The influence of mass imperfections on the evolution of standing waves in slowly rotating spherical bodies

Standing waves can exist as stable vibrating patterns in perfect structures such as spherical bodies, and inertial rotation of the body causes precession (Bryan's effect). However, an imperfection such as light mass anisotropy destroys the standing waves. In this paper, an imperfection is introduced in the form of light mass anisotropy for a vibrating, slowly rotating spherical body. Assuming this light mass imperfection throughout this paper, the effects of slow rotation and light isotropic viscous damping are considered in a system of variables consisting of the amplitudes of principal and quadrature vibrating patterns, the angle of the rotation of the vibrating pattern (called the precession angle) and the phase shift of the vibrating pattern. We demonstrate how a combination of both qualitative and quantitative analysis (using, inter alia, the method of averaging) predicts that the inertial angular rate does not influence changes with time in the amplitudes of the principal and quadrature vibrations or the phase shift. The light mass imperfection causes changes with time which appear to be of a damped oscillatory nature for both the quadrature component as well as the principal component. The precession angular rate appears to depend on the inertial angular rate as well as the quadrature component of the vibration but is not influenced by the damping factor. It is not directly proportional to the inertial angular rate as is the case for a perfect isotropically damped structure. If the quadrature component is not suppressed, then a “capture effect” appears to occur, namely that the precession angle will not grow at a constant rate but is “captured” and shows periodic behaviour. It is evident that the damping factor does not influence changes with time in the phase shift and that the mass imperfection substantially influences these changes. The phase shift appears to be negative, strictly decreasing and unbounded.     

Vibration characteristics of a rotating flexible arm with ACLD treatment

In this paper, the vibration behavior and control of a clamped–free rotating flexible cantilever arm with fully covered active constrained layer damping (ACLD) treatment are investigated. The arm is rotating in a horizontal plane in which the gravitational effect and rotary inertia are neglected. The stress–strain relationship for the viscoelastic material (VEM) is described by a complex shear modulus while the shear deformations in the two piezoelectric layers are neglected. Hamilton's principle in conjunction with finite element method (FEM) is used to derive the non-linear coupled differential equations of motion and the associated boundary conditions that describe the rigid hub angle rotation, the arm transverse displacement and the axial deformations of the three-layer composite. This refined model takes into account the effects of centrifugal stiffening due to the rotation of the beam and the potential energies of the VEM due to extension and bending. Active controllers are designed with PD for the piezosensor and actuator. The vibration frequencies and damping factors of the closed-loop beam/ACLD system are obtained after solving the characteristic complex eigenvalue problem numerically. The effects of different rotating speed, thickness ratio and loss factor of the VEM as well as different controller gain on the damped frequency and damping ratio are presented. The results of this study will be useful in the design of adaptive and smart structures for vibration suppression and control in rotating structures such as rotorcraft blades or robotic arms.     

Vibration of high-speed rotating rings coupled to space-fixed stiffnesses

This study investigates the vibration of high-speed rotating rings coupled to space-fixed discrete stiffnesses. The ring radial and tangential deformations are defined using space-fixed (Eulerian) coordinates, where material particles pass through fixed locations in space. Engineering strain is used in the strain energy expression. The derived nonlinear equations from Hamilton's principle are linearized about the steady non-trivial configuration that results from constant ring rotation. Comparisons are made to other models in the literature that use different assumptions. The governing equations are cast in terms of matrix differential operators that reveal the system's standard gyroscopic system structure. The natural frequencies and vibration modes are calculated over a wide-range of rotation speeds for axisymmetric free rings and a non-axisymmetric ring with a space-fixed discrete stiffness element.     

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.     

Cauchy problems for the conformal vacuum field equations in general relativity

Cauchy problems for Einstein's conformal vacuum field equations are reduced to Cauchy problems for first order quasilinear symmetric hyperbolic systems. The “hyperboloidal initial value” problem, where Cauchy data are given on a spacelike hypersurface which intersects past null infinity at a spacelike two-surface, is discussed and translated into the conformally related picture. It is shown that for conformal hyperboloidal initial data of classH ^S,s≧4, there is a unique (up to questions of extensibility) development which is a solution of the conformal vacuum field equations of classH ^S. It provides a solution of Einstein's vacuum field equations which has a smooth structure at past null infinity.     

