Vertical Inertia Research Articles

Relationship of Critical Flow in Waterfall to Minimum Energy Head

An analytical solution of the nonlinear equations of the theory of a directed fluid sheet (which includes vertical inertia) is used to prove that for constant depth at the waterfall's brink and variable far upstream depth, the energy head in steady flow in a free waterfall is minimized when the flow is critical. This result provides additional physical understanding of the special nature of critical flow in a waterfall and supplies further justification for using the free waterfall as a self-calibrated flow meter, as suggested by Rouse (1936). Also, this proof should not be confused with the more common proof of minimum energy head in a uniform stream because here the flow rate is not maintained constant and the fluid depth contracts as gravity accelerates the flow near the waterfall's brink.

Wave forces on a large, horizontal submerged cylinder

Abstract A numerical boundary integral equation method combined with a non-linear time stepping procedure is used for the calculation of wave forces on a large, submerged, horizontal circular cylinder. As the method is based on potential theory, all computations are performed in the inertia dominated domain, that is, for small Keulegan-Carpenter numbers. Computations are carried out for the Eulerian mean current under wave trough level equal to zero. When the cylinder is moved towards the sea bed the computations show that the inertia coefficients increase significantly, which is associated with a blockage effect. Furthermore, the effect of the wave steepness is reduced when the submergence of the cylinder is increased. In the vicinity of the free water surface the vertical inertia coefficient is highly dependent upon the wave steepness, which tends to reduce it, whereas the horizontal inertia coefficient is only slightly dependent on the wave steepness. Computations are also carried out for cylinder diameters comparable with the wave length. Finally, inertia coefficients computed by the present method are compared with some analytical results by Ogilvie [(1963), First and second order forces on a cylinder submerged under a free surface. J. Fluid Mech. 16, 451–472]. As long as the assumptions leading to Ogilvie's theory are fulfilled (cylinder radius small compared to the wave length), the results are quite similar.

Forces on a circular cylinder in elliptical orbital flows at low Keulegan-Carpenter numbers

Flows around a circular cylinder in elliptical orbital flows have been computed using the random vortex method for situations where the flow is attached and laminar; the Keulegan-Carpenter number is less than or equal to 1·5. The forces are related to the flow characteristics and steady streaming is shown to have a significant influence for ellipticity even as low as 1 8 , depending on the Keulegan-Carpenter number. The interdependence of horizontal and vertical inertia coefficients and of horizontal and vertical drag coefficients on ellipticity may be modelled simply for certain ranges of Keulegan-Carpenter number and ellipticity. Force components at frequencies other than the orbital frequency can occur and cause scatter in values of drag coefficient in successive half cycles for the higher Keulegan-Carpenter numbers investigated; this does not occur in the linear oscillatory case (zero ellipticity). Previous experience suggests that there will be qualitatively similar effects at very high, full-scale Reynolds numbers with turbulent flow.

Inertia regime loading on a submarine pipeline in random waves

An experimental investigation has been carried out on random wave-induced forces on a smooth horizontal submarine pipeline held fixed at various gaps from a plane boundary. The pipeline was subjected to random waves with Pierson-Moskowitz spectrum (P-M spectrum) at various energy levels to achieve different significant wave conditions. Based on the significant wave concept, the pipeline is considered to lie in the predominantly inertia regime of wave loading. The analysis of in-line and transverse forces are carried out in the frequency domain using spectral density method. While the in-line force is analysed considering only the inertia term of the classical Morison equation, a new spectral density model is developed for the transverse forces on a pipeline in random waves. The in-line hydrodynamic coefficient of inertia and the transverse hydrodynamic coefficients of lift and vertical inertia are evaluated in the frequency domain and through the use of least-squares technique. The in-line and transverse hydrodynamic coefficients are found to be a function of gap ratio of the pipeline from the plane boundary and correlate very well with the potential flow results at least within the range of significant Keulegan-Carpenter number or period parameter encountered in this study. Further, the results of the random wave force tests are compared with those of regular waves under similar pipeline conditions.

