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.     

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.     

Dynamic identification of beam axial loads using one flexural mode shape

In the last decades, various methods have been proposed for the experimental evaluation of tensile forces acting in tie-beams of arches and vaults. Moreover, static and dynamic approaches have been formulated to evaluate critical compressive axial forces and flexural stiffness of end constraints. Adopting Euler–Bernoulli beam model, this paper shows that, if bending stiffness and mass per unit length of a beam with constant cross-section are known, the axial force and the flexural stiffness of the end constraints can be deduced by one vibration frequency and three components of the corresponding mode shape. Finally, data conditions are given to assess a physically admissible identification of the unknown parameters.     

Dynamic analysis of an arch due to a moving load

The radial (in-plane) bending-vibration responses of a uniform circular arch under the action of a moving load were investigated by means of the arch (curved beam) elements. Instead of the complex explicit-form shape functions given by the existing literature, the simple implicit-form shape functions associated with the radial (normal), tangential and rotational displacements of the arch element were derived. Based on the relationships between the nodal forces and nodal displacements of an arch element the elemental stiffness matrix was obtained, and based on the equation relating the kinetic energy and nodal velocities the elemental consistent mass matrix was determined. Assembly of the elemental property matrices yields the overall stiffness and mass matrices of the complete circular arch. The analytical free vibration analysis results were used to confirm the reliability of the presented stiffness and mass matrices for the arch element. Then the dynamic responses of a typical segmental circular arch, with constant curvature, due to a concentrated load moving along the circumferential direction were discussed. In addition to the circular arch, a hybrid (curved) beam composed of a circular-arch segment and two identical straight-beam segments was also studied. All numerical results were compared with the finite element solutions based on the conventional straight-beam elements and reasonable agreement was achieved. Influence of the moving speed, centrifugal force and frictional force on the dynamic behaviors of the circular arch and the hybrid beam was investigated.     

Higher order tapered beam finite elements for vibration analysis

In this paper, explicit for mass and stiffness matrices of two higher order tapered beam elements for vibration analysis are presented. One possesses three degrees of freedom per node and the other four degrees of freedom per node. The four degrees of freedom of the latter element are the displacement, slope, curvature and gradient of curvature. Thus, this element adequately represents all the physical situations involved in any combination of displacement, rotation, bending moment and shearing force. The explicit element mass and stiffness matrices eliminate the loss of computer time and round-off-errors associated with extensive matrix operations which are necessary in the numerical evaluation of these expressions. Comparisons with existing results in the literature concerning tapered cantilever beam structures with or without an end mass and its rotary inertia are made. The higher order tapered beam elements presented here are superior to the lower order one in that they offer more realistic representations of the curvature and loading history of the beam element. Furthermore, in general the eigenvalues obtained by employing the higher order elements converge more rapidly to the exact solution than those obtained by using lower order one.     

Measurement of viscoelastic properties from the vibration of a compliantly supported beam

A laboratory method is presented by which the viscoelastic properties of compliant materials are measured over a wide frequency range. The test setup utilizes a flexible beam clamped at one end and excited by a shaker at the free end. A small specimen of a compliant material is positioned to support the beam near its midpoint. The deformation from gravity is minimized since the specimen is not loaded by an attached mass. Forced vibration responses measured at two locations along the beam are used to derive a transfer function from which the dynamic properties of compliant materials are directly determined by use of a theoretical procedure investigating the effects of specimen stiffness on the propagation of flexural waves. Sensitivity of the measured properties to experimental uncertainties is investigated. Young's moduli and associated loss factors are determined for compliant materials ranging from low-density closed-cell foams to high-density damping materials. The method is validated by comparing the measured viscoelastic properties to those from an alternative dynamic test method.     

A tapered beam finite element for rotor dynamics analysis

The stiffness, mass and gyroscopic matrices of a rotating beam element are developed, a cubic function being used for the transverse displacement. Shear deflection is included by use of end nodal variables of shear strain, along with transverse displacement and cross-section rotation; rotatory inertia effects are included in the energy functional to provide a Timoshenko beam formulation. The gyroscopic effects for small perturbations are linearized as a skew symmetric damping matrix. The formulation is implemented by numerical integration for a linearly tapered circular beam. A technique of reduction of the shear nodal variable prior to global assembly is shown to provide little loss in accuracy with reduced system bandwidth. Numerical comparisons for three previously published beam models are included, with results presented for the case of forward and reverse precession to verify the gyroscopic effects. The utility of the element in a general program for rotor dynamics analysis is identified.     

