Leif Kari

Non-Linear Behavior of a Rubber Isolator System Using Fractional Derivatives

A non-linear rubber isolator included in a dynamic system is examined where influences of dynamic amplitude and frequency are investigated through measurements and modeling. The frequency dependence of the isolator is modeled by a fractional calculus element while a frictional component accounts for its amplitude dependence. The model works in the time-domain and simulations of harmonic and non-harmonic motion are compared to measurements. Good agreement is obtained in a wide frequency and amplitude range for a freely oscillating one degree of freedom system, with the isolator acting as a coupling between exciting foundation and mass, and for a single isolator showing the typical amplitude dependence known as the Payne effect. The model is found to be superior to the commonly applied Kelvin–Voigt element in modeling the dynamic isolator properties.

Nonlinear isolator dynamics at finite deformations : An effective hyperelastic, fractional derivative, generalized friction model

In presenting a nonlinear dynamic model of a rubber vibrationisolator, the quasistatic and dynamic motion influences on theforce response are investigated within the time and frequencydomain. It is found that the dynamic stiffness at the frequency ofa harmonic displacement excitation, superimposed upon the longterm isolator response, is strongly dependent on staticprecompression, dynamic amplitude and frequency. The problems ofsimultaneously modelling the elastic, viscoelastic and frictionforces are removed by additively splitting them, modelling theelastic force response by a nonlinear, shape factor basedapproach, displaying results that agree with those of aneo-Hookean hyperelastic isolator at a long term precompression.The viscoelastic force is modeled by a fractional derivativeelement, while the friction force governs from a generalizedfriction element displaying a smoothed Coulomb force. A harmonicdisplacement excitation is shown to result in a force responsecontaining the excitation frequency and its every otherhigher-order harmonic, while using a linearized elastic forceresponse model, whereas all higher-order harmonics are present forthe fully nonlinear case. It is furthermore found that the dynamicstiffness magnitude increases with static precompression andfrequency, while decreasing with dynamic excitationamplitude – eventually increasing at the highest amplitudes due tononlinear elastic effects – with its loss angle displaying amaximum at an intermediate amplitude. Finally, the dynamicstiffness at a static precompression, using a linearized elasticforce response model, is shown to agree with the fully nonlinearmodel except at the highest dynamic amplitudes.

Testing of nonlinear interaction effects of sinusoidal and noise excitation on rubber isolator stiffness

The nonlinear excitation effects on dynamic stiffness and damping of a filled rubber isolator are investigated through measurements. For a single harmonic excitation they are found to exhibit a strong amplitude dependence, following the well-known Payne effect where stiffness is high for small excitation amplitudes and low for large amplitudes while damping displays a maximum at intermediate amplitudes. However, expanding the measurements to a multiple harmonic excitation, the commonly applied superimposition principle of single harmonic responses due to this excitation is shown to be non-valid. On the contrary, it is found that reference stiffness at a small excitation amplitude and high frequency is reduced and damping increased while superimposing a large amplitude low frequency excitation component. Superimposition of low frequency noise signals displays essentially the same influence on the reference characteristics. As a rule of thumb, the largest excitation amplitude over the isolator normally determines the stiffness at the reference frequency while the influence of envelope amplitude is increased as its frequency approaches that of the component of interest.

On the dynamic stiffness of preloaded vibration isolators in the audible frequency range: Modeling and experiments

The nonlinear, preload-dependent dynamic stiffness of a cylindrical vibration isolator is examined via measurements and modeling within an audible frequency range covering 50 to 1000 Hz at various preloads. The stiffness is found to depend strongly on frequency-resulting in peaks and troughs, and on preload-particularly above 500 Hz. The problems of simultaneously modeling the rubber prestrain dependence and its audible short-term response are removed by adopting a nearly incompressible material model, being elastic in dilatation while displaying viscoelasticity in deviation. The latter exhibits a time strain separable relaxation tensor with a single function embodying its time dependence. This function is based on a continuous fractional order derivative model, the main advantage being the minimum number of parameters required to successfully model the rubber properties over a broad structure-borne sound frequency domain, while embodying a continuous distribution of relaxation time. The weak formulations corresponding to the stiffness problem are solved by an updated Lagrangian nonlinear finite-element procedure. The model and measurement results agree strikingly well with static and dynamic measurements throughout the whole frequency domain for the examined preloads.

In vitro contrast-enhanced ultrasound measurements of capillary microcirculation: Comparison between polymer- and phospholipid-shelled microbubbles

The focus of contrast-enhanced ultrasound research has developed beyond visualizing the blood pool and its flow to new areas such as perfusion imaging, drug and gene therapy, and targeted imaging. In this work comparison between the application of polymer- and phospholipid-shelled ultrasound contrast agents (UCAs) for characterization of the capillary microcirculation is reported. All experiments are carried out using a microtube as a vessel phantom. The first set of experiments evaluates the optimal concentration level where backscattered signal from microbubbles depends on concentration linearly. For the polymer-shelled UCAs the optimal concentration level is reached at a value of about 2×10(4)MB/ml, whereas for the phospholipid-shelled UCAs the optimal level is found at about 1×10(5)MB/ml. Despite the fact that the polymer shell occupies 30% of the radius of microbubble, compared to 0.2% of the phospholipid-shelled bubble, approximately 5-fold lower concentration of the polymer UCA is needed for investigation compared to phospholipid-shelled analogues. In the second set of experiments, destruction/replenishment method with varied time intervals ranging from 2ms to 3s between destructive and monitoring pulses is employed. The dependence of the peak-to-peak amplitude of backscattered wave versus pulse interval is fitted with an exponential function of the time γ=A(1-exp(-βt)) where A represents capillary volume and the time constant β represents velocity of the flow. Taking into account that backscattered signal is linearly proportional to the microbubble concentration, for both types of the UCAs it is observed that capillary volume is linearly proportional to the concentration of the microbubbles, but the estimation of the flow velocity is not affected by the change of the concentration. Using the single capillary model, for the phospholipid-shelled UCA a delay of about 0.2-0.3s in evaluation of the perfusion characteristics is found while polymer-shelled UCA provide response immediately. The latter at the concentration lower than 3.6×10(5)MB/ml have no statistically significant delay (p<0.01), do not cause any attenuation of the backscattered signal or saturation of the receiving part of the system. In conclusion, these results suggest that the novel polymer-shelled microbubbles have a potential to be used for perfusion evaluation.

