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.     

Optimal design of thin walled I beams for extreme natural frequency of torsional vibrations

The optimal design of thin-walled I beams so as to extremize the natural frequency of torsional vibration is considered. It is assumed that only one dimension of the cross-section, except for the web height, may be variable in given limits, along the axis of the beam. The optimality condition for the variable dimension is settled by means of Pontryagin's maximum principle. The effect of the constant, axial loads is also included. the solution of the problem formulated is generally found in an iterative way. Some numerical examples of optimization of the I beam with variable widt of flanges are given.     

Nonlinear nonuniform torsional vibrations of bars by the boundary element method

In this paper a boundary element method is developed for the nonuniform torsional vibration problem of bars of arbitrary doubly symmetric constant cross-section taking into account the effect of geometrical nonlinearity. The bar is subjected to arbitrarily distributed or concentrated conservative dynamic twisting and warping moments along its length, while its edges are supported by the most general torsional boundary conditions. The transverse displacement components are expressed so as to be valid for large twisting rotations (finite displacement-small strain theory), thus the arising governing differential equations and boundary conditions are in general nonlinear. The resulting coupling effect between twisting and axial displacement components is considered and torsional vibration analysis is performed in both the torsional pre- or post-buckled state. A distributed mass model system is employed, taking into account the warping, rotatory and axial inertia, leading to the formulation of a coupled nonlinear initial boundary value problem with respect to the variable along the bar angle of twist and to an “average” axial displacement of the cross-section of the bar. The numerical solution of the aforementioned initial boundary value problem is performed using the analog equation method, a BEM based method, leading to a system of nonlinear differential-algebraic equations (DAE), which is solved using an efficient time discretization scheme. Additionally, for the free vibrations case, a nonlinear generalized eigenvalue problem is formulated with respect to the fundamental mode shape at the points of reversal of motion after ignoring the axial inertia to verify the accuracy of the proposed method. The problem is solved using the direct iteration technique (DIT), with a geometrically linear fundamental mode shape as a starting vector. The validity of negligible axial inertia assumption is examined for the problem at hand.     

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.     

FUNDAMENTAL FREQUENCY OF CRACKED BEAMS IN BENDING VIBRATIONS: AN ANALYTICAL APPROACH

This paper presents an analytical approach to the fundamental frequency of cracked Euler-Bernoulli beams in bending vibrations. The flexibility influence function method used to solve the problem leads to an eigenvalue problem formulated in integral form. The influence of the crack was represented by an elastic rotational spring connecting the two segments of the beam at the cracked section. In solving the problem, closed-form expressions for the approximated values of the fundamental frequency of cracked Euler-Bernoulli beams in bending vibrations are reached. The results obtained agree with those numerically obtained by the finite element method.     

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.     

A method of stability analysis for non-linear vibration of beams

In this paper, a method of stability analysis for the large amplitude, steady state response of a non-linear beam under periodic excitation is presented. The stability problem is investigated by studying the behavior of a small perturbation of the steady state response which results in a coupled Hill-type equation. The problem is transformed by the harmonic balance method into an eigenvalue problem of a non-symmetric matrix. The effectiveness and the accuracy of the proposed method for a Mathieu equation are examined and the application to the stability analysis of the non-linear vibrations of a beam is presented.     

A SEMI-ANALYTICAL APPROACH TO THE NON-LINEAR DYNAMIC RESPONSE PROBLEM OF BEAMS AT LARGE VIBRATION AMPLITUDES, PART II: MULTIMODE APPROACH TO THE STEADY STATE FORCED PERIODIC RESPONSE

The semi-analytical approach to the non-linear dynamic response of beams based on multimode analysis has been presented in Part I of this series of papers (Azrar et al., 1999 Journal of Sound and Vibration224, 183-207 [1]). The mathematical formulation of the problem and single mode analysis have been studied. The objective of this paper is to take advantage of applying this semi-analytical approach to the large amplitude forced vibrations of beams. Various types of excitation forces such as harmonic distributed and concentrated loads are considered. The governing equation of motion is obtained and can be considered as a multi-dimensional form of the Duffing equation. Using the harmonic balance method, the equation of motion is converted into non-linear algebraic form. Techniques of solution based on iterative-incremental procedures are presented. The non-linear frequency and the non-linear modes are determined at large amplitudes of vibration. The basic function contribution coefficients to the displacement response for various beam boundary conditions are calculated. The percentage of participation for each mode in the response is presented in order to appraise the relation to higher modes contributing to the solution. Also, the percentage contributions of the higher modes to the bending moment near to the clamps are given, in order to determine accurately the error introduced in the non-linear bending stress estimated by different approximations. Solutions obtained in the jump phenomena region have been determined by a careful selection of the initial iteration at each frequency. The non-linear deflection shapes in various regions of the solution, the corresponding axial force ratios and the bending moments are presented in order to follow the behaviour of the beam at large vibration amplitudes. The numerical results obtained here for the non-linear forced response are compared with those from the linear theory, with available non-linear results, based on various approaches, and with the single mode analysis.     

