# On the axial vibrations of rotating bars

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TL;DR: For axial deformation of rotating rods, it was shown in this article that unless this linear model is valid for large strain, it is impossible to determine whether this instability really exists; because, as the angular speed is increased, the strain becomes large well short of the critical speed.

Abstract: For strain sufficiently small such that Hooke's Law is valid, it is shown that only a linear model for axial deformation of rotating rods can be derived. As discussed in the literature, this linear model exhibits an instability when the angular speed reaches a certain critical value. However, unless this linear model is valid for large strain, it is impossible to determine whether this instability really exists; because, as the angular speed is increased, the strain becomes large well short of the critical speed. Next, axial deformation of rotating rods is analyzed using two strain energy functions to model non-linear elastic behavior. The first of these functions is the usual quadratic strain energy function augmented with a cubic term. With this model it is shown that no instability exists if the non-linearity is stiffening (i.e. if the coefficient of the cubic term is positive), although the strain can become large. If the non-linearity is of the softening variety, then the critical angular speed drops as the degree of softening increases. Still, the strains are large enough that, except for rubber-like materials, a non-linear elastic model is not likely to be appropriate. The second strain energy function is based on the square of the logarithmic strain and yields a softening model. It quite accurately models the behavior of certain rubber rods which exhibit the instability within the validated range of elongation.

17 citations

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TL;DR: In this article, the non-linear equations of motion of a rotating blade undergoing extensional and flapwise bending vibration are derived, including nonlinearities up to O (e3), and the strain-displacement relationship derived is compared with expressions derived by earlier investigators and the errors and the approximations made in some of those are brought out.

Abstract: The non-linear equations of motion of a rotating blade undergoing extensional and flapwise bending vibration are derived, including non-linearities up to O (e3). The strain-displacement relationship derived is compared with expressions derived by earlier investigators and the errors and the approximations made in some of those are brought out. The equations of motion are solved under the inextensionality condition to obtain the influence of the amplitude on the fundamental flapwise natural frequency of the rotating blade. It is found that large finite amplitudes have a softening effect on the flapwise frequency and that this influence becomes stronger at higher speeds of rotation.

8 citations

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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.

571 citations

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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.

64 citations

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TL;DR: In this article, a set of nonlinear equations of equilibrium for an elastic wind turbine or helicopter blades are derived for the case of small strains and moderate rotations (slopes).

Abstract: A set of nonlinear equations of equilibrium for an elastic wind turbine or helicopter blades are presented. These equations are derived for the case of small strains and moderate rotations (slopes). The derivation includes several assumptions which are carefully stated. For the convenience of potential users the equations are developed with respect to two different systems of coordinates, the undeformed and the deformed coordinates of the blade. Furthermore, the loads acting on the blade are given in a general form so as to make them suitable for a variety of applications. The equations obtained in the study are compared with those obtained in previous studies.

57 citations

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TL;DR: In this paper, the non-linear equations of motion of a slender bar rotating at constant angular velocity about a transverse axis are formulated under the assumption that a small perturbed motion occurs about an initially stressed equilibrium configuration.

Abstract: The non-linear equations of motion of a slender bar rotating at constant angular velocity about a transverse axis are formulated. Under the assumption that a small perturbed motion occurs about an initially stressed equilibrium configuration, linearized equations of motion for the longitudinal and flexural deformations of a rotating bar carrying a tip mass are derived. Numerical computations for the natural frequencies of the lowest three modes of free vibration reveal that the values of the extensional frequencies increase monotonically, contrary to previously published results, as the angular velocity of rotation increases.

47 citations

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TL;DR: In this paper, the extensional equations of motion for a cantilever bar rotating about an axis fixed in space are derived, and it is shown that the form of the nonlinear strain-displacement relation is important in determining the nature of the relationship between the frequency of extensional oscillations and the rotational speed.

Abstract: The extensional equations of motion for a cantilever bar rotating about an axis fixed in space are derived. It is shown that the form of the non-linear strain-displacement relation is important in determining the nature of the relationship between the frequency of extensional oscillations and the rotational speed. In particular, the frequency may or may not increase monotonically with rotational speed, depending on the degree of hardening in the effective extensional spring. The determination whether an instability occurs as the rotational speed increases is beyond the limits of engineering beam theory.

12 citations

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