Department of Mechanical Engineering

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Vibration control has always been a challenging problem to the helicopter designer. This paper addresses the problem of the formulation and solution of an active vibration control scheme for helicopters, based on the concept of active control of structural response. First, using a mathematical procedure employing the Fisher information matrix, optimum sensor locations have been identified in a three-dimensional model of a flexible fuselage structure. The reference parameters used in the selection process are the elements of a vector defined as the effective independence distribution vector and the condition number of the Fisher information matrix. It is observed that, irrespective of the excitation frequency, these optimally selected sensor locations experience relatively high levels of vibration. Then, using the measurement from these optimal sensor locations, a multi-input/multi-output control problem has been formulated and solved to obtain the active control forces required for vibration minimization in the helicopter fuselage. It is observed that sensor locations have a significant influence on the level of vibration reduction in a fuselage structure.

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Journal of Aircraft

Selection of sensor locations for active vibration control of helicopter fuselages

Free, out-of-plane vibration of a rotating beam with a non-linear, elastomeric constraint has been investigated. The elastomer is modelled as a parallel combination of spring and damper elements. The stiffness and damping parameters are evaluated by using previous experimental data. The linear analysis is performed by two techniques, one based on a power series expansion and the other based on the Rayleigh-Ritz principle. The results are found to be in excellent agreement. In the non-linear analysis, a numerical-perturbation technique is applied to determine the frequency-amplitude relationship. A parametric study revealing the influence of the non-linear constraint is presented. © 1999 Academic Press.

Journal of Sound and Vibration

Free out-of-plane vibrations of a rotating beam with non-linear elastomeric constraints

Radial thin film flows are obtained by the impingement of circular free liquid jets on surfaces. The surfaces may be in the form of a circular plate, cone or that of a sphere. These flows are governed by the effects of inertia, viscosity, gravity and surface tension. Based on the film Reynolds number and Froude number, a circular hydraulic jump can be obtained in such flows. In this paper a new integral method is proposed for such axisymmetric laminar flows. The boundary layer approximation is used. The equations are solved using a cubic velocity profile, considering the radial hydrostatic pressure gradient in the film flow. In the new approach the coefficients of the cubic profile depend on the pressure gradient and body force terms and are allowed to vary with radial distance. Thus for example, separation can be predicted. The effect of the jet Reynolds number, Froude number and the surface dimension is considered. For flows with the circular hydraulic jump, the region upstream and downstream of the jump is solved separately using the boundary condition at the surface edge. Radial thin film flows are obtained by the impingement of circular free liquid jets on surfaces. The surfaces may be in the form of a circular plate, cone or that of a sphere. These flows are governed by the effects of inertia, viscosity, gravity and surface tension. Based on the film Reynolds number and Froude number, a circular hydraulic jump can be obtained in such flows. In this paper a new integral method is proposed for such axisymmetric laminar flows. The boundary layer approximation is used. The equations are solved using a cubic velocity profile, considering the radial hydrostatic pressure gradient in the film flow. In the new approach the coefficients of the cubic profile depend on the pressure gradient and body force terms and are allowed to vary with radial distance. Thus for example, separation can be predicted. The effect of the jet Reynolds number. Froude number and the surface dimension is considered. For flows with the circular hydraulic jump, the region upstream and downstream of the jump is solved separately using the boundary condition at the surface edge.

International Journal of Heat and Mass Transfer

Integral analysis applied to radial film flows

This paper deals with the stability and transition to turbulence of wall-bounded unsteady velocity profiles with reverse flow. Such flows occur, for example, during unsteady boundary layer separation and in oscillating pipe flow. The main focus is on results from experiments in time-developing flow in a long pipe, which is decelerated rapidly. The flow is generated by the controlled motion of a piston. We obtain analytical solutions for laminar flow in the pipe and in a two-dimensional channel for arbitrary piston motions. By changing the piston speed and the length of piston travel we cover a range of values of Reynolds number and boundary layer thickness. The velocity profiles during the decay of the flow are unsteady with reverse flow near the wall, and are highly unstable due to their inflectional nature. In the pipe, we observe from flow visualization that the flow becomes unstable with the formation of what appears to be a helical vortex. The wavelength of the instability ≃ 3δ + ̄ where δ + ̄ is the average boundary layer thickness, the average being taken over the time the flow is unstable. The time of formation of the vortices scales with the average convective time scale and is ≃ 33/(Δ + ̄u/δ + ̄), where Δu = (umax - umin) and umax, umin and δ are the maximum velocity, minimum velocity and boundary layer thickness respectively at each instant of time. The time to transition to turbulence is ≃ 39/(Δ + ̄u/δ + ̄). Quasi-steady linear stability analysis of the velocity profiles brings out two important results. First that the stability characteristics of velocity profiles with reverse flow near the wall collapse when scaled with the above variables. Second that the wavenumber corresponding to maximum growth does not change much during the instability even though the velocity profile does change substantially. Using the results from the experiments and the stability analysis, we are able to explain many aspects of transition in oscillating pipe flow. We postulate that unsteady boundary layer separation at high Reynolds numbers is probably related to instability of the reverse flow region.