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.     

Normal modes of a continuous system with quadratic and cubic non-linearities

A power series solution is presented for the free vibrations of simply supported beams resting on elastic foundation having quadratic and cubic non-linearities. The time-dependence is assumed harmonic and the problem is posed as a non-linear eigenvalue problem. The spatial variable is transformed into an independent variable that satisfies the boundary conditions. This permits a power series expansion of the beam motion in terms of the new variable. A recurrence relation is obtained from the governing equation and used in conjunction with the Rayleigh energy principle to compute the natural frequencies. The results show that, for a first order approximation, only the lower frequencies and first mode shape are significantly affected by the cubic non-linearity.     

Non-linear control of torsional and bending vibrations of oilwell drillstrings

Drillstring dynamics is highly non-linear in nature and its model can only be described by a set of non-linear differential equations. In addition to this complexity, the drillstring dynamics are not linearly controllable and thus linear control methods are not suitable for suppressing the coupled torsional and lateral vibrations of a rotating drillstring. In this paper a non-linear dynamic inversion control design method is used to suppress the lateral and the torsional vibrations of a non-linear drillstring. It was found that the designed controller is effective in suppressing the torsional vibrations and reducing the lateral vibrations significantly.     

周期薄壁开口梁中的双耦合振动带隙

本文研究了由两种材料组合构成的周期性薄壁开口梁的弯曲和扭转双耦合振动。基于双耦合振动方程，给出了平面波展开法。当填充比不变时，晶格常数是影响带隙相对宽带的一个因素；当晶格常数和填充比不变时，杨氏模量是影响带隙宽带的主要因素，而不是密度。利用有限元法计算了有限周期结构的振动频率响应，在带隙频率范围内，振动衰减40dB左右。这些发现对于声子晶体的应用具有重要意义。     

Optimal design of beams in torsion under periodic loading

The optimal design of beams in torsion under harmonically varying torques is discussed. The analysis covers the cases when the excitation frequency is either less than or greater than the fundamental frequency of the beam. The beams analyzed are in the main assumed to have rectangular cross-section but the theory is easily extended to other section shapes. In each case the problem is stated in variational form with the introduction of constraints through Lagrange multipliers. The mathematical analysis of the various problems presented results in a system of non-linear differential equations with associated boundary conditions. The solutions given for some of the cases provide expressions for the design variable and the response, along the length of the beam, in terms of the forcing frequency and some constants which can be determined for the particular problem. The computed results and data are given in tabular form and some optimum profiles are shown graphically.     

Experimental and numerical investigation of friction-induced vibration of a beam-on-beam in contact with friction

The vibrations generated by friction are responsible for various noises such as squealing, squeaking and chatter. Although these phenomena have been studied for a long time, it is not well-understood. In this study, an experimental and numerical study of friction-induced vibrations of a system composed of two beams in contact is proposed. The experimental system exhibits periodic steady state vibrations of different types. To model and understand this experimental vibratory phenomenon, complex eigenvalue and dynamic transient analyses are performed. In the linear complex eigenvalue analysis, flutter instability occurs via the coalescence of two eigenmodes of the system. This linear study provides an accurate value of the experimental frequency of vibration. To understand what happens physically during friction-induced instability, a dynamic transient analysis that takes account of the non-linear aspect of a frictional contact is performed. In this analysis, friction-induced instability is characterized by self-sustained vibrations and by stick, slip and separation zones occurring at the surface of the contact. The results stemming from this analysis show that good correlation between numerical and experimental vibrations can be obtained (in time and frequency domains). Moreover, time domain simulations permit understanding the physical phenomena involved in two different vibratory behaviours observed experimentally.     

VIBRATION OF A COMPLIANT TOWER IN THREE-DIMENSIONS

The three-dimensional motion of an offshore compliant tower using both rigid and flexible beam models is studied in this paper. The tower is modelled as a beam supported by a torsional spring at the base with a point mass at the free end. The torsional spring constant is the same in all directions. When the beam is considered rigid, the two-degree-of-freedom model is employed. The two degrees constitute the two angular degrees of spherical co-ordinates, and the resulting equations are coupled and non-linear. When the beam is considered as elastic, three displacements are obtained as functions of the axial co-ordinate and time; again with coupled and non-linear equations of motion. The free and the forced responses due to deterministic loads are presented. The free responses of the rigid and elastic beams show rotating elliptical paths when viewed from above. The rate at which the path rotates depends on the initial conditions. When a harmonic transverse loading is applied in one direction, the displacement in that direction shows subharmonic resonance of order 1/2 and 1/3 while the displacement in the perpendicular direction is affected minimally. Next, in addition to the harmonic load in one direction, a transverse load is applied in the perpendicular direction. The transverse load varies exponentially with depth but is constant with time. It is found that the transverse load affects the transverse displacements in the perpendicular direction minimally.     

