Coriolis Acceleration Terms Research Articles

Development of a Three-Dimensional Hydrodynamic Model Based on the Discontinuous Galerkin Method

Though the discontinuous Galerkin method is attracting more and more applications in many fields due to its local conservation, high-order accuracy, and flexibility for resolving complex geometries, only a few three-dimensional hydrodynamic models based on the discontinuous Galerkin method are present. In this study, a three-dimensional hydrodynamic model with a σ-coordinate system in the vertical direction is developed. This model is discretized in space using the discontinuous Galerkin method and advanced in time with the implicit-explicit Runge–Kutta method. Numerical tests indicate that the developed model is convergent and can obtain better results with a smaller computational time when a higher approximation order is adopted. Other tests with exact solutions also indicate that the developed model can well simulate the vertical circulation under the effect of surface wind stress and the flow under the combined effect of wind stress and Coriolis acceleration terms. The simulation results of tidal flow in part of Bohai Bay, China, indicate that the model can be used for the simulation of tidal wave motion in realistic situations.

A universal modelling approach and a response index to the vibration analysis for space parallel robots by means of screw theory

Vibration analysis is very important for evaluating the vibration performance of some space parallel robots (SPRs), but most of the matrix analysis methods are case-by-case treatment and the vibration response is difficult to estimate accurately in the design phase. In order to enrich this field, a universal vibration modelling approach and a vibration response index are proposed in this paper. Utilizing the strong universality of the screw theory, a universal dynamic modelling method is proposed, in which the complicated Lie algebra terms are appear explicitly and the Coriolis acceleration term is intentionally independent of each other. Based on the Rayleigh quotient and considering the effects of gravity and joint clearances, a vibration response index is presented to reflect the vibration amplitude of SPRs under the unit initial conditions. Based on the built experimental prototype and some applications, numerical examples are presented and some performances of the vibration index have been fully discussed, and the results show that the mean error between the mathematical models and experimental results is about 5.4%. When researchers are engaged in vibration research on different SPRs, we expect that the proposed mathematical model and the proposed index will provide a convenient way for vibration modelling and optimization.

A Rotor Looseness‒Rub-Impact‒Resonance Fault Mechanism Modeling

In view of the limitation of the numerical method for coupling faults of rotating machinery, a modeling method for coupling faults of rotor, DeformationMotion Vector Analysis Method, is purposed in order to explain the fault mechanism more thoroughly and universally. Firstly, the “Deformation-Motion” is defined by fault and the relationship between fault deformation and periodic signal is established by Fourier series. Secondly, the “Vector Analysis” analyzes the deformation-motion in the dynamic coordinate system, and establishes the relationship between the fault deformation-motion and the periodic excitation force by using the dynamic and static coordinate systems. Thirdly, the analytical solution is obtained by harmonic balance method, and the coriolis acceleration term is first found in the analytical solution of rotating machinery failure. By further analysis of the analytical solution, we find that the general trend of the natural frequency of the rotor system will gradually approach the excitation source frequencies and eventually lead to the resonance: that is the law of Natural Frequency Migration Resonance Theory. Finally, the validity of the modeling method and the correctness of the analytical solution are verified by engineering practice

The generalized Ekman model for the tropical cyclone boundary layer revisited: The myth of inertial stability as a restoring force

AbstractWe revisit the linear boundary‐layer approximation that expresses a generalized Ekman balance and use it to clarify a range of interpretations in the previous literature on the tropical cyclone boundary layer. Some of these interpretations relate to the reasons for inflow in the boundary layer and others relate to the presumed effects of inertial stability on boundary‐layer dynamics. Inertial stability has been invoked, for example, to explain aspects of boundary‐layer behaviour, including the frontogenetic nature of the boundary layer and its relationship to vortex spin‐up. Our analysis exposes the fallacy of invoking inertial stability as a resistance to radial inflow in the boundary layer. The analysis shows also that the nonlinear acceleration terms become comparable to the linear Coriolis acceleration terms in relatively narrow vortices that are inertially stable above the boundary layer. Estimates of the nonlinear accelerations using the linear solutions are expected to underestimate the actual contribution in a nonlinear boundary‐layer model, cautioning against neglecting the nonlinear terms in diagnostic or prognostic models.

