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.     

Parametric response of viscoelastically supported beams

The stability of viscoelastic beams with an attached mass and viscoelastic end supports under axial and tangential periodic loads is investigated. Viscoelastic end supports are substituted for translational and rotational springs with viscoelastic damping. The regions of instability for simple and combination resonances are obtained from the ordinary Mathieu equation which is obtained from the equation of motion by application of Galerkin's method. In numerical computations, the influences of the direction of loading, the attached mass, the support stiffness, and the damping on the regions of instability for simple and combination resonances are clarified.     

VIBRATION SUPPRESSION OF A BEAM STRUCTURE BY INTERMEDIATE MASSES AND SPRINGS

A method based on the dynamic Green function has been proposed to determine the optimum values of masses and/or springs and their locations on a beam structure in order to confine the vibration at an arbitrary location. In the analysis, the beam is driven by a harmonic external excitation. The added masses on the beam and the springs attached are modelled as simple reactions that provide transverse forces to the beam. These forces act as secondary forces that reduce the response caused by the external force. Numerical simulation shows that the vibration of the beam can be confined in a certain region by the presence of masses and springs in best arrangement. This method is demonstrated for both a simply supported and a cantilever beam. An experimental set-up was designed in which a simply supported beam is excited by an electrodynamic shaker and the response of the beam is measured using an He-Ne laser system. This assures very accurate measurements and avoids any additional loading effects as in the case of accelerometers. Comparisons of the theoretical and the experimental results show good agreement.     

Influence of the torsional vibration of the periodically attached perpendicular beam resonator on the flexural band of a Euler-Bernoulli beam

The bending and torsional vibration of the periodic perpendicular cantilever beam-mass resonators (PCBMR) is idealized as translational and rotational oscillators attached to the main beam. In this paper, the effect of that torsional vibration of the PCBMR on the dynamics of an infinitely long Euler-Bernoulli beam is evaluated. The band-structure is explored by implementing the transfer matrix method in conjunction with Bloch-Floquet's theorem. The combination of the translational and rotational oscillator modifies the relative position of the coupling coefficient in the transfer matrix, which plays a pivotal role in the band-gap formation. The flexural band-structure is highly sensitive to the torsional vibration while the radius of gyration of the tip mass is considerably higher than the length of the PCBMR. Ill-tuning leads to split and reduction of attenuation band to 50%; whereas, around 38% elongation of the attenuation band in the low frequency regime can be achieved by proper tuning.     

Stability analysis with lumped mass and friction effects in elastically supported pipes conveying fluid

This paper presents an accurate finite element procedure for the stability analysis of elastically supported pipes conveying fluid. With consideration of effects of lumped masses, fluid pressure and friction, the equations of motion are derived based on Hamilton's principle for the mass transport system. The kinematics of the pipe is based on Timoshenko beam theory for which the transverse shear deformation and rotary inertia of the pipe are included. The material behaviour of the pipe is described by the Kelvin viscoelastic model. The dynamic stability behaviours obtained by the present work are more conservative as compared with those evaluated by conventional Euler-Bernoulli beam theory. Also, it is found that the lumped masses, fluid pressure and friction will destabilize the system while the elastic support may have either a stabilizing or destabilizing effect depending on its stiffness and location. To demonstrate the validity and accuracy of the technique developed, several numerical examples are illustrated.     

Free vibration analysis of planar curved beams by wave propagation

In this paper, a systematic approach for the free vibration analysis of a planar circular curved beam system is presented. The system considered includes multiple point discontinuities such as elastic supports, attached masses, and curvature changes. Neglecting transverse shear and rotary inertia, harmonic wave solutions are found for both extensional and inextensional curved beam models. Dispersion equations are obtained and cut-off frequencies are determined. Wave reflection and transmission matrices are formulated, accounting for general support conditions. These matrices are combined, with the aid of field transfer matrices, to provide a concise and efficient method for the free vibration problem of multi-span planar circular curved beams with general boundary conditions and supports. The solutions are exact since the effects of attenuating wave components are included in the formulation. Several examples are presented and compared with other methods.     

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.     

Nonlinear vibration of micromachined asymmetric resonators

In this paper, the nonlinear dynamics of a beam-type resonant structure due to stretching of the beam is addressed. The resonant beam is excited by attached electrostatic comb-drive actuators. This structure is modeled as a thin beam-lumped mass system, in which an initial axial force is exerted to the beam. This axial force may have different origins, e.g., residual stress due to micro-machining. The governing equations of motion are derived using the mode summation method, generalized orthogonality condition, and multiple scales method for both free and forced vibrations. The effects of the initial axial force, modal damping of the beam, the location, mass, and rotary inertia of the lumped mass on the free and forced vibration of the resonator are investigated. For the case of the forced vibration, the primary resonance of the first mode is investigated. It has been shown that there are certain combinations of the model parameters depicting a remarkable dynamic behavior, in which the second to first resonance frequencies ratio is close to three. These particular cases result in the internal resonance between the first and second modes. This phenomenon is investigated in detail.     

