Structural synthesis of sandwich beams with outer layers of box-section

It is proved by model measurements that, for sandwich beams constructed from two rectangular tubes and a damping layer glued between them, the following calculation methods can be applied. Static bending and shear stresses as well as deflections of simply supported beams may be calculated by Allen's formulae for sandwich beams with flexurally stiff faces. The first eigenfrequency and the loss factor can be determined by using the diagrams given in reference [1]. For the loss factors Ungar's formula gives a suitable approximation. A minimum cost design procedure is presented for a sandwich beam with constant cross-section. The unknown dimensions of the cross-section are determined which satisfy the design constraints and minimize the material costs. In a numerical example, constraints relating to the maximal dynamic stresses and deflection as well as local buckling of plate elements are considered. In the optimization the backtrack combinatorial discrete programming method is applied. A numerical comparison shows that the material costs of a sandwich beam are lower than those of a homogeneous box one.     

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.     

The steady state out-of-plane response of a Timoshenko curved beam with internal damping

The steady state out-of-plane response of a Timoshenko curved beam with internal damping to a sinusoidally varying point force or moment is determined by use of the transfer matrix approach. For this purpose, the equations of out-of-plane vibration of a curved beam are written as a coupled set of the first order differential equations by using the transfer matrix of the beam. Once the matrix has been determined by numerical integration of the equations, the steady state response of the beam is obtained. The method is applied to free-clamped non-uniform beams with circular, elliptical, catenary and parabolical neutral axes driven at the free end; the driving point impedance and force or moment transmissibility are calculated numerically and the effects of the slenderness ratio, varying cross-section and the function expressing the neutral axis on them are studied.     

Stability of a pretwisted tapered cantilever beam subjected to dissipative and follower forces

Stability of a pretwisted tapered cantilever beam of rectangular cross-section subjected to a follower force at its free end is investigated. The effects of internal and external damping are included in the study. The non-self-adjoint boundary value problem is formulated with the Euler-Bernoulli theory and an associated adjoint boundary value problem is introduced. Approximate values of the critical load are calculated on the basis of a suitable adjoint variational principle for several values of the geometric and material parameters of the beam. The results are shown in graphs.     

Optimal design of beams under flexural vibration

The minimum weight design of a cantilever beam in flexural vibration is considered. The aim is the maximization of a given natural bending frequency (usually the first) for a given beam weight or equivalently the minimization of beam weight for a specified value of a natural frequency. The beams considered are of rectangular section and are subject, in a range of cases presented, to a variety of constraints on lower and upper bounds on the cross-section dimensions or to the specification of a point mass at the end of the beam. Simple bending theory is regarded as applicable to the problem. A variational statement of the problem is made and the necessary conditions for a minimum are obtained as a system of non-linear equations which are solved numerically. Results are given in the form of tables and of figures showing computed optimum profiles. Some experiments on a sample set of beams of equal mass are described briefly. The optimum profile beam was found to have the greatest fundamental frequency, in support of the theoretical predictions.     

PASSIVE DAMPING AUGMENTATION OF A VIBRATING BEAM USING PSEUDOELASTIC SHAPE MEMORY ALLOY WIRES

This paper examines the effectiveness of pseudoelastic shape memory alloy (SMA) wires for passive damping of flexural vibrations of a clamped-free beam with a tip mass. A finite-element model of the system is developed and validated with experimental results. The SMA behavior is modelled using amplitude-dependent complex modulus. Numerical simulations indicate that the damping introduced by the SMA wires will increase for higher excitation-force amplitudes that produce higher strain levels in the SMA wires. Increasing the wire cross-section area provides more damping at low force-excitation amplitudes, but reduced damping at higher amplitudes. The angle between the beam and the SMA wires is an influential parameter, and a value in the 10-20° range was found to introduce maximum damping. The underlying physical mechanisms are examined in detail. System damping depends only mildly on the SMA wire length, and is unaffected by the tip mass.     

An exact solution for the natural frequencies and mode shapes of an immersed elastically restrained wedge beam carrying an eccentric tip mass with mass moment of inertia

In general, the exact solutions for natural frequencies and mode shapes of non-uniform beams are obtainable only for a few types such as wedge beams. However, the exact solution for the natural frequencies and mode shapes of an immersed wedge beam is not obtained yet. This is because, due to the “added mass” of water, the mass density of the immersed part of the beam is different from its emerged part. The objective of this paper is to present some information for this problem. First, the displacement functions for the immersed part and emerged part of the wedge beam are derived. Next, the force (and moment) equilibrium conditions and the deflection compatibility conditions for the two parts are imposed to establish a set of simultaneous equations with eight integration constants as the unknowns. Equating to zero the coefficient determinant one obtains the frequency equation, and solving the last equation one obtains the natural frequencies of the immersed wedge beam. From the last natural frequencies and the above-mentioned simultaneous equations, one may determine all the eight integration constants and, in turn, the corresponding mode shapes. All the analytical solutions are compared with the numerical ones obtained from the finite element method and good agreement is achieved. The formulation of this paper is available for the fully or partially immersed doubly tapered beams with square, rectangular or circular cross-sections. The taper ratio for width and that for depth may also be equal or unequal.     