Reduction of the linear stability problem for general zonal flows with any Rossby and Richardson numbers to a PDE

The stability to linear disturbances of a general laterally and/or vertically sheared zonal flow in thermal wind balance is considered for any Rossby number and any Richardson number. The problem for normal mode perturbations is reduced to a second‐order partial differential equation (PDE) for a single dependent variable, the pressure perturbation, in which the complex phase speed is an eigenvalue. It is shown that the boundary conditions of no normal flow are expressible in terms of the pressure and its first‐order derivatives, and that the PDE can be derived from the linearized equation for conservation of the Ertel potential vorticity. The derivations are performed first for an incompressible Boussinesq fluid in Cartesian coordinates allowing the perturbations to be either hydrostatic or non‐hydrostatic. They are shown to generalize to a perfect gas and to motions on a sphere. A PDE for general (non‐normal mode) linear perturbations can also be derived. The Ertel potential vorticity of the zonal flow is shown to arise in the component of the momentum equation parallel to the zonal flow in a generalized Coriolis acceleration term. An ordinary differential equation (ODE) is also derived for the vertical velocity perturbation to laterally uniform zonal flows which is consistent with previous results. The differences between the singularities of the equations for pressure and vertical velocity perturbations are discussed. Integral constraints on the perturbations are used to prove the standard necessary condition for symmetric instabilities and a generalization of the Charney–Stern–Pedlosky integral constraints.

Parametric resonance analysis of rectangular plates subjected to moving inertial loads via IHB method

This paper deals with the induced instability due to parametric resonance of rectangular plates traversed by inertial loads and lying on elastic foundations. The extended Hamilton's principle is employed to derive the partial differential equation associated with the transverse motion of the plate. Subsequently, this equation is transformed into a set of ordinary differential equations by the Galerkin method. Including vertical, centripetal and Coriolis acceleration terms related to the moving mass transition in the analysis leads to governing equations with time-varying mass, damping and stiffness coefficients. Particularly, the intermittent passage of masses along rectilinear paths, or the motion of an individual mass along an orbiting path, permits to subcategorize the problem as a parametrically excited system with periodic coefficients. By applying the incremental harmonic balance (IHB) method, the stability of the induced plate vibrations is investigated, revealing an emersion of instability tongues in the parameters plane. Semi-analytical results are provided for various boundary conditions of the plate which got verified through direct numerical simulations and other results reported in the literature.

Acoustic nonreciprocity in Coriolis mean flow systems.

One way to break acoustic reciprocity is to have a moving wave propagation medium. If the acoustic wave vector and the moving fluid velocity are collinear, the wave vector shift caused by the fluid flow can be used to break. In this paper, an alternative approach is investigated in which the fluid velocity enters the differential equation of the system as a cross product term with the wave vector. A circular field where the fluid velocity increases radially has a Coriolis acceleration term. In such a system, the acoustic wave enters from the central wall and exits from the perimeter wall. In this paper, the differential equation is solved numerically and the effect of fluid velocity on the nonreciprocity factor is examined.

Breaking time reversal symmetry with coriolis mean flow systems

One way to break time reversal symmetry is to have a moving wave propagation medium. If the acoustic wave vector and the moving fluid velocity are collinear, such as the one created by Fluery et al. (Science, 343, 2014) we can use the wave vector shift caused by the fluid flow to break reciprocity. An alternative approach we have taken is to use a fluid velocity field which enters the differential equation of the system as a cross product term with the wave vector. A circular field in which the fluid velocity increases radially has a Coriolis acceleration term. In this system, the acoustic wave enters from the central wall and exits from the perimeter wall. Equations of conservation of mass and momentum, after linearization can be simplified to: (∂/∂t+V.▽)P+ρC2▽.(u+V) = 0, ρ∂/∂t(u+V)+ρ(V.▽)u+ρ(u.▽)V+ρ(V.▽)V+▽P = 0, in which u is the particle acoustic velocity and V = rωeθis the fluid velocity due to the rotational velocity field. It can be shown that the third term in the second equation can be simplified as:ρωek×u. This term Coriolis acceleration induces nonreciprocity. In this work, we solved the differential equation numerically and investigated the effect of fluid velocity on the nonreciprocity factor.

