Maxwell rheological model for lipid-shelled ultrasound microbubble contrast agents

The present paper proposes a model that describes the encapsulation of microbubble contrast agents by the linear Maxwell constitutive equation. The model also incorporates the translational motion of contrast agent microbubbles and takes into account radiation losses due to the compressibility of the surrounding liquid. To establish physical features of the proposed model, comparative analysis is performed between this model and two existing models, one of which treats the encapsulation as a viscoelastic solid following the Kelvin-Voigt constitutive equation and the other assumes that the encapsulating layer behaves as a viscous Newtonian fluid. Resonance frequencies, damping coefficients, and scattering cross sections for the three shell models are compared in the regime of linear oscillation. Translational displacements predicted by the three shell models are examined by numerically calculating the general, nonlinearized equations of motion for weakly nonlinear excitation. Analogous results for free bubbles are also presented as a basis to which calculations made for encapsulated bubbles can be related. It is shown that the Maxwell shell model possesses specific physical features that are unavailable in the two other models.     

Bending vibration of axially loaded Timoshenko beams with locally distributed Kelvin-Voigt damping

Utilizing the Timoshenko beam theory and applying Hamilton's principle, the bending vibration equations of an axially loaded beam with locally distributed internal damping of the Kelvin-Voigt type are established. The partial differential equations of motion are then discretized into linear second-order ordinary differential equations based on a finite element method. A quadratic eigenvalue problem of a damped system is formed to determine the eigenfrequencies of the damped beams. The effects of the internal damping, sizes and locations of damped segment, axial load and restraint types on the damping and oscillating parts of the damped natural frequency are investigated. It is believed that the present study is valuable for better understanding the influence of various parameters of the damped beam on its vibration characteristics.     

The influence of a wieghardt type elastic foundation on the stability of some beams subjected to distributed tangential forces

In this investigation, the influence of a Wieghardt type elastic foundation on the stability of cantilever and clamped-hinged beams subjected to either a uniformly or a linearly distributed tangential force is considered. In addition to the usual transverse foundation modulus, the Wieghardt model includes the effects of inertia and shear deformation in the foundation. Approximate solutions of the Ritz type are obtained for the pertinent eigenvalue problems, and numerical calculations are reported for various combinations of the internal damping, inertia, transverse foundation modulus and shear foundation modulus parameters. The numerical results reveal that, in general, for a fixed value of the transverse foundation modulus parameter κ, an increase in the shear foundation modulus increases the critical load, whereas an increase in the foundation inertia parameter tends to decrease the critical load. The system consisting of a clamped-hinged beam subjected to a uniformly distributed tangential force loses stability through divergence, provided that the value of κ is sufficiently small. However, when κ becomes large enough, stability will be lost through flutter. In this case, the critical load considered as a function of κ possesses a discontinuity at the transition between divergence and flutter, and its value will either increase or decrease, depending upon the degree of damping in the system.     

On the weakly damped vibrations of a vertical beam with a tip-mass

In this paper the wind-induced, horizontal vibrations of a weakly damped vertical Euler-Bernoulli beam with and without a tip-mass will be studied. The damping is assumed to be boundary damping and global Kelvin-Voigt damping. The boundary damping is assumed to be proportional to the velocity of the beam at the top. The horizontal vibrations of the beam can be described by an initial-boundary value problem. In this paper, the multiple-timescales perturbation method will be applied to construct approximations of the solutions of the problem. Also it will be shown that a combination of boundary damping and Kelvin-Voigt damping can be used to damp the wind-induced vibrations of a vertical beam with tip-mass uniformly.     

A Two-Layer Model for Superposed Electrified Maxwell Fluids in Presence of Heat Transfer

Based on a modified-Darcy-Maxwell model, two-dimensional, incompressible and heat transfer flow of two bounded layers, through electrified Maxwell fluids in porous media is performed. The driving force for the instability under an electric field, is an electrostatic force exerted on the free charges accumulated at the dividing interface. Normal mode analysis is considered to study the linear stability of the disturbances layers. The solutions of thelinearized equations of motion with the boundary conditions lead to an implicit dispersion relation between the growth rate and wave number. These equations are parameterized by Weber number, Reynolds number, Marangoni number, dimensionless conductivities, and dimensionless electric potentials. The case of long waves interfacial stability has been studied. The stability criteria are performed theoretically in which stability diagrams are obtained. Inthe limiting cases, some previously published results can be considered as particular cases of our results. It is found that the Reynolds number plays a destabilizing role in the stability criteria, while the damping influence is observed for the increasing of Marangoni number and Maxwell relaxation time.     

