Non-linear free torsional vibrations of thin-walled beams with bisymmetric cross-section

A finite element method for studying non-linear free torsional vibrations of thin-walled beams with bisymmetric open cross-section is presented. The non-linearity of the problem arises from axial loads generated at moderately large amplitude torsional vibrations due to immovability of end supports. The derivation of the fundamental differential equation of the problem is based on the classical assumption of a thin-walled beam with a non-deformable cross-section. The non-linear eigenvalue problem is solved iteratively by series of linear eigenvalue problems until the required accuracy is obtained. Non-linear frequencies, fundamental mode shapes and axial loads computed for various amplitude of torsional vibrations of thin-walled I beams are included.     

Nonlinear response of shear deformable beams on tensionless nonlinear viscoelastic foundation under moving loads

In this paper, a boundary element method is developed for the geometrically nonlinear response of shear deformable beams of simply or multiply connected constant cross-section, traversed by moving loads, resting on tensionless nonlinear three-parameter viscoelastic foundation, undergoing moderate large deflections under general boundary conditions. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse moving loading as well as to axial loading. To account for shear deformations, the concept of shear deformation coefficients is used. Three boundary value problems are formulated with respect to the transverse displacement, to the axial displacement and to a stress functions and solved using the Analog Equation Method, a Boundary Element based method. Application of the boundary element technique yields a system of nonlinear Differential-Algebraic Equations, which is solved using an efficient time discretization scheme, from which the transverse and axial displacements are computed. The evaluation of the shear deformation coefficient is accomplished from the aforementioned stress function using only boundary integration. Analyses are performed to illustrate, wherever possible, the accuracy of the developed method, to investigate the effects of various parameters, such as the load velocity, load frequency, shear deformation, foundation nonlinearity, damping, on the beam displacements and stress resultants and to examine how the consideration of shear and axial compression affects the response of the system.     

ESTIMATION OF 3D FIELD QUANTITIES AND ENERGY FLUX ASSOCIATED WITH ELASTIC WAVES IN A BAR

A method to estimate dispersion relations and warping associated with elastic wave propagation in a bar is presented. The method is based on Hamilton's principle. It is shown how the theoretical model together with strain measurements can be used to evaluate three dimensional (3D) field quantities like displacements and stresses at an arbitrary position in the bar, as well as energy flux through an arbitrary cross-section of the bar. It is also shown how redundant measurements can be used to increase the accuracy. The method is general and can be applied to any mode of wave propagation, isotropic or anisotropic linearly elastic material, and any cross-sectional geometry. Here, it is applied to longitudinal waves in a split Hopkinson pressure bar with linear elastic isotropic material behaviour and square cross-section. In particular, axial displacement, axial stress and energy flux are evaluated at a free end of the bar in order to test the method. The method is also used to estimate the Poisson ratio of the bar material, by measuring axial and transverse strains at the same axial position.     

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.     

Analytical formulation and evaluation for free vibration of naturally curved and twisted beams

An analytical study for free vibration of naturally curved and twisted beams with uniform cross-sectional shapes is carried out using spatial curved beam theory based on the Washizu's static model. In the governing equations of motion of the beams, all displacement functions and the generalized warping coordinate are defined at the centroid axis and also the effects of rotary inertia, transverse shear deformations and torsion-related warping are included in the proposed model. Explicit analytical expressions are derived for the vibrating mode shapes of a curved, bending-torsional-shearing coupled beam under clamped-clamped boundary condition with the help of symbolic computing package Mathematica, and a process of searching is used to determine the natural frequencies. Comparisons of the present results with the FEM results using beam elements in ANSYS code show good accuracy in computation and validity of the model. Further, the present model is used for cylindrical helical springs with circular cross-section fixed at both ends, and the results indicate that the natural frequencies agree well with the theoretical and experimental results available.     

Optimal design of thin-walled I beams for a given natural frequency of torsional vibrations

A method of extremum weight design of thin-walled I beams for a given natural frequency of torsional vibrations is presented. The effects of warping stresses and constant axial loads are taken into account. The optimality condition for only one (except for the web height) dimension of the cross-section, variable along the axis of the beam, is derived by using Pontryagin's maximum principle. The solution of the problem formulated, with account also taken of the additional geometrical conditions, is obtained in an iterative way. Some numerical examples of optimal design of an I beam with variable flange width, for a specified fundamental frequency, are given.     

