Coupled bending-bending-torsion flutter of a mistuned cascade with nonuniform blades
01 Jan 1982-
TL;DR: In this paper, a set of aeroelastic equations describing the motion of an arbitrarily mistuned cascade with flexible, pretwisted, nonuniform blades is developed using an extended Hamilton's principle.
Abstract: A set of aeroelastic equations describing the motion of an arbitrarily mistuned cascade with flexible, pretwisted, nonuniform blades is developed using an extended Hamilton's principle. The derivation of the equations has its basis in the geometric nonlinear theory of elasticity in which the elongations and shears are negligible compared to unity. A general expression for foreshortening of a blade is derived and is explicity used in the formulation. The blade aerodynamic loading in the subsonic and supersonic flow regimes is obtained from two dimensional, unsteady, cascade theories. The aerodynamic, inertial and structural coupling between the bending (in two planes) and torsional motions of the blade is included. The equations are used to investigate the aeroelastic stability and to quantify the effect of frequency mistuning on flutter in turbofans. Results indicate that a moderate amount of intentional mistuning has enough potential to alleviate flutter problems in unshrouded, high aspect ratio turbofans.
Citations
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TL;DR: In this paper, it was shown that the double-layer capacitance decreases and the electro-osmotic mobility saturates at large voltages, due to steric repulsion and increased viscosity of the condensed layer.
Abstract: The classical theory of electrokinetic phenomena assumes a dilute solution of point-like ions in chemical equilibrium with a surface whose double- layer voltage is of order the thermal voltage, kBT/e = 25mV. In nonlinear 'induced-charge' electrokinetic phenomena, such as ac electro-osmosis, several volts ! 100kBT/e are applied to the double layer, and the theory breaks down and cannot explain many observed features. We argue that, under such a large voltage, counterions 'condense' near the surface, even for dilute bulk solutions. Based on simple models, we predict that the double-layer capacitance decreases and the electro-osmotic mobility saturates at large voltages, due to steric repulsion and increased viscosity of the condensed layer, respectively. The former suffices to explain observed high-frequency flow reversal in ac electro- osmosis; the latter leads to a salt concentration dependence of induced-charge flows comparable to experiments, although a complete theory is still lacking. Electrically driven flows in ionic solutions are finding many new applications in microfluidics (1). The theory of electro-osmosis (2) was developed for slip past a surface in chemical equilibrium, whose double-layer voltage is typically of order kBT/e = 25mV. However, the discovery of ac electro-osmosis (ACEO) at micro-electrodes (3, 4) has shifted attention to a new regime, where the induced double-layer voltage is ! 100kBT/e, oscillating
98Â citations
59Â citations
01 Jan 1981
TL;DR: In this paper, experimental and analytical results are presented for a bending-torsion flutter phenomena encountered during wind-tunnel testing of a ten-bladed, advanced, high-speed propeller (turboprop) model with thin airfoil sections, high blade sweep, low aspect ratio, high solidity and transonic tip speeds.
Abstract: Experimental and analytical results are presented for a bending-torsion flutter phenomena encountered during wind-tunnel testing of a ten-bladed, advanced, high-speed propeller (turboprop) model with thin airfoil sections, high blade sweep, low aspect ratio, high solidity and transonic tip speeds. Flutter occurred at free-stream Mach numbers of 0.6 and greater and when the relative tip Mach number (based on vector sum of axial and tangential velocities) reached a value of about one. The experiment also included two- and five-blade configurations. The data indicate that aerodynamic cascade effects have a strong destabilizing influence on the flutter boundary. The data was correlated with analytical results which include aerodynamic cascade effects and good agreement was found.
34Â citations
01 Jan 1984
TL;DR: In this article, an analytical model for investigating vibration and flutter of mistuned bladed disk assemblies is presented, which accounts for elastic, inertial and aerodynamic coupling between bending and torsional motions of each individual blade.
Abstract: An analytical model for investigating vibration and flutter of mistuned bladed disk assemblies is presented. This model accounts for elastic, inertial and aerodynamic coupling between bending and torsional motions of each individual blade, elastic and inertial couplings between the blades and the disk, and aerodynamic coupling among the blades. The disk was modeled as a circular plate with constant thickness and each blade was represented by a twisted, slender, straight, nonuniform, elastic beam with a symmetric cross section. The elastic axis, inertia axis, and the tension axis were taken to be noncoincident and the structural warping of the section was explicitly considered. The blade aerodynamic loading in the subsonic and supersonic flow regimes was obtained from two-dimensional unsteady, cascade theories. All the possible standing wave modes of the disk and traveling wave modes of the blades were included. The equations of motion were derived by using the energy method in conjunction with the assumed mode shapes for the disk and the blades. Continuities of displacement and slope at the blade-disk junction were maintained. The equations were solved to investigate the effects of blade-disk coupling and blade frequency mistuning on vibration and flutter. Results showed that the flexibility of practical disks such as those used for current generation turbofans did not have a significant influence on either the tuned or mistuned flutter characteristics. However, the disk flexibility may have a strong influence on some of the system frequencies and on forced response.
33Â citations
01 Jan 1984
TL;DR: In this paper, the authors compared wind tunnel results for five eight to ten blade advanced models compared with analytical predictions and found that blade sweep was important in achieving net efficiencies near 80 percent at Mach 0.7 to 0.8 and reducing near-field cruise noise by about 6 dB.
