Effect of angle of attack on the stability of a rotating non-uniform cantilever with a tip mass subjected to dissipative and non-conservative forces

The influence of angle of attack on the stability of a rotating viscoelastic tapered cantilever beam of rectangular cross-section carrying a tip mass and subjected to a circulatory force at its free end is investigated. The effect of external damping is included in the study. The non-self-adjoint boundary value problem is formulated with the Euler-Bernoulli hypothesis and an appropriate adjoint boundary value problem is introduced. Approximate values of the critical flutter load are calculated on the basis of an adjoint variational principle for several values of geometric and material parameters of the system. The results are presented through a series of graphs.     

Vibration of a rotating beam with tip mass

The vibration frequency of a rotating beam with tip mass is investigated. The finite element method is used, a third order polynomial being assumed for the variation of the lateral displacement. The effects of the root radius, the setting angle and the tip mass are incorporated into the finite element model. The results are compared with results from previous authors utilizing Myklestad and extended Galerkin methods. The results show that the setting angle has a significant effect on the first mode frequencies but not on the high frequencies. The tip mass tends to depress the frequencies at low speeds of rotation but it tends to increase the frequencies at high speeds of rotation. The results of this work have applications in wind turbine rotors, helicopter rotors, etc., and the method used here can be extended to investigate the vibration frequency of flexible blade auto cooling fans.     

Lateral displacements of a vibrating cantilever beam with a concentrated mass

The displacement equation for a uniform cross-section, cantilever-type beam carrying a concentrated mass at one end is solved under the most general conditions of an arbitrary distributed lateral load and arbitrary boundary and initial conditions. The method employs complex variable residue theory t0 determine the inversion integral for the Laplacetransformed solution of the boundary value problem. An example problem is solved and the displacement is shown graphically at several points along the beam for two values of the concentrated mass.     

Vibration frequency of a curved beam with tip mass

In flexible blade auto cooling fans, the first vibration frequency is of fundamental importance. These fan blades are usually curved and have a tip mass in the form of a strip along one edge. For the first frequence, the blade can be modelled as a curved beam with a tip mass. This paper reports on an investigation of the vibration frequency of a curved beam with a tip mass, in which both theoretical finite element and experimental methods were used. In the finite element methods, both the normal and tangential displacements are approximated by cubic polynomials to ensure that rigid body displacements are closely represented. The effect of the tip mass is incorporated into the mass matrix. The results show that the curvature has a slight effect on the first mode natural frequencies but has great influence on the higher frequencies, and that the coupling effect between the tip mass and the curvature is insignificant.     

The stability of cantilever panels in uniform incompressible flow

The stability of large aspect ratio cantilever panels in uniform incompressible flow is studied by employing linearized plate and potential flow theories together with asymptotic expressions for the generalized pressures. It is assumed that if the downstream wake effects on the panel stability are insignificant then the results will be almost independent of where these effects are neglected in the analysis. Accordingly two approximations for the pressure, corresponding to different ways of neglecting the wake effects, are compared, and it is concluded that such effects cannot be ignored when the mass ratio is large. The results also demonstrate that a two-dimensional analysis (infinite aspect ratio) may yield an overestimate of the critical flow velocity for finite width panels.     

Variational approaches to the natural frequencies of a clamped uniform beam-column

The transverse flexibility of heavily loaded structural elements may be increased significantly by beam-column effects, particularly when the structures are subjected to large longitudinal accelerations. The determination of the affected vibration and stability characteristics is then of concern to the designers of space-vehicle components and of payload elements, since these may be flexible enough to have critical accelerations in the neighborhood of those attained by modern rockets. For as simple a structural element as the uniform beam subjected to a longitudinal acceleration, the determination of the natural frequencies presents difficulties because the governing differential equation contains variable coefficients. Exact solutions are impossible to obtain.     

The influence of rotatory inertia,tip mass,and damping on the stability of a cantilever beam on an elastic foundation

The stability of a uniform viscoelastic cantilever resting on an elastic foundation, carrying a tip mass, and subjected to a follower force at its free end is investigated. The effects of the rotatory inertia of the beam, the transverse and rotatory inertias of the tip mass, and the foundation modulus, which characterizes a Winkler type of elastic foundation, are included in the partial differential equation of motion and boundary conditions, and the influence of these quantities on the value of the critical flutter load parameter Q_f is sought. The exact forms of the fundamental frequency equations are derived for the cases of a viscoelastic and a purely elastic beam, and these equations are solved numerically for Q_f These numerical results reveal that Q_f depends strongly upon the foundation modulus for the cantilever carrying a tip mass or possessing rather small internal damping. In the absence of damping and a tip mass, the value of Q_f, computed upon the inclusion of the rotatory inertia of the beam in the formulation of the equation of motion, is decreased slightly and continues to decrease in essentially a linear manner as the value of the foundation modulus parameter κ is decreased. Moreover, when the effect of very small internal damping is included, the value of Q_f computed when the rotatory inertia of the beam is neglected increases slowly in an essentially linear fashion as x increases, whereas, when the effect of rotatory inertia is retained, the value of Q_f decreases as κ is increased. Additional numerical results are reported graphically.     

Dynamics modelling of a flexible hub-beam system with a tip mass

This paper presents a finite-element model for a flexible hub-beam system with a tip mass. Both viscous damping and air drag force are introduced into this model. The complete coupling between the system rigid and flexible degrees of freedom is allowed since the start of the formulation and developing the system kinematic variables. Based on deformation theory and geometric constraints, a second order approximation for the displacement field is proposed and the dynamic stiffening is accounted for. Hamilton's principle is utilized in deriving the equations of motion. The corresponding dynamics models of the tip mass and damping forces are developed in a consistent manner through formulating their energy expressions and applying Hamilton's principle. The finite element method is employed for spatial discretization due to its versatility, high accuracy and convergence. Numerical simulations show that the second order term in deformation field can have significant effect on dynamics behavior of flexible multibody systems. It is also shown that the traditional linear model cannot account for dynamic stiffening and may lead to erroneous result in some high-speed systems because the deformation field commonly used in structural dynamics is straight employed in this model. In contrast, the developed model (CCM) based on the second order deformation field can predict valid results. The effects of tip mass and damping on dynamics behavior of the hub-beam system are also discussed.     

Effect of plastic deformation at the crack tip in a crystal on the direction of the tip growth under a mixed load

Evolution of plastic deformation at the tip of a wedge-shaped crack in a crystal under planar strain (modes I and II) was calculated for different cleavage planes, easy-slip systems, angles at the wedge tip, and ratios of the external extension and shear loads. Time distributions are obtained for the plastic deformation, the effective shear stress, the stress intensity factor, and the crack growth direction under monotonic load of the crystal up to a specified limit and further relaxation to establishment of equilibrium distributions under a constant external load. Numerical calculations were performed for an α-Fe crystal.     

Vibration frequencies for a non-uniform beam with end mass

This study deals with the determination of natural frequencies of a non-uniform cantilever beam which carries a concentrated mass at the free end. The effect of the rotary inertia of the end mass has been included. Numerical results for the first five eigenfrequencies are presented for a wide range of values of the beam dimensions and the concentrated mass.