Torsional vibration of crankshaft in an engine–propeller nonlinear dynamical system

Theoretical and experimental studies on torsional vibration of an aircraft engine–propeller system are presented in this paper. Two system models—a rigid body model and a flexible body model, are developed for predicting torsional vibrations of the crankshaft under different engine powers and propeller pitch settings. In the flexible body model, the distributed torsional flexibility and mass moment of inertia of the crankshaft are considered using the finite element method. The nonlinear autonomous equations of motion for the engine–propeller dynamical system are established using the augmented Lagrange equations, and solved using the Runge–Kutta method after a degrees of freedom reduction scheme is applied. Experiments are carried out on a three-cylinder four-stroke engine. Both theoretical and experimental studies reveal that the crankshaft flexibility has significant influence on the system dynamical behavior.     

Triply coupled vibrations of thin-walled open cross-section beams including rotary inertia effects

An exact analytical method is presented for predicting the undamped natural frequencies of beams with thin-walled open cross-sections having no axis of symmetry. The governing differential equations give a characteristic equation of the 12th order with real coefficients. The roots are found numerically and the exact boundary conditions are considered especially for free ends to obtain natural frequencies. The simpler cases of neglecting cross-sectional warping and/or rotary inertia are also dealt with. It is seen that when the effect of rotary inertia is neglected significant errors incur for some boundary conditions, cross-section thicknesses and mode numbers. This is more profound when the warping effect is taken into account.     

Torsional vibration of multi-step non-uniform rods with various concentrated elements

An analytical approach and exact solutions for the torsional vibration of a multi-step non-uniform rod carrying an arbitrary number of concentrated elements such as rigid disks and with classical or non-classical boundary conditions is presented. The exact solutions for the free torsional vibration of non-uniform rods whose variations of cross-section are described by exponential functions and power functions are obtained. Then, the exact solutions for more general cases, non-uniform rods with arbitrary cross-section, are derived for the first time. In order to simplify the analysis for the title problem, the fundamental solutions and recurrence formulas are developed. The advantage of the proposed method is that the resulting frequency equation for torsional vibration of multi-step non-uniform rods with arbitrary number of concentrated elements can be conveniently determined from a homogeneous algebraic equation. As a consequence, the computational time required by the proposed method can be reduced significantly as compared with previously developed analytical procedures. A numerical example shows that the results obtained from the proposed method are in good agreement with those determined from the finite element method (FEM), but the proposed method takes less computational time than FEM, illustrating the present methods are efficient, convenient and accurate.     

Wave structure analysis of guided waves in a bar with an arbitrary cross-section

Both dispersion curves and wave structures, which are displacement distributions on a bar cross-section, are essential for guided wave NDEs. Theoretical dispersion curves and wave structures for a bar with an arbitrary cross-section are derived in this paper using a special modeling technique called the semi-analytical finite element method (SAFEM). The guidelines for guided wave NDEs of bar-like structures are also shown based on wave structure and modal analysis. First, the relationship between the dispersion curves and their corresponding wave structures were obtained for a square rod. Modes with longitudinal vibration have higher group velocities and torsional modes have constant phase and group velocities. Next, the relationship between the dispersion curves and wave structures for a rail are detailed. The rail is used to represent a bar with a complex cross-section. Similar to the square rod results, the rail's longitudinal modes have higher group velocities. However, the rail contains modes with local vibration. Finally, single mode detection and excitation techniques are introduced. A single mode can be obtained by detecting and exciting with a weighted function that corresponds to a specific mode's wave structure.     

Solution to the problem of Nicolai

We consider the problem of Nicolai on dynamic stability of an elastic cantilever rod loaded by an axial compressive force P and a twisting tangential torque L in continuous formulation. The problem is to find the stability region for non-equal principal moments of inertia of the rod in the space of three parameters: P, L and the parameter α for the ratio of principal moments of inertia. New governing equations and boundary conditions, which form the basis for analytical and numerical studies, are derived. An important detail of this formulation is that the pre-twisting of the rod due to the torque L is taken into account. The singular point on the stability boundary at the critical Euler force P_E is recognized and investigated in detail. For an elliptic cross-section of a uniform rod the stability region is found numerically with the use of the Galerkin method and the exact numerical approach. The obtained numerical results are compared with the analytical formulas of the asymptotic analysis.     