Vibrations and buckling of a beam on a variable winkler elastic foundation

Two methods for solving the eigenvalue problems of vibrations and stability of a beam on a variable Winkler elastic foundation are presented and compared. The first is based on using the exact stiffness, consistent mass, and geometric stiffness matrices for a beam on a variable Winkler elastic foundation. The second method is based on adding an element foundation stiffness matrix to the regular beam stiffness matrix, for vibrations and stability analysis. With these matrices, it is possible to find the natural frequencies and mode shapes of vibrations, and buckling load and mode shape, by using a small number of segments. It is concluded that the use of the element foundation stiffness approach yields better convergence at lower computation costs.     

An approximate method for determining the static deflection and natural frequency of a cracked beam

This paper provides an approximate method to determine the stiffness and the fundamental frequency of a cracked beam. The cracked beam is first represented as an un-cracked beam with equivalent reduced sections around the cracks. The effect of the cracks is explained, visualised and quantified using the equivalence concept developed for stepped beams with periodically variable cross-sections. Then an alternative expression of the improved Rayleigh method is provided to calculate the natural frequencies of a beam with a variable stiffness distribution along its length. As the method is insensitive to the assumed mode shapes, it avoids the difficulty in choosing appropriate mode shapes and yields accurate results. This is shown using several examples to compare the results determined using the proposed method and the Finite Element method (FEM). The method greatly simplifies the calculation of cracked beams with complicated configurations, such as a beam with several cracks, a cracked beam with concentrated masses, a beam with cracks close to each other, and a beam with periodically distributed cracks.     

Combined dynamic stiffness matrix and precise time integration method for transient forced vibration response analysis of beams

A method has been developed for determining the transient response of a beam. The beam is divided into several continuous Timoshenko beam elements. The overall dynamic stiffness matrix is assembled in turn. Using Leung's equation, we derive the overall mass and stiffness matrices which are more suitable for response analysis than the overall dynamic stiffness matrix. The forced vibration of the beam is computed by the precise time integration method. Three illustrative beams are discussed to evaluate the performance of the current method. Solutions calculated by the finite element method and theoretical analysis are also enumerated for comparison. In these examples, we have found that the current method can solve the forced vibration of structures with a higher precision.     

Coupling effects on a cantilever subjected to a follower force

In this investigation a solution methodology is presented for studying the stability of a uniform cantilever having a translational and rotational spring at its support, carrying two concentrated masses, one at the support and the other at its tip, and subjected to a follower compressive force at its free end. The analysis is based on Timoshenko's beam theory by considering the cantilever as a continuous elastic system. The coupling effects on the flutter load are fully assessed for a variety of parameters such as translational and rotational springs at the support, translational and rotational inertia of the concentrated masses, and cross-sectional shape, as well as transverse shear deformation and rotatory inertia of the mass of the column.     

Vibration of an excitation system supported flexibly on a viscoelastic sandwich beam at its mid-point

A vibration analysis of an excitation system supported flexibly on a three layer sandwich beam is presented in this paper. The flexibly supported excitation system, which is essentially the primary system, consists of a mass, a spring and a dash-pot. The beam is analyzed separately as a continuous system in a classical way and then its dynamic stiffness at the junction point is combined with that of the primary system to obtain the resultant dynamic stiffness, which in turn is used to develop the expressions for the response of the primary system and the transmissibility provided by the whole system. Both response and transmissibility are evaluated for different geometrical and physical parameters of the sandwich beam. The solution to this problem is also obtained by approximating the sandwich beam by a lumped mass supported on a spring and dash-pot. The results in the two cases are compared. Results obtained from an experimental test-rig substantiate the theoretical results.     

On asymptotics of the solution of the moving oscillator problem

Asymptotic behavior of the solution of the moving oscillator problem is examined for large and small values of the spring stiffness for the general case of non-zero beam initial conditions. In the limiting case of infinite spring stiffness, it is shown that the moving oscillator problem for a simply supported beam is not equivalent, in a strict sense, to the moving mass problem. The two problems are shown to be equivalent in terms of the beam displacements but are not equivalent in terms of stresses (the higher order derivatives of the two solutions differ). In the general case, the force acting on the beam from the oscillator is shown to contain a high-frequency component , which does not vanish and can even grow when the spring coefficient tends to infinity. The magnitude of this force and its dependence on the oscillator parameters can be estimated by considering the asymptotics of the solution for the initial stage of the oscillator motion. It is shown that, for the case of a simply supported beam, the magnitude of the high-frequency force depends linearly on the oscillator eigenfrequency and velocity. The deficiency of the moving mass model is principally that it fails to predict stresses in the supporting structure. For small values of the spring stiffness, the moving oscillator problem is shown to be equivalent to the moving force problem. The discussion is amply illustrated by results of numerical experiments.     