Effect of longitudinal or inplane deformation and inertia on the large amplitude flexural vibrations of slender beams and thin plates

Large amplitude flexural vibrations of slender beams, and thin circular and rectangular plates have been studied when a compatible longitudinal or inplane mode is coupled with the fundamental flexural mode. It is shown that the effect of longitudinal or inplane deformation and inertia is to reduce the non-linearity in the flexural frequency-amplitude relationship. Further, for slender beams and thin plates, the effect of longitudinal or inplane inertia is negligible.     

Forced transverse vibrations of an elastically connected complex simply supported double-beam system

The present paper is devoted to analyzing undamped forced transverse vibrations of an elastically connected complex double-beam system. The problem is formulated and solved in the case of simply supported beams. The classical modal expansion method is applied to ascertain dynamic responses of beams due to arbitrarily distributed continuous loads. Several cases of particularly interesting excitation loadings are investigated. The action of stationary harmonic loads and moving forces is considered. In discussing vibrations caused by exciting harmonic forces, conditions of resonance and dynamic vibration absorption are determined. The beam-type dynamic absorber is a new concept of a continuous dynamic vibration absorber (CDVA), which can be applied to suppress excessive vibrations of corresponding beam systems. A numerical example is presented to illustrate the theoretical analysis.     

FREE VIBRATIONS OF SELF-STRAINED ASSEMBLIES OF BEAMS

This paper examines the natural frequencies and modes of transverse vibration of two simple redundant systems comprising straight uniform Euler-Bernoulli beams in which there are internal self-balancing axial loads (e.g., loads due to non-uniform thermal strains). The simplest system consists of two parallel beams joined at their ends and the other is a 6-beam rectangular plane frame. Symmetric mode vibration normal to the plane of the frame is studied. Transcendental frequency equations are established for the different systems. Computed frequencies and modes are presented which show the effect of (1) varying the axial loads over a wide range, up to and beyond the values which cause individual members to buckle (2) pinning or fixing the beam joints (3) varying the relative flexural stiffness of the component beams. When the internal axial loads first cause any one of the component beams to buckle, the fundamental frequency of the whole system vanishes. The critical axial loads required for this are determined. A simple criterion has been identified to predict whether a small increase from zero in the axial compressive load in any one member causes the natural frequencies of the whole system to rise or fall. It is shown that this depends on the relative flexural stiffnesses and buckling loads of the different members. Computed modes of vibration show that when the axial modes reach their critical values, the buckled beam(s) distort with large amplitudes while the unbuckled beam(s) move either as rigid bodies or with bending which decays rapidly from the ends to a near-rigid-body movement over the central part of the beam. The modes of the systems with fixed joints change very little (if at all) with changing axial load, except when the load is close to the value which maximizes or minimizes the frequency. In a narrow range around this load the mode changes rapidly. The results provide an explanation for some computed results (as yet unpublished) for the flexural modes and frequencies of flat plates with non-uniform thermal stress distributions.     

The effects of large vibration amplitudes on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates. Part I: iterative and explicit analytical solution for non-linear transverse vibrations

The effects of large vibration amplitudes on the first two axisymmetric mode shapes of clamped thin isotropic circular plates are examined. The theoretical model based on Hamilton's principle and spectral analysis developed previously by Benamar et al. for clamped-clamped beams and fully clamped rectangular plates is adapted to the case of circular plates using a basis of Bessel's functions. The model effectively reduces the large-amplitude free vibration problem to the solution of a set of non-linear algebraic equations. Numerical results are given for the first and second axisymmetric non-linear mode shapes for a wide range of vibration amplitudes. For each value of the vibration amplitude considered, the corresponding contributions of the basic functions defining the non-linear transverse displacement function and the associated non-linear frequency are given. The non-linear frequencies associated to the fundamental non-linear mode shape predicted by the present model were compared with numerical results from the available published literature and a good agreement was found. The non-linear mode shapes exhibit higher bending stresses near to the clamped edge at large deflections, compared with those predicted by linear theory. In order to obtain explicit analytical solutions for the first two non-linear axisymmetric mode shapes of clamped circular plates, which are expected to be very useful in engineering applications and in further analytical developments, the improved version of the semi-analytical model developed by El Kadiri et al. for beams and rectangular plates, has been adapted to the case of clamped circular plates, leading to explicit expressions for the higher basic function contributions, which are shown to be in a good agreement with the iterative solutions, for maximum non-dimensional vibration amplitude values of 0.5 and 0.44 for the first and second axisymmetric non-linear mode shapes, respectively.     

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.