Vibration analysis of drillstrings with self-excited stick-slip oscillations

Drillstring vibration is one of the major causes for a deteriorated drilling performance. Field experience revealed that it is crucial to understand the complex vibrational mechanisms experienced by a drilling system in order to better control its functional operation and improve its performance. Sick-slip oscillations due to contact between the drilling bit and formation is known to excite severe torsional and axial vibrations in the drillstring. A dynamic model of the drillstring including the drillpipes and drillcollars is formulated. The equation of motion of the rotating drillstring is derived using Lagrangian approach in conjunction with the finite element method. The model accounts for the torsional-bending inertia coupling and the axial-bending geometric nonlinear coupling. In addition, the model accounts for the gyroscopic effect, the effect of the gravitational force field, and the stick-slip interaction forces. Explicit expressions of the finite element coefficient matrices are derived using a consistent mass formulation. The generalized eigenvalue problem is solved to determine modal transformations, which are invoked to obtain the reduced-order modal form of the dynamic equations. The developed model is integrated into a computational scheme to calculate time-response of the drillstring system in the presence of stick-slip excitations.     

Field reconstruction in an anisotropic elastic medium

For steady-state vibrations of an anisotropic elastic body of finite dimensions, a method of the determination of the vibration energy flows in the body is proposed. The method is based on the measurements of the surface values of the stress and displacement vectors at a part of the boundary. The proposed algorithm of the wave field reconstruction is reduced to solving nonclassical boundary integral equations of the first kind with smooth kernels. The formulation of these equations does not require the determination of fundamental solutions, but represents a conditionally well-posed problem. The numerical realization of the proposed method is based on the Tikhonov regularization method and the idea of the boundary element method. Numerical experiments consisting in the reconstruction of the displacements and stresses at the boundary of a rectangular and a circular domains of austenitic steel are performed in the framework of a planar problem of the orthothropic elasticity theory.     

Effects of cables damage on vertical and torsional eigenproperties of suspension bridges

This paper presents a continuum model for the nonlinear coupled vertical and torsional vibrations of suspension bridges with arbitrary damage in one main cable and, after pursuing a suitable linearization of the equations of motion, an investigation of damage effects on modal parameters. Damage is modeled as a diffused loss of cross-section representing the typical effect of fretting fatigue and it is introduced in the formulation by enforcing relevant literature results providing analytical solution for the static response of damaged suspended cables. The coupled nonlinear equations of motion of the damaged bridge, including the effects of shear deformation, rotary inertia and warping of the cross-section of the girder, are derived by application of Hamilton?s principle. In this way, the equations of motion available in the literature for undamaged suspension bridges are generalized to the presence of arbitrary damage in one main cable and the resulting eigenfrequencies and eigenfunctions are derived in an analytical fashion. An extensive parametric investigation is finally presented to discuss damage effects on eigenfunctions and eigenfrequencies under variation of practically meaningful parameters.     

Triply coupled vibrational band gap in a periodic and nonsymmetrical axially loaded thin-walled Bernoulli–Euler beam including the warping effect

The propagation of triply coupled vibrations in a periodic, nonsymmetrical and axially loaded thin-walled Bernoulli–Euler beam composed of two kinds of materials is investigated with the transfer matrix method. The cross-section of the beam lacks symmetrical axes, and bending vibrations in the two perpendicular directions are coupled with torsional vibrations. Furthermore, the effect of warping stiffness is included. The band structures of the periodic beam, both including and excluding the warping effect, are obtained. The frequency response function of the finite periodic beam is simulated with the finite element method. These simulations show large vibration-based attenuation in the frequency range of the gap, as expected. By comparing the band structure of the beam with plane wave expansion method calculations that are available in the literature, one finds that including the warping effect leads to a more accurate simulation. The effects of warping stiffness and axial force on the band structure are also discussed.     

VIBRATION ANALYSIS OF TWISTED CANTILEVERED CONICAL COMPOSITE SHELLS

The finite element method based on the Hellinger-Reissner principle with independent strain is applied to the vibration problem of cantilevered twisted plates and cylindrical, conical laminated shells. With a small number of elements, the present assumed strain finite element method is validated by convergence tests and numerical tests, comparing with the previous published vibration results for cantilevered conical shell. Computational effort and virtual storage reduce significantly due to good convergence. This study presents the twisting angle effect on vibration characteristics of conical laminated shells. Parameter studies with varying shallowness of cylindrical and conical shells are carried out. As the curvature increases, the fundamental mode shape changes from twisting mode to bending mode. For shells with a large curvature, the fundamental frequency, which is always characterized to bending mode, is almost constant independent of twisting angle. The twisting angle affects greatly twisting frequency and mode shape.     