Damped vibration analysis of an elastically connected complex double-string system

The main purpose of the present paper is to consider theoretically damped transverse vibrations of an elastically connected double-string system. This system is treated as two viscoelastic strings with a Kelvin-Voigt viscoelastic layer between them. A theoretical analysis has been made for a simplified model of the system, in which assumed physical parameters make it possible to decouple the governing equations of motion by introducing the principal co-ordinates. Applying the method of separation of variables and the modal expansion method, exact analytical solutions for damped free and forced responses of the system subjected to arbitrarily distributed transverse continuous loads are determined in the case of arbitrary magnitude of linear viscous damping. It is important to note that the solutions obtained are explicitly expressed in terms of parameters characterizing the physical properties of the system under discussion. For the sake of completeness of the analysis, solutions for undamped free and forced vibrations are also formulated.     

Mass optimization of non-conservative cantilever beams with internal and external damping

An adjoint variational principle has been developed for a non-conservatively loaded cantilever beam with Kelvin-Voigt internal and linear external damping and is applied to a beam with a linearly distributed tangential load acting along the centerline of the beam. Relative mass optimization for beams of both rectangular and circular crosssections is considered from a graphical standpoint and from the viewpoint of a computer optimization routine with data given and discussed in both instances. In going to a Rosenbrock optimization routine for beams of rectangular cross-section with a minimum tip thickness constraint imposed it was quite clear that mass ratio reductions in the range 14·9 % to 38 % are possible and that the values of internal and external damping appear influential in determining just how much of a mass reduction is possible. Similarly, for beams of circular cross-section a Rosenbrock optimization routine with a minimum tip diameter constraint imposed showed that mass ratio reductions of the order of 27 % are possible.     

On the propagation of linear transverse acoustic waves in isotropic media with mechanical relaxation phenomena due to viscosity and a tensorial internal variable: II. Some cases of special interest (Poynting-Thomson,Jeffreys, Maxwell,Kelvin-Voigt,Hooke and Newton media)

The propagation of linear transverse acoustic waves in isotropic media in which mechanical relaxation phenomena occur was considered in a previous paper. In particular expressions for the velocity and attenuation of the waves were obtained and the limiting cases of waves with high and low frequencies were discussed. In the present paper we investigate the propagation of linear transverse acoustic waves in Poynting-Thomson, Jeffreys, Maxwell, Kelvin-Voigt, Hooke and Newton media. We show that the dispersion relations for these waves may be considered as degeneracies of the dispersion relation which we derived in the general case of a viscoanelastic medium with memory. In particular we investigate the explicit dependence of the dispersion relations on the thermodynamic parameters and the phenomenological coefficients.     

Effect of Longitudinal Magnetic Field on Vibration Characteristics of Single-Walled Carbon Nanotubes in a Viscoelastic Medium

The effect of longitudinal magnetic field on vibration response of a sing-walled carbon nanotube (SWCNT) embedded in viscoelastic medium is investigated. Based on nonlocal Euler-Bernoulli beam theory, Maxwell’s relations, and Kelvin viscoelastic foundation model, the governing equations of motion for vibration analysis are established. The complex natural frequencies and corresponding mode shapes in closed form for the embedded SWCNT with arbitrary boundary conditions are obtained using transfer function method (TFM). The new analytical expressions for the complex natural frequencies are also derived for certain typical boundary conditions and Kelvin-Voigt model. Numerical results from the model are presented to show the effects of nonlocal parameter, viscoelastic parameter, boundary conditions, aspect ratio, and strength of the magnetic field on vibration characteristics for the embedded SWCNT in longitudinal magnetic field. The results demonstrate the efficiency of the proposed methods for vibration analysis of embedded SWCNTs under magnetic field.     