Time domain analysis of random wave forces on pipelines in the inertia regime

An experimental investigation on random wave induced forces on a smooth submarine pipeline, fixed horizontally near a plane boundary, is carried out in the inertia-dominated regime. The water particle kinematics at the centre line of the pipe are generated using linear numerical filters, derived from the time history of water surface elevation. The pipeline was subjected to Pierson-Moskowitz spectrum (P-M spectrum) at various energy levels to achieve different significant wave conditions. The in-line hydrodynamic coefficient of inertia and the transverse hydrodynamic coefficients of lift and vertical inertia are evaluated utilizing the measured forces and through the use of least-squares method in the time domain. These hydrodynamic coefficient are found to be a function of gap ration of the pipeline from the plane boundary and correlate very well with the potential flow results within the range of the Keulegan-Carpenter number or period parameter investigated.

Oscillations over basins of variable depth

This paper is concerned with oscillations of a body of water in basins of variable depth, employing a system of linearized equations which can be obtained from the theory of a directed fluid sheet for an incompressible, homogeneous, inviscid fluid (Green & Naghdi 1976a, 1977). For free oscillations over a level bottom, an assessment of the range of validity of the linearized theory is made by an appropriate comparison with a corresponding well-known exact solution (Lamb 1932). This assessment indicates an ‘intermediate’ range of validity for the linearized theory not covered by usual classical approximations for long waves. Encouraged by this assessment, we apply the linear theory of a directed fluid sheet to basins of variable depth; and, in particular, consider a class of basin profiles whose equilibrium depth (along its width) varies in one direction only. By a method of asymptotic integration, a general solution is obtained which is relatively simple and accounts for the effect of vertical inertia. The solution is sinusoidal in time, periodic along the breadth direction and involves Bessel functions of the first order in the width direction. For two special basin profiles, detailed comparisons are made between the predictions of the asymptotic solution (i.e. the frequencies in the lowest modes of oscillations) with corresponding results obtained by other procedures.

Studies on wave-subsea pipeline interaction

Estimation of wave induced loadings on subsea pipelines involves the evaluation of the hydrodynamic coefficients of drag, inertia, vertical inertia and lift. These coefficients vary with the methodology adopted and the order of wave theory used to compute the water particle kinematics. This paper presents a new and simple method to determine the hydrodynamic coefficients for subsea pipelines in the drag and inertial flow regime. The results are reported for three relative clearances of a pipeline model from the simulated ocean floor.

鋼構造骨組の耐震設計用動力学モデルに関する研究 : その 1・鉛直地動が地震応答に及ぼす影響に関する一考察

As the basic research for the study of developing a kinematic model, the investigation reported in this paper is an attempt to clarify the effect of vertical ground motion on the seismic response, which is usually ignored in current earthquake resistant design practice. To examine the effect of vertical ground motion on the seismic response, the inelastic response of steel frames subjected to the combination of horizontal and vertical components of the earthquake was compared with that to a horizontal components of the earthquake. The inclusion of vertical ground motion can result in a significant increase in the ductility requirements of the upper story members, of which the design is controlled by the vertical loading. It also seems to be probable that the inelastic deformation of the lower story columns in high-rise building may be affected by vertical ground motion in terms of increasing axial forces. It is concluded that effects of vertical ground motion on the seismic response are large enough to warrant consideration in earthquake resistant design practice. It should be noted that the inelastic interaction between horizontal inertia forces caused by horizontal ground motion and vertical inertia forces by vertical ground motion may play an important role to explain the effects of vertical ground motion on the seismic response.

Nonlinear electroelastic equations cubic in the small field variables

The nonlinear differential equations and boundary conditions containing terms up to cubic in the small field variables are obtained from general rotationally invariant nonlinear electroelastic equations derived previously. The electroelastic equations cubic in the small field variables are considerably more tractable than the general electroelastic equations and are applicable in the description of such phenomena as the dependence of wave velocities on wave amplitudes and resonant frequencies on vibration amplitudes in addition to a host of other nonlinear phenomena. The nonlinear constitutive equations for an isotropic purely elastic solid containing terms up to cubic in the small mechanical displacement gradients are presented. The nonlinear equations for the extensional motion of thin isotropic plates containing terms up to cubic in the small mechanical displacement gradients in the plane of the plate are obtained and the influence of the vertical inertia is included, in addition to the extensional stiffness and inertia, when the plating is attached to an electroelastic solid. Subject Classification: 40.24.