The effect of rotary inertia and shear deformation on the frequency and normal mode equations of uniform beams carrying a concentrated mass

New frequency equations for transverse vibrations of Timoshenko beams carrying a concentrated mass at an arbitrary point along the beam are given. Normal mode equations for the hinged-hinged beam are given and the orthogonality condition is presented for beams with hinged, clamped or free ends. A numerical example is given and frequency charts show the effects of varying the size and location of the concentrated mass.     

Instability of vibrations of a mass that moves uniformly along a beam on a periodically inhomogeneous foundation

The stability of vibrations of a mass that moves uniformly along an Euler-Bernoulli beam on a periodically inhomogeneous continuous foundation is studied. The inhomogeneity of the foundation is caused by a slight periodical variation of the foundation stiffness. The moving mass and the beam are assumed to be always in contact. With the help of a perturbation analysis it is shown analytically that vibrations of the system may become unstable. The physical phenomenon that lies behind this instability is parametric resonance that occurs because of the periodic (in time) variation of the foundation stiffness under the moving mass. The first instability zone is found in the system parameters within the first approximation of the perturbation theory. The location of the zone is strongly dependent on the spatial period of the inhomogeneity and on the weight of the moving mass. The larger this period is and/or the smaller the mass, the higher the velocity is at which the instability occurs.     

Optimal design of acoustical sandwich panels with a genetic algorithm

An optimization study is performed to design a sandwich panel with a balance of acoustical and mechanical properties at minimal weight. An acoustical model based on higher-order sandwich beam theory is used with mechanical analysis of the maximum deflection at the center of the sandwich panel under a concentrated force. First, a parametric study is performed to determine the effects of individual design variables on the sound transmission loss of the sandwich panel. Next, by constraining the acoustical and mechanical behavior of the sandwich panel, the area mass density of the sandwich panel is minimized using a genetic algorithm. The sandwich panels are constructed from eight face-sheet and sixteen core materials, with varying thicknesses of the face sheets and the core. The resulting design is a light-weight, mechanically efficient sound insulator with strength and stiffness comparable to sandwich structures commonly used in structural applications.     

Transient response of an elastic beam with discrete impedances from decelerating boundaries

Analyzed is transient response of an elastic beam from decelerating boundaries. It is shown that passive attenuation of peak transient deceleration is possible by coupling the beam to a tuned one-degree-of-freedom (1-dof) oscillator. For a fixed oscillator mass, optimum spring stiffness yields the lowest deceleration response. Furthermore, over a finite range of spring stiffness, the second coupled system mode disappears yielding an additional sharp reduction in transient deceleration response. The disappearance of the second mode is independent of boundary conditions from the weakest simple supports to the strongest clamped ends.     

Vibration and stability of elastically supported beams carrying an attached mass under axial and tangential loads

The vibration and stability of an elastically supported beam carrying an attached mass and subjected to axial and tangential compressive loads are investigated. The analysis is based on the Timoshenko beam theory and the effects of the attached mass are expressed with Dirac delta functions. The influences of the support stiffness, the direction of loading, and the slenderness ratio on the natural frequency and critical load of a beam are discussed.     

Dynamic contact analysis of a tensioned beam with a moving mass–spring system

The dynamic contact problem of a tensioned beam with clamped-pinned ends is analyzed when the beam contacts a moving mass–spring system. The contact and contact loss conditions are expressed in terms of constraint equations after considering the dynamic contact between the beam and the moving mass. Using these constraints and equations of motion for the beam and moving mass, dynamic contact equations are derived and then discretized using the finite element method, which is based on the Lagrange multiplier method. The time responses for the contact forces are computed from these discretized equations. The contact force variations and contact loss are investigated for the variations of the moving mass velocity, the beam tension, the moving mass, and the stiffness of the moving mass–spring system. In addition, the possibility of contact loss and safe contact conditions between the moving mass and the tensioned beam are also studied.     

A CONCENTRATED MASS ON THE SPINNING UNCONSTRAINED BEAM SUBJECTED TO A THRUST

The dynamic stability of a spinning unconstrained beam subjected to a pulsating follower forceP₀ +P₁cos Ωt is analyzed. A concentrated mass is located at an arbitrary location on the beam, and the stability of the beam is studied with the mass at various locations. The beam is analyzed using the Timoshenko-type shear deformation theory with the rotary inertia. Hamilton's principle is used to derive the equations of motion, and the spinning speed of the beam with various non-dimensional parameters subjected to a pulsating follower force is investigated. The finite element method is applied to analyze the spinning beam model, and the method of multiple scales is used to investigate the dynamic stability characteristics. A pulsating follower force is applied, and then the stability regions are changed with the transitions of the stability area in many regions. The results show that the concentrated mass increases the dynamic stability of the spinning unconstrained beam subjected to a thrust. As the spinning speed of the beam is increased, the instability regions are reduced, but various slight instability regions are additionally developed.