Shear and rotatory inertia effect on Beck's column

In this paper the influence of transverse shear deformation and rotatory inertia upon the flutter load of Beck's column with various support characteristics for a variety of slenderness ratios and cross-sectional shapes is presented. The analysis is based on Cowper's formulae for establishing Timoshenko's shear coefficient K'. From this investigation it is found that the inclusion of these parameters may have an appreciable destabilizing effect in the case of a fully fixed cantilever, and particularly in the case of a partially fixed cantilever with an attached mass at the support. This occurs especially in columns with low critical slenderness ratios and thin cross-sections. Moreover, it is noticed that the flutter frequency— for flutter loads obtained by coalescing either of the first and second or second and third flexural eigenfrequencies-never exceeds the precise value 11·01l… of Beck's column.     

Dynamic stability of elastically supported pipes conveying pulsating fluid

The effect of support flexibility on the dynamic behaviour of pipes conveying fluid is investigated for both steady and pulsatile flows. The pipes are built-in at the upstream end and supported at the other by both a translational and a rotational spring. For the steady flow condition, the critical flow velocities, the frequencies and flow induced damping patterns that are associated with the different vibration modes of selected pipe systems are determined as functions of the flow velocity. The results from steady flow cases show that the pipes may first lose stability by either buckling or flutter, depending on the values of the rotational and translational spring constants and their relative magnitudes. In the case of pulsatile flow, the Floquet theory is utilized for the stability analysis of the selected pipe-fluid systems. Numerical results are presented to illustrate the effects of the amount of translational and rotational resiliences at the elastic support on the regions of parametric and combination resonances of the pipes. The results more of the interesting aspects of the behaviour of non-conservative systems.     

Vibration analysis of scanning thermal microscope probe nanomachining using Timoshenko beam theory

In this paper, the effect of thermal vibration on the resonant frequency of transverse vibration of scanning thermal microscope (SThM) cantilever probe is analyzed using the Timoshenko beam theory, including the effects of rotary inertia and shear deformation. The thermal vibration effect can be considered as an axial force and is dependent of temperature distribution of the probe. In this analysis, the temperature is assumed to be distributed in accordance with the constant, linear, and quadratic models along the probe length. The Rayleigh–Ritz method is used to solve the vibration problem of the probe. The numerical results show that the frequency obtained with the constant model is the highest, while it is the lowest for the quadratic model. The frequency of vibration modes of the probe increases with increasing the temperature of the probe. As the ratio of probe length to its thickness increases, the frequency of vibration modes decreases. In addition, the effects of rotary inertia and shear deformation on the frequency are significant, especially in higher order modes and smaller values of the ratio of the probe length to its thickness.     

Sensitivity analysis and optimization of nodal point placement for vibration reduction

A method is developed for sensitivity analysis and optimization of nodal point locations in connection with vibration reduction. A straightforward derivation of the expression for the derivative of nodal locations is given, and the role of the derivative in assessing design trends is demonstrated. An optimization process is developed in which added lumped masses on the structure are used as design variables to move the node to a preselected location—for example to where a low response amplitude is required or to a point which makes the mode shape nearly orthogonal to the force distribution, thereby minimizing the generalized force. The optimization formulation leads to values for added masses that adjust a nodal location while minimizing the total amount of added mass required to do so. As an example, the node of the second mode of a cantilever box beam model of a rotor blade is relocated to coincide with the centroid of a prescribed force distribution, thereby reducing the generalized force substantially without adding excessive mass. A comparison with an optimization formulation that directly minimizes the generalized force indicates that nodal placement gives essentially a minimum generalized force when the node is appropriately placed.     

On the vibration and stability of a free-free beam subjected to end-loads

In this paper the vibration and stability of a free-free beam subjected to direction-controlled axial loads at its ends are investigated. The eigencurves and mode shapes of the beam are presented for various values of the directional control parameter. It is found that the behaviour of the free-free beam subjected to compressive axial loads is unstable for any direction parameter—except for the follower loading case. However, the same beam subjected to tensile loads is stable.     

On the lateral stability of a cantilever beam subjected to a non-conservative load

In this paper vibration and stability of a cantilever beam subjected to vertical and follower loads are investigated. The eigencurves of the beam are presented for various values of the ratio of follower load to vertical and follower loads. The divergence and flutter instability loads of the beam are given for a wide range of the ratio.     