Abstract: Advanced high-speed propellers offer large performance improvements for aircraft that cruise in the Mach 0.7 to 0.8 speed regime. At these speeds, studies indicate that there is a 15 to near 40 percent block fuel savings and associated operating cost benefits for advanced turboprops compared to equivalent technology turbofan powered aircraft. Recent wind tunnel results for five eight to ten blade advanced models are compared with analytical predictions. Test results show that blade sweep was important in achieving net efficiencies near 80 percent at Mach 0.8 and reducing nearfield cruise noise by about 6 dB. Lifting line and lifting surface aerodynamic analysis codes are under development and some results are compared with propeller force and probe data. Also, analytical predictions are compared with some initial laser velocimeter measurements of the flow field velocities of an eightbladed 45 swept propeller. Experimental aeroelastic results indicate that cascade effects and blade sweep strongly affect propeller aeroelastic characteristics. Comparisons of propeller near-field noise data with linear acoustic theory indicate that the theory adequately predicts near-field noise for subsonic tip speeds but overpredicts the noise for supersonic tip speeds.
31Â citations
References
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01 Dec 1974
TL;DR: In this article, the Hamilton's principle and the Newtonian method are used to derive the equations of motion for long, straight, slender, homogeneous, isotropic beams undergoing moderate displacements.
Abstract: The equations of motion are developed by two complementary methods, Hamilton's principle and the Newtonian method. The resulting equations are valid to second order for long, straight, slender, homogeneous, isotropic beams undergoing moderate displacements. The ordering scheme is based on the restriction that squares of the bending slopes, the torsion deformation, and the chord/radius and thickness/radius ratios are negligible with respect to unity. All remaining nonlinear terms are retained. The equations are valid for beams with mass centroid axis and area centroid (tension) axis offsets from the elastic axis, nonuniform mass and stiffness section properties, variable pretwist, and a small precone angle. The strain-displacement relations are developed from an exact transformation between the deformed and undeformed coordinate systems. These nonlinear relations form an important contribution to the final equations. Several nonlinear structural and inertial terms in the final equations are identified that can substantially influence the aeroelastic stability and response of hingeless helicopter rotor blades.
576Â citations
01 Feb 1957
TL;DR: In this article, the differential equations of motion for the lateral and torsional deformations of twisted rotating beams are developed for application to helicopter rotor and propeller blades, and the generality is such that previous theories involving various simplifications are contained as subcases to the theory presented in this paper.
Abstract: The differential equations of motion for the lateral and torsional deformations of twisted rotating beams are developed for application to helicopter rotor and propeller blades. No assumption is made regarding the coincidence of the neutral, elastic, and mass axes, and the generality is such that previous theories involving various simplifications are contained as subcases to the theory presented in this paper. Special attention is given the terms which are not included in previous theories. These terms are largely coupling-type terms associated with the centrifugal forces. Methods of solution of the equations of motion are indicated by selected examples.
281Â citations
TL;DR: In this article, the existence and uniqueness of extremely simple system flutter modes are proved for blade rows consisting of identical blades equally spaced about a common rotor, with no loss of generality whatsoever, in terms of a single "equivalent blade."
Abstract: The problems associated with the prohibitive number of possible system modes for a fluttering compressor or turbine blade row are eliminated by the development that comprises the present report. The existence and uniqueness of extremely simple system flutter modes are proved for blade rows consisting of identical blades equally spaced about a common rotor. These simple system modes, if properly interpreted, have the effect of reducing by a factor of n the number of degrees of freedom necessary to analyze an w-bladed configuration. Stated differently, the system of n blades may be considered, with no loss of generality whatsoever, in terms of a single "equivalent blade." The proof holds under any type of flow and any and all types of interblade coupling, so long as a linear analysis is permissible. Moreover, since it is the flutter-inception point that is of interest in predicting critical velocity or rotational speed, it may well be that the conclusions developed apply even to the onset of stall flutter. Practical application of the method to stall-flutter calculations would, of course, require the availability of aerodynamic stallflutter coefficients. The development is carried out first under the assumption of infinite rotor inertia or, in other words, constant rotor velocity. This restriction is then relaxed, and the treatment is expanded to permit torsional oscillations of the rotor itself. I t is proved that under certain conditions the assumption of infinite rotor inertia introduces no error whatsoever.
177Â citations
TL;DR: In this article, a closed-form analytical solution is obtained by using a double application of the Wiener-Hopf technique for predicting the unsteady flow in a supersonic cascade with subsonic axial flow velocity.
Abstract: Linearized theory is used to predict the unsteady flow in a supersonic cascade with subsonic axial flow velocity. A closed-form analytical solution is obtained by using a double application of the Wiener-Hopf technique. Although numerical and semianalytical solutions of this problem have already appeared in the literature, this paper contains the first completely analytical solution. It has been stated in the literature that the blade source should vanish at the infinite duct resonance condition. The present analysis shows that this does not occur. This apparent discrepancy is explained in the paper.
85Â citations
01 Jan 1978
TL;DR: In this paper, the second-degree nonlinear equations of motion for the coupled flapwise bending, lagwise bending and axial extension of an untwisted, torsionally rigid, nonuniform, rotating beam having an arbitrary angle of precone with the plane perpendicular to the axis of rotation are derived using Hamilton's principle.
Abstract: In an attempt both to unify and extend the analytical basis of several aspects of the dynamic behavior of flexible rotating beams, the second-degree nonlinear equations of motion for the coupled flapwise bending, lagwise bending, and axial extension of an untwisted, torsionally rigid, nonuniform, rotating beam having an arbitrary angle of precone with the plane perpendicular to the axis of rotation are derived using Hamilton's principle. The derivation of the equations is based on the geometric nonlinear theory of elasticity and the resulting equations are consistent with the assumption that the strains are negligible compared to unity. No restrictions are imposed on the relative displacements or angular rotations of the cross sections of the beam other than those implied by the assumption of small strains. Illustrative numerical results, obtained by using an integrating matrix as the basis for the method of solution, are presented both for the purpose of validating the present method of solution and indicating the range of applicability of the equations of motion and the method of solution.
64Â citations