Free vibration of a rotating non-uniform beam with arbitrary pretwist, an elastically restrained root and a tip mass

The governing differential equations for the coupled bending-bending vibration of a rotating beam with a tip mass, arbitrary pretwist, an elastically restrained root, and rotating at a constant angular velocity, are derived by using Hamilton's principle. The frequency equation of the system is derived and expressed in terms of the transition matrix of the transformed vector characteristic governing equation. The influence of the tip mass, the rotary inertia of the tip mass, the rotating speed, the geometric parameter of the cross-section of the beam, the setting angle, and the pretwist parameters on the natural frequencies are investigated. The difference between the effects of the setting angle on the natural frequencies of pretwisted and unpretwisted beams is revealed.     

ON THE FREQUENCY RESPONSE FUNCTION OF A DAMPED CANTILEVER SIMPLY SUPPORTED IN-SPAN AND CARRYING A TIP MASS

This paper deals with the determination of the frequency response function of a cantilevered Bernoulli-Euler beam which is viscously damped by a single damper. The beam is simply supported in-span and carries a tip mass. The frequency response function is obtained through a formula that was established for the receptance matrix of discrete linear systems subjected to linear constraint equations, by considering the simple support as a linear constraint imposed on generalized co-ordinates. The comparison of the numerical results obtained via a boundary value problem formulation justifies the approach used here.     

Acoustic radiation from a viscoelastic beam impacted by a steel sphere

The acoustic radiation from a viscoelastic beam impacted by a steel sphere has been studied both theoretically and experimentally. Transverse vibrations of free-free viscoelastic beams have been analyzed by employing the modal analysis technique and an approximate method, with the Hertz theory used to evaluate impact forces. The wave equation was solved to determine the acoustic pressure radiated from impacted beams of circular and elliptical cross-sections. The theoretical predictions are compared with the experimental results for the radiation from PMMA beams of circular and rectangular cross-sections. It is shown that for beams of circular cross-sections the theoretical and experimental results are in good agreement and that for beams of rectangular cross-sections the radiation is well predicted by modeling them as beams with elliptical cross-sections.     

The effects of rotation,tip mass,and hub radius on the stability of beck's column

The stability of a cantilever beam subjected to a follower force at its free end and rotating at a uniform angular velocity is investigated. The beam is assumed to be offset from the axis of rotation, carries a tip mass at its free end, and undergoes deflection in a direction perpendicular to the plane of rotation. The equations of motion are formulated within the Euler-Bernoulli and Timoshenko beam theories for the case of a Kelvin model viscoelastic beam. The associated adjoint boundary value problems are derived and appropriate adjoint variational principles are introduced. These variational principles are used for the purpose of determining approximately the values of the critical flutter load of the system as it depends upon its damping parameters, tip mass and its rotary inertia, hub radius, and speed of rotation. The variation of the critical flutter load with these parameters is revealed in a series of several graphs. The numerical results show that the critical load can be reduced significantly due to (a) the transverse and rotary inertia of the tip mass and (b) increasing values of the internal damping parameter associated with the transverse shear deformation of the rotating beam.     

Non-linear free vibrations of stepped thickness beams

The non-linear free vibrations of stepped thickness beams are analyzed by assuming sinusoidal responses and using the transfer matrix method. The numerical results for clamped and simply supported, one-stepped thickness beams with rectangular cross-section are presented and the effects of the beam geometry on the non-linear vibration characteristics are discussed. The results are also compared with those obtained by a Galerkin method in which the linear mode function of the beam is used. The use of a Galerkin method seems to considerably overestimate the non-linearity of the stepped thickness beam in certain cases.     

Free vibration of thin walled open section beams with constrained damping treatment

Free vibration characteristics of a thin walled, open cross-section beam, with constrained damping layers at the flanges, are investigated. Both uncoupled transverse vibration and the coupled bending-torsion oscillations, of a beam of a top-hat section, are considered. Numerical results are presented for natural frequencies and modal loss factors in the first two modes of simply supported and clamped-clamped beams. For the uncoupled mode the constrained damping treatment is more effective than an unconstrained one, but for the coupled mode the effect is just the opposite.     

Determination of the steady state response of a Timoshenko beam of varying cross-section by use of the spline interpolation technique

The steady state response of an internally damped Timoshenko beam of varying cross-section to a sinusoidally varying point force is determined by use of the spline interpolation technique. For this purpose, with the beam divided into small elements, the response of each element is expressed by a quintic spline function with unknown coefficients. The response is obtained by determining these coefficients so that the spline function satisfies the equation of motion of the beam at each dividing point and also satisfies the boundary conditions at both ends. In this case, the slope due to pure bending of the beam is conveniently adopted as the function essentially expressing the response, from which the transverse deflection, driving point impedance, transfer impedance and force transmissibility of the beam are derived. The method is applied to cantilever beams with linearly, parabolically and exponentially varying rectangular cross-sections; these responses of the beams are calculated numerically and the effects of the varying cross-section on them are studied.     