Rotating bodies in general relativity

Synge's approximation method is used in order to obtain the gravitational field of a massive body with an axis of symmetry around which it is rotating steadily. The method is carried out to include the second approximation. This means that terms of orderm ² are retained as significant, and there is an error of orderm ³ in the field equations,m being the mass of the body.     

Use of modal energy transfer as a new damping device with controllable bandwidth

This paper presents a new design of nonlinear dynamic absorber (NDA) using the phenomenon of modal energy transfer between the symmetric mode and the anti-symmetric mode of a curved beam. It can reduce the resonance vibration of a primary structure with a controllable operational frequency range. The energy transfer is initiated by an autoparametric vibration and the excitation force required is lowest when the ratio of the resonance frequencies of the first symmetric mode (ω₁) and first anti-symmetric mode (ω₂) is close to 2.The resonance frequency of the first anti-symmetric mode (ω₂) can be altered to control the operational frequency range. The autoparametric vibration response can be used to create an energy-dissipative region with a controllable bandwidth. It is also possible to create a non-dissipative region in between two dissipative regions. This is useful for providing damping for a conventional dynamic absorber without adding high damping material. The damping is due to the dissipation of energy to anti-symmetric mode. Numerical calculations indicate that the resonance vibration of a primary structure can be successfully reduced using this approach. The results are verified with experimental data.     

Does Charge Contribute to the Frame Dragging of?Spacetime?

Electrically charged systems bound by a strong gravitational force can sustain a huge amount of electric charge (up to 10²⁰ C) against Coulomb repulsion. General relativistically such systems form a stable hydrostatic configuration both in the non-rotating and rotating cases. Here we study the effects of electric charge (electric energy density) on the spacetime outside a rotating electrically charged system bound by a strong gravitational force. In particular we investigate the effect of charge density on frame-dragging of spacetime in the exterior region. Using the coupled Einstein-Maxwell equations it is found that in the slow rotation approximation charge accumulation not only acts like an additional mass, thus modifying the spherically symmetric part of the spacetime, the electric charge also contributes directly to the dragging of spacetime. A modified Lense-Thirring formula for the spacetime frame dragging frequency is obtained and its implication for rotating charged compact stars is discussed.     

Corrected Entropy of BTZ Black Holes

In this paper, corrected entropy of a class of BTZ black holes, include charge and rotation, studied. We obtain corrected Bekenstein-Hawking entropy and find that effect of charge viewed at order A ^?2, and effect of rotation viewed at order A ^?6, therefore Q and J don’t have contribution in corrected entropy lower than the second order. We also write the first law of black hole thermodynamics for the case of charged rotating BTZ black hole.     

A plane symmetric solution of Einstein's field equations of general relativity containing zero-rest-mass scalar fields

An exact static solution of Einstein's field equations of general relativity in the presence of zero-rest-mass scalar fields has been obtained when both the metric tensor gijand the zero-rest-mass scalar field φexhibit plane symmetry in the sense of Taub [9]. Our solution generalizes the empty space-time solution with plane symmetry previously obtained by Taub to the situation when static zero-rest-mass scalar fields are present. The static plane symmetric solutoins of Einstein's field equations in the presence of massive scalar fields, and the difference between the massless and non-massless scalar fields are being investigated, and will be published separately later on. We also hope to discuss non-static plane symmetric solutions of Einstein's field equations in the presence of scalar fields in future.     

Derivation of the Gross-Pitaevskii Equation for Rotating Bose Gases

We prove that the Gross-Pitaevskii equation correctly describes the ground state energy and corresponding one-particle density matrix of rotating, dilute, trapped Bose gases with repulsive two-body interactions. We also show that there is 100% Bose-Einstein condensation. While a proof that the GP equation correctly describes non-rotating or slowly rotating gases was known for some time, the rapidly rotating case was unclear because the Bose (i.e., symmetric) ground state is not the lowest eigenstate of the Hamiltonian in this case. We have been able to overcome this difficulty with the aid of coherent states. Our proof also conceptually simplifies the previous proof for the slowly rotating case. In the case of axially symmetric traps, our results show that the appearance of quantized vortices causes spontaneous symmetry breaking in the ground state.     