Acoustic, hydrodynamic and thermal modes in a supersonic cold jet

Large-eddy simulation data for a Mach 1.3 round jet are decomposed into acoustic, hydrodynamic and thermal components using Doak’s momentum potential theory. The decomposed fields are then analysed to examine the properties of each mode and their dynamics based on the transport equation for the total fluctuating enthalpy. The solenoidal fluctuations highlight hydrodynamic components of the jet and capture the shear layer growth and breakdown process. The acoustic mode exhibits a jittering coherent wavepacket structure in the turbulent region and consequent highly directional downstream radiation. The expected radial decay rates,$r^{-6}$for hydrodynamic and$r^{-2}$for acoustic, are recovered and closely follow the universal radiation spectra in the sideline and downstream directions. The scalogram of the acoustic mode in the near-acoustic-field region is consistent with that of the pressure perturbation signal in the acoustic-frequency range, but effectively removes the hydrodynamic and thermal content. The time-resolved and mean behaviour of terms in the total fluctuating enthalpy equation is analysed in detail. A large-scale intermittent event in the near-acoustic field is shown to be associated with an intrusion of vortices from the shear layer into the core of the jet. Acoustic sources are created when the resulting negative fluctuations in the solenoidal component interact with positive fluctuations in the Coriolis acceleration term. The latter are associated with regions of high vorticity on the inner side of the shear layer. In contrast, sinks result from the interaction of solenoidal momentum fluctuations with positive entropy gradients along entrainment streaks.

터보 압축기 다단 회전축계의 진동 및 안정성 연구

The mathematical modeling on the free vibration and stability of a multi-stepped shaft of turbo compressor is performed in this study. The multi-stepped shaft is modeled as a non-uniform Timoshenko beam supported by anisotropic bearings. It is assumed that the shaft is spinning with constant speed about its longitudinal axis and subjected to a conservative axial force induced by front and rear impellers attached to the shaft. The structural model incorporates non-classical features such as transverse shear and rotary inertia. A structural coupling between vertical and lateral motions is induced by Coriolis acceleration terms. The governing equations are derived via Hamilton’s variational principle and the equations are transformed to the standard form of an eigenvalue problem. The implications of combined gyroscopic effect, conservative axial force, bearing stiffness and damping are revealed and a number of pertinent conclusions are outlined. In this study analytical results are compared with those from ANSYS finite element analysis and experimental modal testing.

Explicit Dynamic Formulation for n-R Planar Manipulators with Frictional Interaction between End-Effecter and Environment

This paper includes modeling and dynamic analysis of the planar manipulators which consist of n revolute joints and its end-effecter may possess a frictional contact with an arbitrary surface. The dynamic analysis has been done by using the Lagrangian method. The main contribution of this work is presenting the equations of motion of the system in an explicit and straightforward manner. To this end, the absolute angles of the links are chosen as the generalized coordinates to constitute the Lagrangian, and by this way the Coriolis acceleration terms included the product of the joint velocities, will be hidden in structure of the equations of motion. It is very interested in the simplified results that show a clear conformity with the interpretations of the Euler's equation of motion of the rigid body. In following, the results are applied to a case study model to explain the deriving the equations of motion by our formulation. An inchworm robot included an articulated mechanism is considered to this end.

The vertical spanning strip wall as a coupled rocking rigid body assembly

The equation of motion of a one way (vertical) spanning strip wall, as an assembly of two rigid bodies, is presented. Only one degree of freedom is needed to completely describe the wall response as the bodies are assumed to be perfectly rectangular and are allowed to rock but not to slide horizontally. Furthermore, no arching action occurs since vertical motion of the upper body is not restrained. Consequently, the equation of motion is nonlinear, with non constant coefficients and a Coriolis acceleration term. Phenomena associated with overburden to self weight ratio, motion triggering, impulsive energy dissipation, amplitude dependency of damping and period of vibration, and scale effect are discussed, contributing to a more complete understanding of experimental observations and to an estimation of system parameters based on the wall characteristics, such as intermediate hinge height and energy damping, necessary to perform nonlinear time history analyses. A comparison to a simple standing, or parapet, wall is developed in order to better highlight the characteristics of this assembly.