Frequency analysis of finite beams on nonlinear Kelvin-Voight foundation under moving loads

The vibration of an Euler-Bernoulli beam, resting on a nonlinear Kelvin-Voight viscoelastic foundation, traversed by a moving load is studied in the frequency domain. The objective is to obtain the frequency responses of the beam and the effects of different parameters on the system response. The parameters include the magnitude and speed of the moving load and the foundation nonlinearity and its damping coefficient. The solution is obtained by using the Galerkin method in conjunction with the multiple scales method (MSM). The governing nonlinear partial differential equations of motion are discretized into sets of nonlinear ordinary differential equations. Subsequently, the solution is calculated for different harmonics by using the MSM as one of the powerful perturbation techniques. The steady-state responses of the main harmonic as well as its two super-harmonics are then obtained. As a case study, a conventional railway track is dynamically simulated and the jump phenomenon in the response is observed for three harmonics. Moreover, a thorough stability analysis of the system is carried out.     

Phonon damping by translation-rotation coupling in orientationally disordered molecular crystals

A simple model for the translation-rotation coupling in orientationally disordered molecular crystals (ODIC) is used to investigate the effect of the coupling to the damping of acoustic phonons in an fcc-ODIC composed of tetrahedral molecules. The influence of the coupling is calculated as a function of molecular and lattice parameters in the framework of linear response theory. The results are applied to the plastic phase I of solid CD₄, revealing a strong damping of short-wavelength phonons. This is consistent with experimental and molecular dynamics observations.     

Stability of damped membranes and plates with distributed inputs

This paper proves the stability of boundary and distributed damped membranes and Kirchhoff plates under distributed inputs. Distributed viscous or Kelvin-Voigt damping ensures a weakly bounded response to a bounded transverse loading for pinned membranes and clamped plates. Damping on part of the boundary can also weakly stabilize the forced response, provided the damped and undamped boundary normals satisfy certain conditions. For example, damping on half and one side of the boundary is sufficient for circular and rectangular domains, respectively.     

Dissipation and dispersion properties of microinhomogeneous media

The propagation of a harmonic elastic wave in a microinhomogeneous (defect-containing) medium is considered in the framework of the rheological model that reperesents the medium in the form of a one-dimensional chain of masses connected by purely elastic elements and by Kelvin-Voigt viscoelastic elements. Analytical expressions are derived for the dissipation and dispersion characteristics of this medium for various distributions of the parameters of the viscoelastic elements. The dissipation and dispersion properties are found to obey the Kramers-Kronig relations. It is also shown that the damping decrement of the wave is almost constant, and the phase velocity monotonically increases in a sufficiently wide range of parameters of the viscoelastic elements in a wide frequency band. The derived expressions for the dispersion and dissipation are used to simulate the propagation of broadband pulses in this kind of medium.     

The influence of rotatory inertia,tip mass,and damping on the stability of a cantilever beam on an elastic foundation

The stability of a uniform viscoelastic cantilever resting on an elastic foundation, carrying a tip mass, and subjected to a follower force at its free end is investigated. The effects of the rotatory inertia of the beam, the transverse and rotatory inertias of the tip mass, and the foundation modulus, which characterizes a Winkler type of elastic foundation, are included in the partial differential equation of motion and boundary conditions, and the influence of these quantities on the value of the critical flutter load parameter Q_f is sought. The exact forms of the fundamental frequency equations are derived for the cases of a viscoelastic and a purely elastic beam, and these equations are solved numerically for Q_f These numerical results reveal that Q_f depends strongly upon the foundation modulus for the cantilever carrying a tip mass or possessing rather small internal damping. In the absence of damping and a tip mass, the value of Q_f, computed upon the inclusion of the rotatory inertia of the beam in the formulation of the equation of motion, is decreased slightly and continues to decrease in essentially a linear manner as the value of the foundation modulus parameter κ is decreased. Moreover, when the effect of very small internal damping is included, the value of Q_f computed when the rotatory inertia of the beam is neglected increases slowly in an essentially linear fashion as x increases, whereas, when the effect of rotatory inertia is retained, the value of Q_f decreases as κ is increased. Additional numerical results are reported graphically.     

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.     