Stationary Electromagnetic Fields of a Slowly Rotating Magnetized Neutron Star in General Relativity

Following the general formalism presented by Rezzolla, Ahmedov and Miller, ⁽¹⁾ we here derive analytic solutions of the electromagnetic fields equations in the internal and external background spacetime of a slowly rotating highly conducting magnetized neutron star. The star is assumed to be isolated and in vacuum, with a dipolar magnetic field not aligned with the axis of rotation. Our results indicate that the electromagnetic fields of a slowly rotating neutron star are modified by general relativistic effects arising from both the monopolar and the dipolar parts of the gravitational field. The results presented here differ from the ones discussed by Rezzolla, Ahmedov and Miller ⁽¹⁾ mainly in that we here consider the interior magnetic field to be dipolar with the same radial dependence as the external one. While this assumption might not be a realistic one, it should be seen as the application of our formalism to a case often discussed in the literature.     

Gravitational redshift of gravitational clocks

The gravitational redshift of gravitational clocks in a given weak gravitational field, within general relativity, is considered. Because the clocks in question have structure and dynamics determined by gravitational interactions, the full machinery of Einstein's equations must be used. Three specific examples are treated: (i) the redshift of the angular frequency of a rotating relativistic star in the gravitational field of a distant body; (ii) the redshift of the angular velocity of a slowly rotating black hole surrounded by an axisymmetric ring of matter; and (iii) the redshift of the observed orbital period of a nearly circular, post-Newtonian binary system in the field of a distant, third body. In all three cases the redshift is the same as if the clocks were non-gravitational, thereby governed by the Einstein equivalence principle. For an observer at infinity, the redshift $\text{Z ≡}\text{Δv}\text{v}$ is given by $\text{Z =}\text{?Gm}\text{Rc}{}^2$, where m is the mass of the distant object and R its distance from the clock in question. The result is independent of how relativistic the clock may be. The significance of this conclusion for the binary pulsar PSR 1913 + 16, where the gravitational redshift of the pulsar's frequency caused by the gravitational field of its companion is an observable effect, is discussed. The extent to which this result is a manifestation of the strong equivalence principle, satisfied, as far is known, only by general relativity, is also noted.     

Finite element modelling for the investigation of in-plane modes and damping shims in disc brake squeal

Squeal propensity of the in-plane modes and the constrained-layer type damping shims for disc brake system is investigated by using the finite element method. Theoretical formulation is derived for a rotating disc in contact with two stationary vibrating pads attached to the damping shim components. By the conversion from the theoretical to FE brake model, the full equations of motion for the actual disc brake system describes the disc rotation, the in-plane friction characteristics and damping shims in association with squeal vibration. It is concluded from the results that the in-plane torsion modes can be generated by the negative friction slope, but they cannot be controlled by the damping shims. The in-plane radial mode is also investigated and found to be very insensitive in squeal generation.     

Spherical Symmetric Perfect Fluid Collapse in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>, <Emphasis Type="Italic">T</Emphasis>) Gravity

This paper contains the study of spherically symmetric perfect fluid collapse in the frame work of f(R, T) modified theory of gravity. We proceed our work by considering the non-static spherically symmetric background in the interior and static spherically symmetric background in the exterior regions of the star. The junction conditions between exterior and interior regions are presented by matching the exterior and interior regions. The field equations are solved by taking the assumptions that the Ricci scalar as well as the trace of energy-momentum tensor are to be constant, for a particular f(R, T) model. By inserting the solution of the field equations in junction conditions, we evaluate the gravitational mass of the collapsing system. Also, we discuss the apparent horizons and their time formation for different possible cases. It is concluded that the term f(R ₀, T ₀) behaves as a source of repulsive force and that’s why it slowdowns the collapse of the matter.     