Finite Earthquake-Response of Inelastic Structure

A numerical procedure based on an absolute minimum principle has been developed for investigating the dynamic response of an inelastic structure at finite deformation. The inelastic response of a wide flange steel column, which is built in at the base and attached to a heavy mass at the top, to combined vertical and horizontal earthquake-type excitations, has been investigated. It has been shown that when the weight of the tip mass is much less than the Euler buckling load of the column, the dynamic response of the structure subjected to combined vertical and horizontal excitation does not deviate much from that caused by the horizontal excitation alone. However, when the tip mass is comparatively large, the vertical inertia force of the tip mass caused by the vertical excitation may influence the resultant motion of the structure under a combination of vertical and horizontal excitations, and consequently, the mode and time of collapse of the structure.

Dynamic Analysis of Coupled Shear Walls and Sandwich Beams

A study is made of the free vibration of planar coupled shear walls, a common lateral load-resisting configuration in building construction where two walls are coupled together by a system of discrete spandrel beams. The differential equations and boundary conditions are obtained by the variational method, and by assuming that the spandrels can be replaced by a continuous system of laminae, or small beams. Natural frequencies and mode shapes are determined, and the results are presented in a number of figures from which the natural frequencies of any coupled shear wall can be extracted. The importance of including vertical displacement in the analysis is discussed, and a study of the effect of neglecting the vertical inertia term is given. These cases are illustrated with graphs and with one specific example. Investigations are also made of the asymptotic behavior of the system as the spandrels become weak, as they become stiff, and as the frequencies become large. Finally, the theory of sandwich beams is presented and compared to that for coupled shear walls. It is observed that for most cases of constant properties, the differential equations (and boundary conditions) reduce to the same mathematical form for both theories.

New Concept To Design Structures Which CanWithstand Earthquake Forces

Any structure has structural and non structural masses. Those masses are usually integrated together When an earthquake strikes, those huge masses create destructive lateral and vertical inertia forces proportionate to the masses and their accelerations, regardless if they are structural or nonstructural masses This new concept divides the structure to two parts The part which is connected to the ground is called the main supporting structure It should be made as rigid as possible and as massless as possible to increase its fundamental natural frequency to a level higher than the earthquake frequency if possible. The rest of the secondary structural and nonstructural masses are supported on top of the main supporting structure, such that they can swing and move laterally to certain extent under springs restrains. The main supporting structure will have small inertia forces because of its smaller masses. Its high rigidity will enable it to withstand such small inertia forces. The swinging-hanging structural and non structural masses will have extremely low frequency, thus will have very small inertia forces, which can be resisted by the strength of any regular structural elements

An approximate method of calculating a counterweight for the balancing of vertical inertia forces

The work of S.N. Bernshtein is used to find a first order harmonic which accomplishes the best uniform approximation (in the sense of Chebyshev) to certain types of truncated periodic functions. The method is applied to the partial balancing by counterweights of a unidirectional inertia force.

Asymmetrical Lap and Other Non-Linearities in Valve-Controlled Hydraulic Actuators

The results of a theoretical investigation into the effect of asymmetrical lap on the open-loop performance of valve-controlled hydraulic actuators are presented. The governing differential equations of the system are developed in a non-dimensional form. The effects of supply lap and exhaust lap on actuator peak pressures, ram amplitude and cavitation, as well as the effect of inertia load on actuator peak pressure and output amplitude, are shown. When the open-loop frequency response of the hydraulic servomechanism is attempted experimentally the ram drifts until it hits its stops. In previous theoretical analyses this effect has not been noted. This paper demonstrates that the inclusion of a term ignored in previous work highlights this effect. This same term is shown to give a kind of feedback when the system drives a vertical inertia load.