An analytical method for free vibration analysis of functionally graded beams with edge cracks

In this paper, an analytical method is proposed for solving the free vibration of cracked functionally graded material (FGM) beams with axial loading, rotary inertia and shear deformation. The governing differential equations of motion for an FGM beam are established and the corresponding solutions are found first. The discontinuity of rotation caused by the cracks is simulated by means of the rotational spring model. Based on the transfer matrix method, then the recurrence formula is developed to get the eigenvalue equations of free vibration of FGM beams. The main advantage of the proposed method is that the eigenvalue equation for vibrating beams with an arbitrary number of cracks can be conveniently determined from a third-order determinant. Due to the decrease in the determinant order as compared with previous methods, the developed method is simpler and more convenient to analytically solve the free vibration problem of cracked FGM beams. Moreover, free vibration analyses of the Euler–Bernoulli and Timoshenko beams with any number of cracks can be conducted using the unified procedure based on the developed method. These advantages of the proposed procedure would be more remarkable as the increase of the number of cracks. A comprehensive analysis is conducted to investigate the influences of the location and total number of cracks, material properties, axial load, inertia and end supports on the natural frequencies and vibration mode shapes of FGM beams. The present work may be useful for the design and control of damaged structures.     

VIBRATION AND STABILITY CONTROL OF SMART COMPOSITE ROTATING SHAFT VIA STRUCTURAL TAILORING AND PIEZOELECTRIC STRAIN ACTUATION

A dual approach based on both structural tailoring and piezoelectric strain actuation, aimed at controlling the free vibration and stability of a spinning circular shaft subjected to axial forces is presented. Due to the involvement in these structural systems of gyroscopic forces and, consequently, of the possible occurrence of divergence and flutter instabilities, the dual control methodology shows a high degree of efficiency toward postponement of the occurrence of these instabilities. The structural model of the shaft as considered in this paper is based on an advanced thin-walled beam that includes the effects of transverse shear, anisotropy of constituent materials, rotatory inertias, etc. The displayed results reveal the synergistic implications of the application of this dual technology toward the enhancement of the dynamic response characteristics and expansion of the domain of stability of these systems.     

Vibration characteristics of a rotating flexible arm with ACLD treatment

In this paper, the vibration behavior and control of a clamped–free rotating flexible cantilever arm with fully covered active constrained layer damping (ACLD) treatment are investigated. The arm is rotating in a horizontal plane in which the gravitational effect and rotary inertia are neglected. The stress–strain relationship for the viscoelastic material (VEM) is described by a complex shear modulus while the shear deformations in the two piezoelectric layers are neglected. Hamilton's principle in conjunction with finite element method (FEM) is used to derive the non-linear coupled differential equations of motion and the associated boundary conditions that describe the rigid hub angle rotation, the arm transverse displacement and the axial deformations of the three-layer composite. This refined model takes into account the effects of centrifugal stiffening due to the rotation of the beam and the potential energies of the VEM due to extension and bending. Active controllers are designed with PD for the piezosensor and actuator. The vibration frequencies and damping factors of the closed-loop beam/ACLD system are obtained after solving the characteristic complex eigenvalue problem numerically. The effects of different rotating speed, thickness ratio and loss factor of the VEM as well as different controller gain on the damped frequency and damping ratio are presented. The results of this study will be useful in the design of adaptive and smart structures for vibration suppression and control in rotating structures such as rotorcraft blades or robotic arms.     

Validity and Accuracy of Resonance Shift Prediction Formulas for Microcantilevers: A Review and Comparative Study

This article provides a review of methods of predicting mass-induced resonance shifts in microcantilevers. It combines a review of factors that influence resonance frequency shifts, such as material properties, size effects, and support compliance with a comparative study of accuracy of predicting resonance shifts due to mass adsorption. The applicability and accuracy of widely used formulas to correlate mass addition with resonance shift are assessed through comprehensive comparison with experimental measurements and numerical methods. The methods include both distributed parameter and lumped parameter formulations. The applications include distributed added masses, tip masses, and added mass at arbitrary locations along a cantilever span.     

BIFURCATIONS AND CHAOS OF AN IMMERSED CANTILEVER BEAM IN A FLUID AND CARRYING AN INTERMEDIATE MASS

The concern of this work is the local stability and period-doubling bifurcations of the response to a transverse harmonic excitation of a slender cantilever beam partially immersed in a fluid and carrying an intermediate lumped mass. The unimodal form of the non-linear dynamic model describing the beam-mass in-plane large-amplitude flexural vibration, which accounts for axial inertia, non-linear curvature and inextensibility condition, developed in Al-Qaisia et al. (2000Shock and Vibration7 , 179-194), is analyzed and studied for the resonance responses of the first three modes of vibration, using two-term harmonic balance method. Then a consistent second order stability analysis of the associated linearized variational equation is carried out using approximate methods to predict the zones of symmetry breaking leading to period-doubling bifurcation and chaos on the resonance response curves. The results of the present work are verified for selected physical system parameters by numerical simulations using methods of the qualitative theory, and good agreement was obtained between the analytical and numerical results. Also, analytical prediction of the period-doubling bifurcation and chaos boundaries obtained using a period-doubling bifurcation criterion proposed in Al-Qaisia and Hamdan (2001 Journal of Sound and Vibration244, 453-479) are compared with those of computer simulations. In addition, results of the effect of fluid density, fluid depth, mass ratio, mass position and damping on the period-doubling bifurcation diagrams are studies and presented.