Piezoelectric resonators with mechanical damping and resistance in current conduction

A novel design method for high Q piezoelectric resonators was presented and proposed using the 3-D equations of linear piezoelectricity with quasi-electrostatic approximation which include losses attributed to mechanical damping in solid and resistance in current conduction. There is currently no finite element sofware for estimating the Q of a resonator without apriori assumptions of the resonator impedance or damping. There is a necessity for better and more realistic modeling of resonators and filters due to miniaturization and the rapid advances in frequency ranges in telecommunication. We presented new three-dimensional finite element models of quartz and barium titanate resonators with mechanical damping and resistance in current conduction. Lee, Liu and Ballato’s 3-D equations of linear piezoelectricity with quasi-electrostatic approximation which include losses attributed to mechanical damping in solid and resistance in current conduction were formulated in a weak form and implemented in COMSOL. The resulting finite element model could predict the Q and other electrical parameters for any piezoelectric resonator without apriori assumptions of damping or resistance. Forced and free vibration analyses were performed and the results for the Q and other electrical parameters were obtained. Comparisons of the Q and other electrical parameters obtained from the free vibration analysis with their corresponding values from the forced vibration analysis were found to be in excellent agreement. Hence, the frequency spectra obtained from the free vibration analysis could be used for designing high Q resonators. Results for quartz thickness shear AT-cut and SC-cut resonators and thickness stretch poled barium titanate resonators were presented. An unexpected benefit of the model was the prediction of resonator Q with energy losses via the mounting supports.     

Stability analysis of partially loaded Leipholz column carrying a lumped mass and resting on elastic foundation

In the present study, the stability of a cantilever column resting on an elastic foundation under the action of a uniformly distributed tangential load is discussed. A Winkler type elastic foundation is considered. Moreover, the effect of a lumped mass located in an arbitrary position on the stability of the system when the column is subjected to a partially distributed follower force is investigated. The equations of motion are obtained using the extended Hamilton's principle and the influences of the lumped mass and applied load are included in the equations using the generalized functions theories. Applying the Ritz technique, the resulting equations are transformed into a general eigenvalue problem. The effects of several design parameters such as foundation elastic modulus, ratio of the lumped mass to the column's mass, position of the lumped mass and the distribution model of the follower force are examined. The validity of the present analysis is confirmed by comparing the results with those obtained in literature and excellent agreement is observed. The numerical results reveal that the load distribution length and model have significant effects on the flutter boundaries of the system.     

Damping performance of two simple oscillators coupled by a visco-elastic connection

A simple dynamic system composed of two linear oscillators is employed to analyze the passive control performance that can be achieved through a visco-elastic damper connecting two adjacent free-standing structures. By extension, the model may also describe the energy dissipation which can be obtained by an internal coupling between two quasi-independent sub-systems composing a single complex structure. Two alternatives are evaluated for the linear coupling by considering either the serial or the parallel spring–dashpot arrangement known as the Kelvin–Voigt and the Maxwell damper model, which may synthetically reproduce the constitutive behavior of different industrial devices. The complex eigenvalues of the coupled system are parametrically analyzed to determine the potential benefits realized by different combinations of the coupling stiffness and damping coefficient. A design strategy to assess these parameters is outlined, driven by the relevant observation that a perfect tuning of the natural frequencies always corresponds, in the parameter space, to the maximum modal damping for one of the resonant modes, independent of the damper model. The effectiveness of the proposed strategy is discussed for different classes of the controlled system, depending on the mass and stiffness ratio of the component oscillators. As a major result, different design parameter charts for the two damper models are carried out and compared to each other. Performance indexes are introduced to quantitatively evaluate the passive control performance with respect to the mitigation of the system forced response under harmonic and seismic ground excitation. The analyses confirm the validity of the design strategy for a well-balanced mitigation of the displacement and acceleration response in both the oscillators.     

Dirac Quasinormal Modes of Static f(R) de Sitter Black Holes

Quasinormal modes(QNMs) for Dirac perturbations of f(R) black holes(BHs) are described in this paper,involving two types of f(R) solution: f(R)(Schwarzschild) BHs and f(R)(Maxwell) BHs. With the finite difference method, the stability of the f(R) black holes(BHs) is analysed and the threshold range of f(R)(Schwarzschild) BHs and f(R)(Maxwell) BHs is defined respectively. The results show that due to the presence of the correction factor R0, the damping rate of Dirac field decreases. Meanwhile, the influence of angular quantum number values |k| on the f(R) BHs is investigated. The results indicate that the QNMs oscillation becomes tenser and damping speed slowly decreases with|k| increasing. Furthermore, under the Dirac perturbation, the stability of f(R) solutions can be reflected in the manner of Dirac QNMs. The relationships between the QNMs and the parameters(|k|, charge Q and mass m) are discussed in massless, and massive cases, by contrast to the classical BHs.