Journal of Fluid Mechanics

Transition of unsteady velocity profiles with reverse flow

Experiments indicate that turbulent free convection over a horizontal surface (e.g. Rayleigh-Bénard convection) consists of essentially line plumes near the walls, at least for moderately high Rayleigh numbers. Based on this evidence, we propose here a two-dimensional model for near-wall dynamics in Rayleigh-Bénard convection and in general for convection over heated horizontal surfaces. The model proposes a periodic array of steady laminar two-dimensional plumes. A plume is fed on either side by boundary layers on the wall. The results from the model are obtained in two ways. One of the methods uses the similarity solution of Rotem & Classen (1969) for the boundary layer and the similarity solution of Fuji (1963) for the plume. We have derived expressions for mean temperature and temperature and velocity fluctuations near the wall. In the second approach, we compute the two-dimensional flow field in a two-dimensional rectangular open cavity. The number of plumes in the cavity depends on the length of the cavity. The plume spacing is determined from the critical length at which the number of plumes increases by one. The results for average plume spacing and the distribution of r.m.s. temperature and velocity fluctuations are shown to be in acceptable agreement with experimental results.

A model for near-wall dynamics in turbulent Rayleigh-Bénard convection

In this paper, we address the quality of the solution derivatives which are recovered from finite element solutions by local averaging schemes. As an example, we consider the Zienkiewicz-Zhu superconvergent patch-recovery scheme (the ZZ-SPR scheme), and we study its accuracy in the interior of the mesh for finite element approximations of solutions of Laplace's equation in polygonal domains. We will demonstrate the following: (1) In general, the accuracy of the derivatives recovered by the ZZ-SPR or any other local averaging scheme may not be higher than the accuracy of the derivatives computed directly from the finite element solution. (2) If the mesh is globally adaptive (i.e. it is nearly equilibrated in the energy-norm) then we can, practically always, gain in accuracy by employing the recovered derivatives instead of the derivatives computed directly from the finite element solution. (3) It is possible to guarantee that the recovered solution-derivatives have higher accuracy than the derivatives computed directly from the finite element solution, in any patch of elements of interest, by employing a mesh which is adaptive only with respect to the patch of interest (i.e. it is nearly equilibrated in a weighted energy-norm). (4) In practice, we are often interested in obtaining highly accurate derivatives (or heat-fluxes, stresses, etc.) only in a few critical regions which are identified by a preliminary analysis. A grid which is adaptive only with respect to the critical regions of interest may be much more economical for this purpose because it may achieve the desired accuracy by employing substantially fewer degrees of freedom than a globally adaptive grid which achieves comparable accuracy in the critical regions.

Computer Methods in Applied Mechanics and Engineering

Pollution error in the h-version of the finite element method and the local quality of the recovered derivatives

In References 1-3 we presented a computer-based theory for analysing the asymptotic accuracy (quality of robustness) of error estimators for mesh-patches in the interior of the domain. In this paper we review the approach employed in References 1-3 and extend it to analyse the asymptotic quality of error estimators for mesh-patches at or near a domain boundary. We analyse two error estimators which were found in References 1-3 to be robust in the interior of the mesh (the element residual with P-order equilibrated fluxes and (P+1) degree bubble solution or (p + 1) degree polynomial solution (ERpB or ERpPp+1; see References 1-3) and the Zienkiewicz-Zhu Superconvergent Patch Recovery (ZZ-SPR; see References 4-7) and we show that the robustness of these estimators for elements adjacent to the boundary can be significantly inferior to their robustness for interior elements. This deterioration is due to the difference in the definition of the estimators for the elements in the interior of the mesh and the elements adjacent to the boundary. In order to demonstrate how our approach can be employed to determine the most robust version of an estimator we analysed the versions of the ZZ estimator proposed in References 9-12. We found that the original ZZ-SPR proposed in References 4-1 is the most robust one, among the various versions tested, and some of the proposed 'enhancements' can lead to a significant deterioration of the asymptotic robustness of the estimator. From the analyses given in References 1-3 and in this paper, we found that the original ZZ estimator (given in References 4-7) is the most robust among all estimators analysed in References 1-3 and in this study. © 1997 by John Wiley & Sons, Ltd.

International Journal for Numerical Methods in Engineering

A model study of the quality of a posteriori error estimators for finite element solutions of linear elliptic problems, with particular reference to the behavior near the boundary

A new turbulence model has been proposed for the axisymmetric plume. In this model the equations for the turbulent kinetic energy (k), its dissipation (ε), temperature fluctuations (t′2), and intermittency (γ) have been solved. The equation for the dissipation (ε) accounts for the effect of entrainment, which in turn requires the solution of a transport equation for the intermittency. The effect of buoyancy on Reynolds stress and radial heat flux has been modelled by the algebraic stress model (ASM). For the buoyancy production of turbulent kinetic energy a simple model (which assumes it to be proportional to √kt′2) has been used. A detailed comparison of the predictions has been made with measurements. The present k - ε - t′2 - γ model is shown to simulate accurately the intermittency and the effects of buoyancy on mean and turbulent quantities. © 1997 by Elsevier Science Inc.

Applied Mathematical Modelling

A new turbulence model for the axisymmetric plume

The axially translating flexible beam with a prismatic joint can be modelled by using the Euler-Bernoulli beam equation together with the convective terms. In general, the method of separation of variables cannot be applied to solve this partial differential equation. In this paper, a non-dimensional form of the Euler Bernoulli beam equation is presented, obtained by using the concept of group velocity, and also the conditions under which separation of variables and assumed modes method can be used. The use of clamped-mass boundary conditions leads to a time-dependent frequency equation for the translating flexible beam. A novel method is presented for solving this time-dependent frequency equation by using a differential form of the frequency equation. The assumed model Lagrangian formulation of dynamics is employed to derive closed form equations of motion. It is shown by using Lyapunov's first method that the dynamic responses of flexural modal variables become unstable during retraction of the flexible beam, while the dynamic response during extension of the beam is stable. Numerical simulation results are presented for the uniform axial motion induced transverse vibration for a typical flexible beam. © 1996 Academic Press Limited.