Three-Dimensional Vibration Analysis of Rotating Laminated Composite Blades

This work offers the first known three-dimensional continuum vibration analysis for rotating, laminated composite blades. A cornerstone of this work is that the dynamic energies of the rotating blade are derived from a three-dimensional elasticity-based, truncated quadrangular pyramid model incorporating laminated orthotropicity, full geometric nonlinearity using an updated Lagrangian formulation and Coriolis acceleration terms. These analysis sophistications are included to accommodate the nonclassical directions of modern blade designs comprising thin, wide chord-lifting surfaces of laminated composite construction. The Ritz method is used to minimize the dynamic energies with displacements approximated by mathematically complete polynomials satisfying the vanishing displacement conditions at the blade root section exactly. Several tables and graphs are presented that describe numerical convergence studies showing the validity of the assumed displacement polynomials used herein. Nondimensional frequency data are presented for various rotating, truncated quadrangular pyramids, serving as first approximations of practical blades employed in aircraft engines and fans. A wide scope of results explain the influence of a number of parameters coined to rotating, laminated composite blade dynamics, namely aspect ratio (a/b), chord ratio (c/b), thickness ratio (b/h), variable thickness distribution (h1/ht), blade pretwist angle (φ0), composite fiber orientation angle (θ), and angular velocity (Ω). Additional examples are given that elucidate the significance of the linear and nonlinear kinematics used in the present three-dimensional formulation along with the importance of the Coriolis acceleration terms included in the analysis.

Two-Dimensional, Nonequilibrium, Wet-Steam Calculations for Nozzles and Turbine Cascades

The paper describes a method of computing nonequilibrium, steady flows of wet steam in two- and quasi-three-dimensional turbine cascades. The mixture conservation equations are solved in an Eulerian reference frame using an inviscid time-marching method that includes the effects of the centrifugal and Coriolis acceleration terms in rotating blade rows. Nucleation and growth of water droplets are computed by integrating the relevant equations along true streamlines in a Lagrangian reference frame. Steam properties are computed using equations that display commercial steam table accuracy for pressures below 10 bar. Special procedures for grouping the range of droplet sizes present are described that allow an accurate representation of the droplet size distribution to be retained without requiring a large increase in CPU time. All types of wet-steam flow, including those involving secondary nucleation, can be computed. Examples are presented that display the sensitivity of the calculation procedure in computing nucleation affected by the shock- and expansion-wave structure in the region of a turbine blade trailing edge. Typical CPU time requirements for nonequilibrium solutions involving primary or secondary nucleations are about three times those for perfect gas calculations.

On the out-of-plane vibrations of thick rotating rings

Equations of motion for the out-of-plane vibrations of rotating thick circular rings are derived. The effects of shear deformation and rotatory inertia are included in the formulation. Natural frequencies are obtained and indicate the effects of the rotation, the shear deformation, and the rotatory inertia. In particular, the relative contributions of the centrifugal and Coriolis acceleration terms to the natural frequencies are examined.

Seismic response of a structure subjected to rotational base excitation

Abstract A modal superposition method which can perform the seismic analysis of a structure subjected to translational and rotational base excitation is presented. Discussed are two different approaches to derive the equations of motion. In the first approach, the reference axes are fixed in space. While in the second approach, they are rigidly fixed at the base of the structure. For rotational base excitation, it is shown that the application of second approach results in equation of motion with asymmetric, time-dependent coefficient matrix due to presence of the Coriolis acceleration term. Analytic integration is used to integrate the modal equations of motion derived by the first approach. Most of the mode shapes have to be included in the time history analysis. The modal superposition method is applied to the seismic analysis of a building subjected to translational and rotational excitation. The displacement results and the computer cost of this analysis are compared with those of using the direct integration method. The computer cost associated with the modal superposition method is lower than that associated with the direct integration method.