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Differential equations of motion for combined flapwise bending, chordwise bending, and torsion of twisted nonuniform rotor blades

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.

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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.

576 citations

Journal ArticleDOI
TL;DR: In this article, the authors focus on the design of small blower tunnels, and most of the information is applicable to wind tunnels in general, but most of their work is focused on wind tunnels with axial fans.
Abstract: Even with today’s computers, a wind tunnel is an essential tool in engineering, both for model tests and basic research. Since the 1930s, when the strong effect of free-stream turbulence on shear layers became apparent, emphasis has been laid on wind tunnels with low levels of turbulence and unsteadiness. Consequently most high performance wind tunnels were designed as closed-circuit types (Fig. 1(a)) to ensure a controlled return flow. However, as will be seen below, it is possible with care to achieve high performance from an open-circuit tunnel, thus saving space and construction cost. ‘Blower’ tunnels (with the fan at entry to the tunnel, Fig. 1(b)) facilitate large changes in working section arrangements; to cope with the resulting large changes in operating conditions, a centrifugal fan is preferable to an axial one. For ease of changing working sections the exit diffuser is often omitted from small blower tunnels, at the cost of a power factor greater than unity. This paper concentrates on the design of small blower tunnels but most of the information is applicable to wind tunnels in general.

398 citations

Journal ArticleDOI
TL;DR: A multibody dynamics approach to the modeling of rotorcraft systems and the algorithms used to integrate the resulting equations of motion with maximum efficiency and robustness are discussed.

199 citations


Cites methods from "Differential equations of motion fo..."

  • ...Historically, the classical approach to rotor dynamics has been to use a modal reduction approach, as pioneered by Houbolt and Brooks [3]....

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20 May 1998
TL;DR: In this paper, a force balance method for calculating section loads of a bearingless rotor rotor was proposed, and the results were compared with the results from a full-scale wind tunnel test.
Abstract: Recent developments of the dynamics models for the comprehensive analysis CAMRAD II are described, specifically advanced models of the geometry and material for the beam component, and a force balance method for calculating section loads. Calculations are compared with measurements for beams undergoing large deflection. Bearingless rotor stability and bending loads calculations are compared with the results from a full-scale wind tunnel test. With a reasonable number of beam elements representing the rotor blade, any large deflection effects are captured by the rigid body motion (which is always exact), and a second-order model of the beam element elastic motion is adequate. The deflection method gives unacceptable results for the structural loads in practical cases, and even with uniform blade properties. The force balance method described here gives good results for blade load, without requiring a large number of nodes.

197 citations

References
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Book
01 Jan 1934
TL;DR: The theory of the slipline field is used in this article to solve the problem of stable and non-stressed problems in plane strains in a plane-strain scenario.
Abstract: Chapter 1: Stresses and Strains Chapter 2: Foundations of Plasticity Chapter 3: Elasto-Plastic Bending and Torsion Chapter 4: Plastic Analysis of Beams and Frames Chapter 5: Further Solutions of Elasto-Plastic Problems Chapter 6: Theory of the Slipline Field Chapter 7: Steady Problems in Plane Strain Chapter 8: Non-Steady Problems in Plane Strain

20,724 citations

01 Jun 1955
TL;DR: In this paper, a Rayleigh energy approach utilizing the bending mode of the nonrotating beam in the determination of the bending frequency of the rotating beam was presented, which is evaluated and is found to give good practical results for helicopter blades.
Abstract: Note presenting a Rayleigh energy approach utilizing the bending mode of the nonrotating beam in the determination of the bending frequency of the rotating beam, which is evaluated and is found to give good practical results for helicopter blades. Charts are presented for the rapid estimation of the first three bending frequencies for rotating and nonrotating cantilever and hinged beams.

53 citations