samples, and (b) formulation for glue characteristics. Several samples of layered arrangement of elastomer and metal strips were used in the experiments. Dynamic and static loading experiments were performed. All these experimental results were used in developing nonlinear empirical models for the elastomer characteristics. Furthermore creep–fatigue test was performed to explain certain observed behavior in the elastomer characteristics. Concluding part of the paper discusses empirical formulation of the layered sample considering elastomer and adhesive layers as basic elements, thus evolving a method to calculate adhesive properties.     

DYNAMIC BEHAVIOR OF THE FULL ROTORSTOP RUBBING: NUMERICAL SIMULATION AND EXPERIMENTAL VERIFICATION

The dynamic behavior of a rotor rubbing, especially rubbing fully with a motion-limiting stop is investigated by numerical and experimental methods. In the dynamic simulation, the sinuous excitation force with low frequency excites the large whirl of the unbalanced rotor and thus causes the rubbing between the rotor and the stop. The simple Coulomb friction model and the multiple segments linear spring model are used to reveal the nature of the rubbing forces. The torque equation of the rotor is built to extract the rotating speed during partial and full rubbing. The stable partial rubbing motion demonstrates that the stop limits the violent vibration amplitude of the rotor effectively. The rubbing experiments confirm the idea of using the inner type of stop to suppress the violent backward whirl with low frequency. When the amplitude of the excitation force exceeds a certain value, the full rubbing occurs with serious continuous friction. During full rubbing, the center of the rotor moves counter-clockwise and whips in the amplitude exceeding the rotor/stop gap dramatically. Moreover, the whip frequency is much higher than the frequencies of the excitation and the unbalance force. And then the rotor rotation is broken quickly by the stop. The predicted dynamical behavior is verified by the rubbing experiments. The relation between the stop/bearing stiffness ratio and such dynamical behavior as the initiation of the rubbing, the over-limit ratio and the contact-ratio is discussed.     

Vibration and damping analysis of annular plates with constrained damping layer treatments

The natural frequencies and modal loss factors of annular plates with fully and partially constrained damping treatments are considered. The equations of free vibration of the plate including the transverse shear effects are derived by a discrete layer annular finite element method. The extensional and shear moduli of the viscoelastic material layer are described by the complex quantities. Complex eigenvalues are then found numerically, and from these, both frequencies and loss factors are extracted. The effects of viscoelastic layer stiffness and thickness, constraining layer stiffness and thickness, and treatment size on natural frequencies and modal loss factors are presented. Numerical results also show that the longer constrained damping treatment in radial length does not always provide better damping than the shorter ones.     

Measurement of viscoelastic properties from the vibration of a compliantly supported beam

A laboratory method is presented by which the viscoelastic properties of compliant materials are measured over a wide frequency range. The test setup utilizes a flexible beam clamped at one end and excited by a shaker at the free end. A small specimen of a compliant material is positioned to support the beam near its midpoint. The deformation from gravity is minimized since the specimen is not loaded by an attached mass. Forced vibration responses measured at two locations along the beam are used to derive a transfer function from which the dynamic properties of compliant materials are directly determined by use of a theoretical procedure investigating the effects of specimen stiffness on the propagation of flexural waves. Sensitivity of the measured properties to experimental uncertainties is investigated. Young's moduli and associated loss factors are determined for compliant materials ranging from low-density closed-cell foams to high-density damping materials. The method is validated by comparing the measured viscoelastic properties to those from an alternative dynamic test method.     

Analytical model for nanoscale viscoelastic properties characterization using dynamic nanoindentation

In the last few decades, nanoindentation has gained widespread acceptance as a technique for materials properties characterization at micron and submicron length scales. Accurate and precise characterization of material properties with a nanoindenter is critically dependent on the ability to correctly model the response of the test equipment in contact with the material. In dynamic nanoindention analysis, a simple Kelvin–Voigt model is commonly used to capture the viscoelastic response. However, this model oversimplifies the response of real viscoelastic materials such as polymers. A model is developed that captures the dynamic nanoindentation response of a viscoelastic material. Indenter tip-sample contact forces are modelled using a generalized Maxwell model. The results on a silicon elastomer were analysed using conventional two element Kelvin–Voigt model and contrasted to analysis done using the Maxwell model. The results show that conventional Kelvin–Voigt model overestimates the storage modulus of the silicone elastomer by ~30%. Maxwell model represents a significant improvement in capturing the viscoelastic material behaviour over the Voigt model.     

Vibration of a rotating shaft with randomly varying internal damping

A simple Jeffcott rotor is considered with both external and internal damping. Coefficient of internal damping is subject to temporal random variations which may occasionally bring the rotor into the domain of dynamic instability. The corresponding sporadic outbreaks in the rotor's vibrational response (whirl) are studied by applying the Krylov-Bogoliubov averaging method to the complex equation of motion and using parabolic approximation for the random coefficient of the internal damping. This results in an explicit analytical solution for the radius of whirl which may be used for predicting reliability of the rotor. Furthermore, a convenient procedure is described for interpreting measured on-line test data for the rotor. Namely, the mean value of the coefficient of internal damping as well as its standard deviation and mean frequency of temporal variations may be estimated directly from the trace of whirl radius which exhibits spontaneous random outbreaks in response.     

Internal friction due to negative stiffness in the indium–thallium martensitic phase transformation

Internal friction and dynamic shear modulus in an indium–21?at.% thallium alloy were measured as functions of frequency and cooling rate using broadband viscoelastic spectroscopy during the martensitic transformation which occurs in this material occurs around 50°C. Microstructural evolution of martensitic bands was captured using time-lapse optical microscopy. The amplitude of damping peaks due to the temperature-induced transformation in the polycrystalline alloy was found to exceed those reported by others for single crystals of similar alloy compositions, in contrast to the usual reduction in damping in polycrystals. The high temperature portion of the damping peak occurs before martensitic bands are observed; therefore this portion cannot be due to interfacial motion. Constrained negative stiffness of the grains can account for this damping, as well as for amplification of internal friction peaks in these polycrystals and for sigmoid-shaped anomalies in the shear modulus at high cooling rates. Surface features associated with a previously unreported pre-martensitic phenomenon are seen at temperatures above martensite-start.