Large amplitude vibration of an initially stressed cross ply laminated plates

In this paper, nonlinear equations of large amplitude vibration for a laminated plate in a general state of nonuniform initial stress are derived. The equations include the effects of transverse shear and rotary inertia. Using these derived governing equations, the large amplitude vibration behaviour of an initially stressed cross-ply laminated plate is studied. The initial stress is taken to be a combination of pure bending stress plus an extensional stress in the plane of the plate. The Galerkin method is used to reduce the governing nonlinear partial differential equations to ordinary nonlinear differential equations and the Runge-Kutta method is used to obtain the nonlinear to linear frequencies. The frequency responses of nonlinear vibration are sensitive of the vibration amplitude, aspect ratio, thickness ratio, modulus ratio, stack sequence, layer number and state of initial stresses. The effects of various parameters on the large amplitude free vibrations are presented.     

Aero-thermo-mechanical characteristics of imperfect shape memory alloy hybrid composite panels

A nonlinear finite element model is provided to predict the static aero-thermal deflection and the vibration behavior of geometrically imperfect shape memory alloy hybrid composite panels under the combined effect of thermal and aerodynamic loads. The nonlinear governing equations are obtained using Marguerre curved plate theory and the principle of virtual work taking into account the temperature-dependence of material properties. The effect of large deflection is included in the formulation through the von Karman nonlinear strain-displacement relations. The thermal load is assumed to be a steady-state constant-temperature distribution, whereas the aerodynamic pressure is modeled using the quasi-steady first-order piston theory. The Newton-Raphson iteration method is employed to obtain the nonlinear aero-thermal deflections, while an eigenvalue problem is solved at each temperature step and static aerodynamic load to predict the free vibration frequencies about the deflected equilibrium position. Finally, the nonlinear deflection and free vibration characteristics of a composite panel are presented, illustrating the effects of geometric imperfection, temperature rise, aerodynamic pressure, boundary conditions and shape memory alloy fiber embeddings on the panel response.     

Free vibrations of curved box girders

The problem of coupled free vibrations of curved thin-walled girders of non-deformable asymmetric cross-section is examined in this paper. The general governing differential equations are derived for quadruple coupling between the two flexural, tangential and torsional vibrations. An approximate solution for the case of triple coupling between the two flexural and the torsional vibrations is given for a simply supported girder, uniform specific gravity of the material of the box being assumed. Section warping is considered but axial forces, rotary inertia and structural damping are neglected. A parametric study is conducted to investigate the effect of relevant parameters on natural frequencies. Eigenfunctions satisfying the orthogonality condition are given. The solution derived herein for the general case is also shown to cover a variety of special cases of straight and curved girders with doubly symmetric or singly symmetric cross-sections.     

Nonlinear dynamics of galloping-based piezoaeroelastic energy harvesters

The normal form is proposed as a tool to analyze the performance and reliability of galloping-based piezoaeroelastic energy harvesters. Two different harvesting systems are considered. The first system consists of a tip mass prismatic structure (isosceles 30° or square cross-section geometry) attached to a multilayered cantilever beam. The only source of nonlinearity in this system is the aerodynamic nonlinearity. The second system consists of an equilateral triangle cross-section bar attached to two cantilever beams. This system is designed to have structural and aerodynamic nonlinearities. The coupled governing equations for the structure’s transverse displacement and the generated voltage are derived and analyzed for both systems. The effects of the electrical load resistance and the type of harvester on the onset speed of galloping are quantified. The results show that the onset speed of galloping is strongly affected by the load resistance for both types of harvesters. The normal form of the dynamic system near the onset of galloping (Hopf bifurcation) is then derived. Based on the nonlinear normal form, it is demonstrated that smaller levels of generated voltage or power are obtained for higher absolute values of the effective nonlinearity. For the first harvesting system, the results show a supercritical Hopf bifurcation for both isosceles 30° or square cross-section geometries. The nonlinear normal form shows that the isosceles triangle section (30°) is more efficient than the square section. For the second harvesting system, the normal form is used to identify the values of the nonlinear torsional spring which changes the harvester’s instability. It is demonstrated that this critical value of the nonlinear torsional spring depends strongly on the load resistance.