Dynamic transfer stiffness measurements of vibration isolators in the audible frequency range

An indirect measurement method for blocked dynamic transfer stiffness of vibration isolators in the audible frequency range, up to 1000 Hz, including static preload and all six degrees of freedom is presented, Techniques for improving the stiffness accuracy are discussed in some detail, To suppress (unwanted) coupling effects between different degrees of freedom an improved excitation and terminating arrangement is adopted. Source correlation technique and stepped sine excitation are applied, increasing the signal-to-noise ratio. Computationally, a heavy blocking mass is replaced by its effective mass in the high frequency region, while using an overdetermined stiffness equation system. This is possible by applying various blocking masses, measuring acceleration at several positions and repeating the measurements. The method applied to a cylindrical vibration isolator at four axial preloads, results in smooth stiffness magnitude and phase curves, displaying antiresonances, resonances and the expected preload dependence, The test rig flanking transmission is shown to be negligible, while applying an auxiliary isolator decoupled test set-up, embedded in a heavy rigid frame construction, The stiffness error due to non-vanishing motion of the blocking mass is also shown to be negligible,

Magneto-sensitive rubber in a noise reduction context – exploring the potential

Abstract In a noise reduction context, magneto-sensitive (MS) rubber is likely to become a reality in the very near future. This conclusion is reached from the following: a review of the rapidly growing literature on the subject, a discussion around experimentally obtained data on magneto-sensitive rubber, and finally a computer simulation of a MS rubber isolator which seeks to illustrate the utility and great potential of this smart material within the audible frequency range. In contrast to normal rubber, magneto-sensitive rubber contains small iron particles that respond to externally applied magnetic fields, consequently altering the mechanical properties of the rubber. This response increases for small strains strengthening further the link to structure-borne sound applications where displacement amplitudes are usually small; this is borne out by vibration measurements in a running car engine, included for the purpose of placing experimental data on MS rubber in a real context.

Smart audio frequency energy flow control by magneto-sensitive rubber isolators

A magneto-sensitive rubber isolator inserted between a source and an infinite plate is modelled in the audible frequency range, and the energy flow into the plate with the rubber subjected to a magnetic field applied perpendicular to the axial displacement is calculated. Subsequently the result is compared to the corresponding energy flow for zero magnetic induction; upon the application of an external magnetic field the rubber becomes stiffer, thus shifting the internal resonances of the isolator. This is a fast and reversible process enabling adaption of the isolator to rapidly changing audio frequency conditions by simply turning on and off a magnetic field. In the application example considered, the energy flow into the plate at the first internal dynamic peak stiffness frequency is reduced by approximately 7?dB?a large difference in a sound and vibration context?by inducing magnetic saturation of the rubber.

Torsion stiffness of a rubber bushing : a simple engineering design formula including amplitude dependence

An engineering design formula for the torsion stiffness of a filled rubber bushing in the frequency domain, including the amplitude dependence, is presented. It is developed by applying a novel separable elastic, viscoelastic, and friction material model to an equivalent strain of the strain state inside the bushing, thus leading to an equivalent shear modulus that is inserted into an analytical formula for the torsion stiffness. The rubber model is the result of extending the force-displacement relation established in a sound rubber component model to the stress-strain level. Unlike other simplified methods, this procedure takes into account the variation in the properties inside the bushing owing to non-homogeneous strain states. Moreover, as this formula depends on the bushing geometry in addition to the material properties, it is a fast engineering tool to design the most suitable rubber bushing to fulfil user requirements. Furthermore, it is shown - by dividing the considered bushing into several slices, consequently each equivalent shear modulus is closer to the true value - that the approach of working with only one equivalent shear modulus for the whole bushing is accurate enough.

Influence of carbon black and plasticisers on dynamic properties of isotropic magnetosensitive natural rubber

Abstract The dynamic shear modulus of magnetosensitive (MS) natural rubber composites is experimentally studied, where influences of carbon black, plasticiser and iron particle concentrations are investigated at various dynamic shear strain amplitudes and external magnetic fields within the lower structure borne frequency range. The iron particles embedded in natural rubber are irregularly shaped and randomly distributed; the plasticisers simplify the iron particle blending process, while carbon black reduces the production costs and improves the mechanical properties. The results show that the relative MS effect on the shear modulus magnitude increases with increased plasticiser and iron particle concentration and decreases with increased carbon black concentration. Furthermore, their relative contributions are quantified. Consequently, the study provides a basis for optimising the composition of MS natural rubber to meet a variety of requirements, including those of vibration isolation, a promising application area for MS materials.