An ultrasonic measurement model using a multi-Gaussian beam model for a rectangular transducer

To date, ultrasonic measurement models have primarily treated systems where circular transducers are used. Recently, however, a highly efficient ultrasonic beam model for a rectangular transducer has also become available where the transducer is represented as a superposition of a relatively few Gaussian beams. Thus, using the multi-Gaussian beams, we developed ultrasonic measurement models for systems where a rectangular transducer is employed. In this paper, we describe the developed models including the beam model, the efficiency factor for a rectangular transducer and far-field scattering models for some standard scatterers. Furthermore, the accuracy of the proposed model is verified by the comparison of the model-based predictions to the experimental measurements.     

On the design of structural junctions for the purpose of hybrid passive-active vibration control

A theoretical investigation of wave scattering and the active modification of wave scattering at structural junctions is presented. A resonant and a non-resonant Euler-Bernoulli beam are coupled, and an external force is introduced at the junction. The external force is intended for feedforward control in order to manipulate the scattering properties at the junction. The purpose of the investigated control law is to make the junction non-reflective in the case of an incident bending wave. The control effort and the resulting power flow are investigated for different properties of the beams. By introducing damping in the resonant beam all incidence wave power is absorbed either passively, in the resonant beam, or actively, by the force. The results form the basis for a discussion of the possible benefits of using such a configuration for hybrid passive-active vibration control. The results show that for certain ratios of bending stiffness and mass the presented hybrid passive-active solution may offer advantages compared to purely passive or purely active solutions.     

Three dimensional dynamics of pretwisted beams: A spectral-Tchebychev solution

This paper presents the application of the spectral-Tchebychev (ST) technique for solution of three-dimensional dynamics of unconstrained pretwisted beams with general cross-section (including both straight and curved cross-sections). In general, the dynamic response of pretwisted beams presents three-dimensional (3D) motions, including coupled bending–bending–torsional–axial motions. As such, accurately solving pretwisted beam dynamics requires a 3D solution approach. In this work, the integral boundary value problem based on the 3D linear elasticity equations is solved numerically using the 3D-ST approach. To simplify evaluation of the volume integrals, the boundaries are simplified by applying two coordinate transformations to render the pretwisted beam with curved cross-section into an equivalent straight beam with rectangular cross-section. Three sample pretwisted beam problems with rectangular, curved, and airfoil cross-sections at different twist rates are solved using the presented approach. In each case, the convergence of the solution is analyzed, and non-dimensional natural frequencies and mode shapes are compared to those from a finite-element (FE) solution. Furthermore, cross-sectional stress and displacements are obtained from the 3D-ST solution. Lastly, the non-dimensional natural frequencies from the 3D-ST and a 1D/2D solutions are compared. It is concluded that the 3D-ST solution can capture the three-dimensional dynamic behavior of pretwisted beams as accurately as an FE solution, but for a fraction of the computational cost. Furthermore, it is shown that 1D/2D solution can lead to significant errors at high twist rates, and thus, the 3D-ST solution should be preferred.     

Interference of circularly polarized light: contrast and application in fabrication of three-dimensional periodic microstructures

The contrast of interference pattern formed by two circularly polarized waves and by a linearly polarized wave and a circularly polarized one is discussed. The results are compared with that by two linear beams. It shows that the use of circular light in holographic fabrication of three-dimensional periodic microstructures may remove the necessity of beam ratio and polarization optimization needed in the interference of three linear noncoplanar beams and improve the uniform contrast of resultant pattern simultaneously.     

Coupled bending and torsional vibration of a beam with in-span and tip attachments

In this paper, the coupled flexural-torsional free and forced vibrations of a beam with tip and/or in-span attachments are studied. First, a mathematical model is established, which consists of a beam with several tip attachments, i.e, a tip mass of non-negligible dimensions, a linear spring grounding the tip mass, and a torsional spring connected at the end of the beam. The modal functions of this model and the orthogonality condition among them are derived. For the purpose of verification the properties of the tip attachments are changed, and the numerical results obtained are compared with those given in the relevant literature. Effects of tip mass and distributed mass in-span on natural frequencies and modes are investigated for two cantilever beams with different cross sections. An application of the orthogonality condition in the case of a beam with tip mass is also presented for a forced vibration example.     

Transverse vibrations of pre-twisted sandwich beams

A set of equations of motion governing the bending and extensional displacements of a pre-twisted sandwich beam of rectangular cross-section are derived by using Hamilton's principle. The middle viscoelastic core is assumed to deform mainly through the classical shearing mechanism. The eigenvalues and loss factors of simply supported pre-twisted sandwich beams are computed by using the variational method. Analysis of the results revealed that pre-twisting the beam increases the real part of the eigenvalue by as much as 20% while reducing the loss factor by as much as 30 %. The loss factor of very soft, thickcored beams is especially sensitive to even small angles of pre-twist: e.g., a 22· 5° pre-twist may reduce the loss factor by as much as 80%. The effect of pre-twist is, however, shown not to be appreciable for soft, thin-cored beams. In any case, pre-twisting of the beam has a detrimental effect on the maximum loss factor that one can obtain for a specific size of the beam when only the shear parameter of the beam is changed.