All flow-stationary cylindrically symmetric solutions of the Einstein field equations for a rotating isentropic perfect fluid

In previous papers by the author a class of flow-stationary cylindrically symmetric solutions of the Einstein field equations for a rotating isentropic perfect fluid was found. The present paper shows how all such solutions may be obtained by methods very similar to those used previously. The solutions depend on one variable and contain one completely arbitrary function f of that variable. The choice of a definite form of f corresponds to fixing the equation of state. After this is fixed, the enthalpy per unit rest-energy of the fluid, H, is determined by a linear homogeneous differential equation of second order, and all the other components of the metric are algebraically determined in terms of f and H.     

Coupling of vibration,rotation, and electron spin in multiplet states

The effective Hamiltonian of Van Vleck (1) for energy calculations of rotating molecules in multiplet states, is extended to include the effect of vibrational angular momentum, within a given nondegenerate electronic state. H_C and H_S operators are defined, to account for vibration-rotation (Coriolis) and spin-vibration interactions, respectively. Such interactions are discussed for degenerate E vibrational states of symmetric tops, as well as for the occurrence of vibrational angular momentum involving different vibrational states, with accidentally low energy separation. Following the classification of Hund's case (b) for the spin-rotation wave function (coupled representation), we find that H_C operators do not mix different F components, whereas H_S operators give also matrix elements between F components differing by one unit in the quantum number N, for given J.     

Effect of helical force on the stationary convection in a rotating ferrofluid

In this paper we studied the onset of instability in a horizontal layer of a rotating ferrofluid in the presence of the helical force. The analytical expression of the Rayleigh number of the system is determined as a function of the dimensionless numbers obtained. Then, the effect of each dimensionless parameter is studied. The helical force, the binary parameter ψ then the magnetic parameters M₁, M₃ and ψ_m accelerate the onset of stationary convection whereas the rotation and the magnetic parameter M₂ delay it. Also all the magnetic parameters, the binary parameter and the rotation cause the convection rolls to shrink while only the helical force increases the size of these structures.     

Extended perturbation formulae for Jahn-Teller molecules

The effect of the Jahn-Teller interaction on a symmetric top molecule with a threefold axis of symmetry in a ^2S + 1E state has been investigated by perturbation theory. Contributions up to sixth order are included. Explicit formulae for various quantities have been derived on the assumption that there is only one Jahn-Teller active mode of vibration; both linear and quadratic Jahn-Teller interactions are considered. The quantities concerned are (i) the vibronic energy levels, (ii) the orbital quenching factor d_t, (iii) the correction to the A-rotational constant, (iv) the correction to the spin-spin dipolar coupling term, and (v) the correction to the spin-rotation coupling constant ε_aa. Because the perturbation expansion converges slowly, the results are only applicable to molecules subject to a weak Jahn-Teller effect. There are several examples of this type of molecule which have been studied experimentally.     

Lagrangian evolution of velocity increments in rotating turbulence: The effects of rotation on non-Gaussian statistics

The effects of rotation on the evolution of non-Gaussian statistics of velocity increments in rotating turbulence are studied in this paper. Following the Lagrangian evolution of the velocity increments over a fixed distance on an evolving material element, we derive a set of equations for the increments which provides a closed representation for the nonlinear interaction between the increments and the Coriolis force. Applying a restricted-Euler-type closure to the system, we obtain a system of ordinary differential equations which retains the effects of nonlinear interaction between the velocity increments and the Coriolis force. A priori tests using direct numerical simulation data show that the system captures the important dynamics of rotating turbulence. The system is integrated numerically starting from Gaussian initial data. It is shown that the system qualitatively reproduces a number of observations in rotating turbulence. The statistics of the velocity increments tend to Gaussian when strong rotation is imposed. The negative skewness in the longitudinal velocity increments is weakened by rotation. The model also predicts that the transverse velocity increment in the plane perpendicular to the rotation axis will have positive skewness, and that the skewness will depend on the Rossby number in a non-monotonic way. Based on the system, we identify the dynamical mechanisms leading to the observations.